would the formulas for area, volume of pseudosphere be affected by much if infinity were defined as 10^500 #837 Correcting Math

[Archimedes Plutonium]

Let me for a moment change my tactic; instead of trying to get 10^500
as a special number, let me assume it is infinity from the start.

Now the formulas for area and volume of the pseudosphere are :
surface area is
4*pi*R*R while its volume is
(2/3)*pi*R*R*R

So now we define pseudosphere cutaway as a euclidean planar cut.

And we define the boundary of finite versus infinite at
10^500 where that number is infinity itself and all numbers higher.

Now we ask the question of the pseudosphere of radius 1 where it ends
at 10^500 of its poles as a Euclidean
planar cut.

And now we ask, how much of a tiny piece of area do we have missing
from the formula 4*pi*R*R and again ask how much volume is missing
from (2/3)*pi*R*R*R

So when we do the integral, the infinity is replaced by the number
10^500.

So the formulas would no longer be equal to the above but have a tiny
subtraction of missing area and missing volume.

Now maybe that is the specialness of the number 10^500 in that the
missing portions of area and volume, by the selection of infinity as
10^500 is a portion that is
special elsewhere.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Let me for a moment change my tactic; instead of trying to get 10^500
as a special number, let me assume it is infinity from the start.
surface area is
4*pi*R*R while its volume is
(2/3)*pi*R*R*R
So now we define pseudosphere cutaway as a euclidean planar cut.
And we define the boundary of finite versus infinite at
10^500 where that number is infinity itself and all numbers higher.
Now we ask the question of the pseudosphere of radius 1 where it ends
at 10^500 of its poles as a Euclidean
planar cut.
And now we ask, how much of a tiny piece of area do we have missing
from the formula 4*pi*R*R and again ask how much volume is missing
from (2/3)*pi*R*R*R
So when we do the integral, the infinity is replaced by the number
10^500.
So the formulas would no longer be equal to the above but have a tiny
subtraction of missing area and missing volume.
Now maybe that is the specialness of the number 10^500 in that the
missing portions of area and volume, by the selection of infinity as
10^500 is a portion that is
special elsewhere.

Of course now, the formula for surface area of pseudosphere would turn
from that of
surface area = (4*pi*R*R) - k where k is a tiny bit of area so that
the pseudosphere is
almost equal to the surface area of the associated sphere of radius R
but a tiny bit
off. Likewise, the same goes for volume; volume = ((2/3)*pi*R*R*R) - j

Now this subtraction is the par normal for the course of mathematics
in that of truncated
regular polyhedra or even regular polygons with their associated
circle, in that they have
area equal to associated circle or sphere after the subtracted factor
is added back on.

The mess up occurred with pseudosphere, because it was the first time
we had infinite
stretch, and the mistake was made in thinking that "infinite stretch
of old math infinity"
is no different from finite stretch.

Once we define the boundary between finite stretch and infinite
stretch as that of 10^500,
then we can proceed without the huge errors of reasoning.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Let me try a different tactic, since I am as yet unable to show the
uniqueness of the number
10^500 as a boundary between finite and infinite.

Let me still use the pseudosphere and this time let me use the tiny
world, the microworld
of numbers. Instead of big numbers, let me use the tiniest numbers.

Now in the Pseudosphere it has a equator disc for radius and of course
the poles stretch
out to infinity of the old math infinity of endlessness.

Now we can draw a triangle into the pseudosphere of where the three
points are the
center of the pseudosphere and a point on the tip of the equator disc
and finally the
0 point. Mind you we are going into the small numbers. So this is a
pseudosphere of
radius 1.

Now if the old math is correct, there are always a new number between
any two given numbers
as the axiom of absolute-betweenness. And there is a geometry axiom
that says only one line
is parallel to a given line from a point outside the line. Now what
this exercise is going to show is that the axioms of geometry of those
two mentioned are contradictory to one another.

So as the pseudosphere starts of the triangle from the center to the
tip of the equator, then the
two sides that approach 0 point. Of course, in the pseudosphere, the
curve (tractrix) cannot intersect any point on the x-axis, not even
the 0 point.

Now we throw away the pseudosphere and we simply construct a triangle
in Euclidean Geometry that has the parallel postulate contradict the
Betweenness postulate and vice versa.

What we do is construct successively thinner triangles with points on
the x-axis as 1 and ever
smaller line segments at x= 1 and infinitely smaller y component at
x=1. We construct a scalene triangle that is shaped like a needle and
thinner and thinner of a needle. We can do this because of the axiom
of betweenness. And the third point vertex of this needle shaped
triangle is the 0 point. But since we can make the two vertices at
x=1, make them infinitely smaller, means that the vertex at 0 is never
a vertex, but two parallel lines. So we have
achieved what the pseudosphere infinite stretch is, in that the curve
approaches 0 but never
intersects with 0.

Now the reason I bring up this alternative to the macroworld 10^500,
is that I am hopeful of
some insight that 10^-500 has some unique special property which
10^-499 does not have.
So if I can find something special with 10^-500, I can thence claim
the same thing for the
macroworld 10^500 is special.

Can we say that division breaks apart at 10^-500?

There is something special with a unique number between 0 and 1 in
probability theory. Correct
me if wrong for my memory is vague on this issue. I believe that if we
used only rational numbers between 0 and 1, there is only one rational
number that can be retrieved from the division of any set of rational
numbers in that interval by either "pi or e".

Another way of saying this is for the Macroworld of pseudosphere and
sphere and 10^500.
There is only one number when using only the integers for which
multiplication of any set
of those integers and either "pi or e" in which the product is a
integer. Now we can relate
to the famous equation e^i*2pi = 1 in which we have pi and e and
integers with an integer
product.

So the question is, since the pseudosphere has "e" and the sphere has
"pi", is the number
10^500 special because (e^i*2pi) x k = 10^500.

Yesterday I spoke of defining the pseudosphere cutaway as equivalent
to the full-pseudosphere. Where we had the full pseudosphere of radius
1 matched by a cutaway pseudosphere of radius somewhere between 1 and
2. So here we can sense something
special of some large numbered radius, where the matched pseudosphere
is not some
fractional number nor irrational number but involves the whole integer
of 10^500.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Let me try a different tactic, since I am as yet unable to show the
uniqueness of the number
10^500 as a boundary between finite and infinite.

Alright, a little help from some unlikely quarters. Last night, late
on TV was watching the BBC
world news and it had a picture of a disturbing woman in U.K. who was
near a garbage can
and a cat nearby. Where the women lifted the cat, (don't remember if
it was by the tail or
neck?) and throwing the poor kitty into the garbage can.

So then I was reminded that biology has pseudospheres and spheres,
where the DNA is packaged or packed inside a cell (sphere) and where
DNA is pseudospheric. Or the cat
tail as the tail of a pseudosphere.

So now, if any mathematician believes that you can have infinite
stretch yet finite area
or finite volume, then biology would have evolved such a beastie
because having infinite
stretch would surely confer the most biological survival into the next
generations.

But I want to use the analogy in a different manner.

If we use the interval 0 to 1 and infinite stretch in that interval
then 10^-500 means
we have a measure of finite versus infinite as that of subtract
10^-500 from 1.

So for example if the boundary were 1/3 then we have a remainder of
2/3 of the space.

With 10^500 we have no comparison.

So with a comparison then we can make progress on why 10^500 is
special.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
Let me try a different tactic, since I am as yet unable to show the
uniqueness of the number
10^500 as a boundary between finite and infinite.

Alright, a little help from some unlikely quarters. Last night, late
on TV was watching the BBC
world news and it had a picture of a disturbing woman in U.K. who was
near a garbage can
and a cat nearby. Where the women lifted the cat, (don't remember if
it was by the tail or
neck?) and throwing the poor kitty into the garbage can.
So then I was reminded that biology has pseudospheres and spheres,
where the DNA is packaged or packed inside a cell (sphere) and where
DNA is pseudospheric. Or the cat
tail as the tail of a pseudosphere.
So now, if any mathematician believes that you can have infinite
stretch yet finite area
or finite volume, then biology would have evolved such a beastie
because having infinite
stretch would surely confer the most biological survival into the next
generations.
But I want to use the analogy in a different manner.
If we use the interval 0 to 1 and infinite stretch in that interval
then 10^-500 means
we have a measure of finite versus infinite as that of subtract
10^-500 from 1.
So for example if the boundary were 1/3 then we have a remainder of
2/3 of the space.
With 10^500 we have no comparison.
So with a comparison then we can make progress on why 10^500 is
special.

Alright, I am playing around here, since I have not found the final
solution on this.
So let us take the microworld infinity of Old Math where there is an
infinity between
1 and 0 and the numbers never cease as they approach 0. But in new
math, numbers
cease at 10^-500 as the boundary, that math cannot continue beyond
10^-500 nor above
10^500 in the macroworld because Physics ceases to exist at those
boundaries.

If you are a new reader and joining this thread for the first time,
wondering why I am
flustered. It is because I am looking for pure math to show that
10^500 and 10^-500 are
special numbers as the boundary. I use pseudosphere and sphere to draw
out this
specialness, but have not found it as of yet.

So now, consider a mouse as a pseudosphere from 0 to 1 and where the
head and body
are in the 1 region and the tail is approaching 0 but is infinitely
long since the numbers
are infinitely abundant as they approach 0. And consider a cat whose
head and body are
10^500 and then has a tail that goes infinitely long after 10^500.

This is playing around, hoping to gain some insight.

Now suppose the cat catches one of these mice with its infinite tail
on the mouse and is
eating the mouse with infinite tail.

In the earlier post I said that suppose the boundary between finite
and infinite from 0 to 1
was 2/3, instead of 10^-500. Then there is a portion with distance
that we can call the infinite
portion of 1/3 and the finite portion of 2/3. We cannot say that for
the macroworld of 10^500
as the infinite portion is not bounded.

But let us inject a boundary by saying the finite portion of the
Macroworld is 2/3 and the infinite
portion is 1/3 so that would entail what? That we have infinity region
as 10^500 to 10^750 ??

Perhaps a better avenue or road to take, is to consider biology, that
the small exists to fill into
the large and the two are dependent on one another. So that a DNA
molecule must be packed
into a cell and that alot of cells must be organized to create a big
creature. So a tree of cells
is a large number for volume and the DNA of each tree cell is a
packing into a small number
for volume. So that the small demands of the large to have a boundary
and not go on endlessly and likewise for the large has the same
demand. So the two are dependent on one
another.

So that if the microworld has a demand of 10^-500 leaving the portion
for infinity to be
10^-500 and the portion for finite to be 1 - (10^-500), thus, forcing
that the macroworld
has finite for 10^500 and infinite region for 10^500 x 10^500 =
10^1000 So finite occupies
from 0 to 10^500 and infinity occupies 10^500 up to 10^1000.

Now I probably am making some calculation and proportion mistakes, but
that is not important.
for what I am seeking is whether there is a proportion that involves
the number 10^-500 and
10^500 that is special in mathematics.

This reminds me of the golden ratio.

So what I am wondering is that if we look for a boundary between
finite and infinite, it will definitely have to exist both in the
microworld and macroworld and it will have to have a
ratio between that of 0 and 1, and that ratio will then transfer to a
ratio of finite region for
macroworld and infinite region for macroworld. So, finally, is 10^500
a special number because
it serves as the Golden Ratio, or golden rectangle in the interval 1
to 0 and then the macroworld? Is 10^500 special because it is unique
in determining in this ratio of intervals?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
from 0 to 10^500 and infinity occupies 10^500 up to 10^1000.
Now I probably am making some calculation and proportion mistakes, but
that is not important.
for what I am seeking is whether there is a proportion that involves
the number 10^-500 and
10^500 that is special in mathematics.
This reminds me of the golden ratio.
So what I am wondering is that if we look for a boundary between
finite and infinite, it will definitely have to exist both in the
microworld and macroworld and it will have to have a
ratio between that of 0 and 1, and that ratio will then transfer to a
ratio of finite region for
macroworld and infinite region for macroworld. So, finally, is 10^500
a special number because
it serves as the Golden Ratio, or golden rectangle in the interval 1
to 0 and then the macroworld? Is 10^500 special because it is unique
in determining in this ratio of intervals?

Alright, maybe I found what is special about 10^500 ( or a number in
that neighborhood).

We have the golden ratio, (1+sqrt5) /2 = 1.618... And we have the
Golden Spiral and
we have the golden logarithmic spiral

The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, . . .


Now let me back up and start that over again as this:

0, 10^-500, 10^-500, 2/10^500, 3/10^500, . . .

Now, let me multiply that backed-up Fibonacci sequence by the number,
you guessed it,
10^500.

Question is, is there a 10^500 and then later on a number 10^1000 in
that sequence?
As I said, it maybe a number in the vicinity of 10^500 but it has the
property of inverse
within the Fibonacci sequence. Whether it is a unique inverse or
whether it is just the
first inverse in the Fibonacci sequence is a different open question
at this moment.

So, has anyone taken the Fibonacci sequence out to the vicinity of
10^500 and can tell
me what two Fibonacci numbers surround below and above 10^500, then I
would want to
know the same answer to the question of 10^1000.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
from 0 to 10^500 and infinity occupies 10^500 up to 10^1000.
Now I probably am making some calculation and proportion mistakes, but
that is not important.
for what I am seeking is whether there is a proportion that involves
the number 10^-500 and
10^500 that is special in mathematics.
This reminds me of the golden ratio.
So what I am wondering is that if we look for a boundary between
finite and infinite, it will definitely have to exist both in the
microworld and macroworld and it will have to have a
ratio between that of 0 and 1, and that ratio will then transfer to a
ratio of finite region for
macroworld and infinite region for macroworld. So, finally, is 10^500
a special number because
it serves as the Golden Ratio, or golden rectangle in the interval 1
to 0 and then the macroworld? Is 10^500 special because it is unique
in determining in this ratio of intervals?

Alright, maybe I found what is special about 10^500 ( or a number in
that neighborhood).
We have the golden ratio, (1+sqrt5) /2 = 1.618... And we have the
Golden Spiral and
we have the golden logarithmic spiral
The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, . . .
0, 10^-500, 10^-500, 2/10^500, 3/10^500, . . .
Now, let me multiply that backed-up Fibonacci sequence by the number,
you guessed it,
10^500.
Question is, is there a 10^500 and then later on a number 10^1000 in
that sequence?
As I said, it maybe a number in the vicinity of 10^500 but it has the
property of inverse
within the Fibonacci sequence. Whether it is a unique inverse or
whether it is just the
first inverse in the Fibonacci sequence is a different open question
at this moment.
So, has anyone taken the Fibonacci sequence out to the vicinity of
10^500 and can tell
me what two Fibonacci numbers surround below and above 10^500, then I
would want to
know the same answer to the question of 10^1000.

Alright, let me expand on this. We have the Golden Number Sequence
(let us call it
that instead of Fibonacci since we have "golden everything else with
this") as this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

provided I did not make a arithmetic mistake we have that as the
Golden Number
Sequence.

Now notice that 3^2 comes closest to 9 in that of 8 in the sequence.

So now, what I am conjecturing is that there is a number in that
sequence that is nearby
to 10^500 and when we square that number, we get another Golden Number
Sequence number nearby that of 10^1000.

Now obviously we have a square of a golden number that is shy of equal
by 1 unit in that
of 2^2 = 4 is shy by 1 of 5 and we have 3^2 = 9 is shy by 1 unit of 8.

So is there a quick proof that no golden numbers is of the form a^2 =
b where a and b are
golden numbers?

And if there is no proof, then is the number nearby 10^500 just such a
number??

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[quasi]

On Thu, 26 Aug 2010 16:34:17 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium
Alright, let me expand on this. We have the Golden Number Sequence
(let us call it
that instead of Fibonacci since we have "golden everything else with
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
provided I did not make a arithmetic mistake we have that as the
Golden Number
Sequence.
Now notice that 3^2 comes closest to 9 in that of 8 in the sequence.
So now, what I am conjecturing is that there is a number in that
sequence that is nearby to 10^500

No, nowhere close.

The Fibonacci number closest to 10^500 is F_2394, which is less than
10^500 by more than 7.3 x 10^498, and the next Fibonacci number,
F_2395, is greater than 10^500 by more than 4.9 x 10^499.

Post by Archimedes Plutonium
and when we square that number, we get another Golden Number
Sequence number nearby that of 10^1000.

No.

Firstly, it's been proved that there is no Fibonacci number greater
than 144 which is a perfect square.

Here is the reference (from Wikipedia):

J H E Cohn (1964). "Square Fibonacci Numbers Etc". Fibonacci Quarterly
2: pp. 109–113. http://math.la.asu.edu/~checkman/SquareFibonacci.html.

Secondly, the closest Fibonacci number to 10^1000 is F_4787, which is
greater than 10^1000 by more than 1.8 x 10^999, and the previous
Fibonacci number, F_4786, is less than 10^1000 by more than 2.6 x
10^999. In other words, 10^1000 is not anywhere close to a Fibonacci
number.

Post by Archimedes Plutonium
Now obviously we have a square of a golden number that is shy of equal
by 1 unit in that
of 2^2 = 4 is shy by 1 of 5 and we have 3^2 = 9 is shy by 1 unit of 8.
So is there a quick proof that no golden numbers is of the form a^2 =
b where a and b are
golden numbers?

I don't know if the proof is quick, but the theorem referenced above
(that no Fibonacci number greater than 144 is a perfect square) makes
it clear that there are no Fibonacci numbers a,b, with a>1, such that
b = a^2.

Post by Archimedes Plutonium
And if there is no proof, then is the number nearby 10^500 just such a
number??

But there _is_ a proof.

quasi

[Transfer Principle]

Post by quasi
On Thu, 26 Aug 2010 16:34:17 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium
So now, what I am conjecturing is that there is a number in that
sequence that is nearby to 10^500

No, nowhere close.
The Fibonacci number closest to 10^500 is F_2394, which is less than
10^500 by more than 7.3 x 10^498, and the next Fibonacci number,
F_2395, is greater than 10^500 by more than 4.9 x 10^499.

Hmmm, I was thinking about what quasi writes here for a
moment, and I wonder, if 10^500 isn't "close" to a
Fibonacci number, perhaps there exist another power of
10 which is close to a Fibonacci number. (We know by
Carmichael's Theorem, of course, that no power of ten
other than 10^0 can exactly equal a Fibonacci.)

I've heard somewhere that the fractional parts of an
irrational number (like log_10(phi), for instance), are
equally distributed in the unit interval. Therefore, for
every positive real epsilon, there ought to exist a
Fibonacci number F_n (with n>2, to avoid the trivial case
F_n=1) such that the fractional part of log_10(F_n) is less
than epsilon.

Therefore, there ought to exist a Fibonacci number F_n and
a power 10^d such that (F_n-10^d)/10^d is less than epsilon.

Therefore, for every natural number m, there ought to exist
a nontrivial Fibonacci number whose decimal representation
begins with 1 followed by m zeroes (followed by the rest of
the digits).

And AP might be interested in values of n and d for
certain small values of epsilon. This sounds like the sort
of sequence that might appear in Sloane (i.e., values of n
or d for declining values of epsilon), but I wasn't able to
find such a sequence.

[Archimedes Plutonium]

Post by Transfer Principle

Post by quasi
On Thu, 26 Aug 2010 16:34:17 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium
So now, what I am conjecturing is that there is a number in that
sequence that is nearby to 10^500

No, nowhere close.
The Fibonacci number closest to 10^500 is F_2394, which is less than
10^500 by more than 7.3 x 10^498, and the next Fibonacci number,
F_2395, is greater than 10^500 by more than 4.9 x 10^499.

Hmmm, I was thinking about what quasi writes here for a
moment, and I wonder, if 10^500 isn't "close" to a
Fibonacci number, perhaps there exist another power of
10 which is close to a Fibonacci number. (We know by
Carmichael's Theorem, of course, that no power of ten
other than 10^0 can exactly equal a Fibonacci.)
I've heard somewhere that the fractional parts of an
irrational number (like log_10(phi), for instance), are
equally distributed in the unit interval. Therefore, for
every positive real epsilon, there ought to exist a
Fibonacci number F_n (with n>2, to avoid the trivial case
F_n=1) such that the fractional part of log_10(F_n) is less
than epsilon.
Therefore, there ought to exist a Fibonacci number F_n and
a power 10^d such that (F_n-10^d)/10^d is less than epsilon.
Therefore, for every natural number m, there ought to exist
a nontrivial Fibonacci number whose decimal representation
begins with 1 followed by m zeroes (followed by the rest of
the digits).
And AP might be interested in values of n and d for
certain small values of epsilon. This sounds like the sort
of sequence that might appear in Sloane (i.e., values of n
or d for declining values of epsilon), but I wasn't able to
find such a sequence.

This reminds me of my proof of the Riemann Hypothesis where I go
through every
Natural Number via the Fibonacci sequence by multiplying every term in
the sequence
by 2, then 3, then 4 etc etc, yielding the Generalized Fibonacci
sequence.

So in one of those sequences we have sticking out in broad daylight
the exact number
10^500. And perhaps in one of those sequences, that number 10^500 is
the sole unique
perfect square.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Transfer Principle

Post by quasi
On Thu, 26 Aug 2010 16:34:17 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium
So now, what I am conjecturing is that there is a number in that
sequence that is nearby to 10^500

No, nowhere close.
The Fibonacci number closest to 10^500 is F_2394, which is less than
10^500 by more than 7.3 x 10^498, and the next Fibonacci number,
F_2395, is greater than 10^500 by more than 4.9 x 10^499.

Hmmm, I was thinking about what quasi writes here for a
moment, and I wonder, if 10^500 isn't "close" to a
Fibonacci number, perhaps there exist another power of
10 which is close to a Fibonacci number. (We know by
Carmichael's Theorem, of course, that no power of ten
other than 10^0 can exactly equal a Fibonacci.)
I've heard somewhere that the fractional parts of an
irrational number (like log_10(phi), for instance), are
equally distributed in the unit interval. Therefore, for
every positive real epsilon, there ought to exist a
Fibonacci number F_n (with n>2, to avoid the trivial case
F_n=1) such that the fractional part of log_10(F_n) is less
than epsilon.
Therefore, there ought to exist a Fibonacci number F_n and
a power 10^d such that (F_n-10^d)/10^d is less than epsilon.
Therefore, for every natural number m, there ought to exist
a nontrivial Fibonacci number whose decimal representation
begins with 1 followed by m zeroes (followed by the rest of
the digits).
And AP might be interested in values of n and d for
certain small values of epsilon. This sounds like the sort
of sequence that might appear in Sloane (i.e., values of n
or d for declining values of epsilon), but I wasn't able to
find such a sequence.

This reminds me of my proof of the Riemann Hypothesis where I go
through every
Natural Number via the Fibonacci sequence by multiplying every term in
the sequence
by 2, then 3, then 4 etc etc, yielding the Generalized Fibonacci
sequence.
So in one of those sequences we have sticking out in broad daylight
the exact number
10^500. And perhaps in one of those sequences, that number 10^500 is
the sole unique
perfect square.

In my boastering I made a mistake, for it looks as though an algebraic
proof of Carmichael
theorem is far easier than a geometrical involving semiperimeters.

To do this proof I need the Generalized Fibonacci Sequence and notate
them as such:

F_1 = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

F_2 = 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466,
754, . . .

F_3 = 0, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, . . .

F_4 = 0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 576, . . .

F_5 = 0, 5, 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, . . .

Alright, I hope I did not make any arithmetic mistakes above for I am
forgoing an earlier bedtime, as this math proof has me challenged.
I cannot go to sleep with a proof on mind. So if I made any arithmetic
errors or omissions, please overlook them.

Now I have not seen at all the Carmichael theorem, only what L. Walker
has spoken about it and saying it is "complex" as in complicated.

I suspect mine will be logarithmically easier in comparison and the
reason I say that
is because I doubt that Carmichael used the generalized Fibonacci
sequence and
used only the usual sequence. Sometimes a math proof is easiest when
done on the
General form of the statement rather than a specialized form. So I
probably have the
easiest proof of this theorem which I suspect is this statement:

Carmichael Statement: the number 144 is the only unique perfect square

Post by Archimedes Plutonium
1 in the

F_1 sequence. Only that Carmichael would have simply said Fibonacci
sequence,
oblivious to the general sequences.

Proof: Notice that the general sequences have a perfect square early
on in the sequence,
where F_1 has 144 as the 13th term. In F_2 we have 4 and 16. In F_3 we
have 9 as the 5th
term. In F_4 we have 4 and in F_5 we have 25.

Now notice that these perfect squares come early on and then it is our
burden to prove these are the only perfect-squares in those sequences.

Now notice that what governs the sequences is the phi number of
(1+sqrt5)/ 2 and this is
an irrational number. And notice that the division of the first few
terms in each sequence
rarely gets to 1.61803 of 1.6180339887498948482.. But notice that
when the first appearance of two consecutive
terms yields a 1.61803, that there are no longer and never any perfect
squares in the sequence.
Now the reason for that disappearance is because the irrationality of
sqrt5 in phi has taken ahold of the numbers. In the first few terms of
the sequence are sloppy with alot of spill-way
in them such as 4/2 =2 and then 6/4 =1.5 so we have not reached yet
where we have 1.61803 appearing in the division in F_1 nor F_2 nor F_3
nor F_4 nor F_5. So that the proof of Carmichael theorem is really a
proof that talks about a Spill-way or a Lee-way that allows for
some shenanigans of numbers to be perfect squares but once the
division of consecutive numbers is solidly planted inside of the phi
number as 1.61803, there is no more play room
or spill way to allow shenanigans.

Now in F_1 we have the unique perfect square >1 of 144 and we have
144/89 gives 1.617
and where previously we had 89/55 = 1.61818

And 144 in F_1 maybe the deepest penetration of spill-way room in the
Generalized Fibonacci
sequences.

So the proof is rather easy, but rather uncharming for it talks about
a messy situation of a few beginning terms of a sequence. It reminds
me of the hallway entrance to houses where the hallway is rather more
dirty than the rest of the house simply because of the entrance
traffic
and the leeway for mess.

So the force of the proof is that once the sqrt5 of phi is present in
1.61803 then no more perfect square terms can exist in that
environment.

Now mine is a direct proof and I would bet that Carmichael's is
indirect.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Transfer Principle]

Post by Transfer Principle
And AP might be interested in values of n and d for
certain small values of epsilon. This sounds like the sort
of sequence that might appear in Sloane (i.e., values of n
or d for declining values of epsilon), but I wasn't able to
find such a sequence.

As I think about this for a moment, I decided simply to look
for the power of 10 near a Fibonacci myself.

Had it not been for the sqrt(5) in the denominator, this
would reduce to finding the continued fraction of the
irrational number log_10(phi). So instead, I tried looking
for values of log_10(phi)n-log_10(5)/2 near an integer in the
AP range (i.e., 500-1000, corresponding to 10^500 to 10^1000).

And one such value stuck out -- namely 617:

Wolfram Alpha Input:
fibonacci 2954

Result:
1.00000989*10^617

Not only that, but let's recall that some posters have
criticized AP's value 10^500 as the finite-infinite boundary
because the RSA-2048 values exceed this boundary. Well, let's
look at 2^2048, the upper bound of the RSA-2048 numbers:

Wolfram Alpha Input:
2^2048

Result:
3.2317006*10^616

And we notice that this is within an order of magnitude of that
Fibonacci near-power of 10 we just found!

So I would recommend that AP choose 10^617 as the boundary
between finite and infinite rather than 10^500. For 10^617 is
close in ratio to a Fibonacci number, and furthermore, it is
larger than 2^2048 so that RSA numbers, which are useful in
science (though in cryptography, not physics), are finite.

But of course, AP is unlikely to adopt this boundary unless he
can tie it somehow to Atom Totality. I, to repeat, am not an
Atom Totalitarian, and so I don't need to refer to a plutonium
atom to choose a largest finite number, but AP does.

Well, elsewhere in this thread, AP asks about the numbers
19^(22^2) and 22^(22^2). Well, let's try something here:

Wolfram Alpha Input:
19^22^2

Result:
8.2554901*10^618

And this is within just three orders of magnitude of the RSA
and Fibonacci numbers that we've found!

So I recommend that 19^22^2 be chosen as the boundary. It
satisfies all of AP's desiderata -- it's near the Fibonacci
near-power of 10, it's larger than the numbers regularly used
in science (including cryptography), and it's related to the
Atom Totality theory (involving 19 and 22).

[Archimedes Plutonium]

Post by Transfer Principle

Post by Transfer Principle
And AP might be interested in values of n and d for
certain small values of epsilon. This sounds like the sort
of sequence that might appear in Sloane (i.e., values of n
or d for declining values of epsilon), but I wasn't able to
find such a sequence.

As I think about this for a moment, I decided simply to look
for the power of 10 near a Fibonacci myself.
Had it not been for the sqrt(5) in the denominator, this
would reduce to finding the continued fraction of the
irrational number log_10(phi). So instead, I tried looking
for values of log_10(phi)n-log_10(5)/2 near an integer in the
AP range (i.e., 500-1000, corresponding to 10^500 to 10^1000).
fibonacci 2954
1.00000989*10^617
Not only that, but let's recall that some posters have
criticized AP's value 10^500 as the finite-infinite boundary
because the RSA-2048 values exceed this boundary. Well, let's

L. Walker, please talk about this RSA and what it means. Does it have
some
Physics meaning? Or is where computers power has reached but has not
gone
beyond? What is RSA-2048?

Post by Transfer Principle
2^2048
3.2317006*10^616

Does that have some Physics behind it?

Post by Transfer Principle
And we notice that this is within an order of magnitude of that
Fibonacci near-power of 10 we just found!
So I would recommend that AP choose 10^617 as the boundary
between finite and infinite rather than 10^500. For 10^617 is
close in ratio to a Fibonacci number, and furthermore, it is
larger than 2^2048 so that RSA numbers, which are useful in
science (though in cryptography, not physics), are finite.

So it is a number in which cryptography has reached but not exceeded
and that if we were to ask cryptographers in a century later they
would
have a new number?

Post by Transfer Principle
But of course, AP is unlikely to adopt this boundary unless he
can tie it somehow to Atom Totality. I, to repeat, am not an
Atom Totalitarian, and so I don't need to refer to a plutonium
atom to choose a largest finite number, but AP does.
Well, elsewhere in this thread, AP asks about the numbers
19^22^2
8.2554901*10^618
And this is within just three orders of magnitude of the RSA
and Fibonacci numbers that we've found!
So I recommend that 19^22^2 be chosen as the boundary. It
satisfies all of AP's desiderata -- it's near the Fibonacci
near-power of 10, it's larger than the numbers regularly used
in science (including cryptography), and it's related to the
Atom Totality theory (involving 19 and 22).

LWalk, please tell what 22^(22)(22) is?

The atomic number of elements of 100 to 109 have 253 nucleons to 266
nucleons.

Can someone tell what 266! is equal to? And can someone tell what
factorial is
close to 10^617


Now here is an odd question that may have some value or relevance.

Remember I called the Generalized Fibonacci Sequence as F_1, F_2,
F_3, ... as the entire
collection of these sequences. Much like the Hensel p-adics is not
just 2-adics but 3-adics, 4-adics, 5-adics, ad infinitum. And it is
silly of math to think the p-adics dwells with just 2-adics
just as silly as thinking the phi and golden ratio and golden log
spiral is dealt with in
just the one sequence of 0, 1, 1, 2, 3, 5, 8, .... when we have F_2 as
0, 2, 2, 4, 6, 10, . . . and
these sequences ad infinitum.

But now, here is a new twist on all of this. Instead of just those
plain numbers what about
a Fibonacci sequence of the factorial so we have F_1! as equal to 0,
1!, 1!, 2!, 3!, 8!, . . .
And many questions will abound from this extension. An immediate
question on my mind
anyway is whether this extension allows for a logarithmic spiral since
it seems to eliminate
rectangles in whirling squares, and if so, what kind of spiral
is it? And the same questions for F_2!.

So what is the deep internal geometrical link between rectangles in
whirling squares and that of the logarithmic spiral that is produced
by those rectangles in whirling squares? Yet with the factorial
Fibonacci do we get any logarithmic spiral? We know that factorial
means all possible
arrangements of n things. So does all possible arrangements of n
things also have a logarithmic spiral arrangement?

I ask this mostly, because the atomic number of elements 100 to 109
have nucleons with coulomb interactions of 253! to that of 266!
respectively. So is there a sort of Fibonacci
sequence of coulomb interactions of nucleons in elements of the
periodic table. And how many
of these sequences do we cover every chemical element?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

(snipped)

Post by Archimedes Plutonium

Post by Transfer Principle
Not only that, but let's recall that some posters have
criticized AP's value 10^500 as the finite-infinite boundary
because the RSA-2048 values exceed this boundary. Well, let's

L. Walker, please talk about this RSA and what it means. Does it have
some
Physics meaning? Or is where computers power has reached but has not
gone
beyond? What is RSA-2048?

I would be extremely delighted if RSA means something about Physics of
signalling
in that you cannot, due to the planck constant use EM waves to make a
signal
at RSA-2048. Does it mean that you cannot have information or messages
sent
with EM waves at RSA-2048.

Is that what it means L Walk, that it is a boundary of Physics where
you can no
longer communicate because the EM waves are in the Uncertainty
Principle range?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
But now, here is a new twist on all of this. Instead of just those
plain numbers what about
a Fibonacci sequence of the factorial so we have F_1! as equal to 0,
1!, 1!, 2!, 3!, 8!, . . .
And many questions will abound from this extension. An immediate
question on my mind
anyway is whether this extension allows for a logarithmic spiral since
it seems to eliminate
rectangles in whirling squares, and if so, what kind of spiral
is it? And the same questions for F_2!.
So what is the deep internal geometrical link between rectangles in
whirling squares and that of the logarithmic spiral that is produced
by those rectangles in whirling squares? Yet with the factorial
Fibonacci do we get any logarithmic spiral? We know that factorial
means all possible
arrangements of n things. So does all possible arrangements of n
things also have a logarithmic spiral arrangement?

So let me see if this Factorial Fibonacci can yield a logarithmic
spiral.


F_1! is 0, 1!, 1!, 2!, 3!, 8!, 40326!, . . .

So can I arrange this sequence into a rectangle of whirling squares?

Oddly enough, I think it is possible and would be like a Fractal
Rectangle of
Whirling Squares.

The logarithmic spiral that results from it is very slow to make its
first revolution
once it goes beyond 8!

Is there anything like this in math literature? I doubt it.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Archimedes Plutonium wrote:

F_1! is 0, 1!, 1!, 2!, 3!, 8!, 40326!, . . .

I forgotten whether some people define 0! as equal to 1. Whether it is
1 or not
is crucial to this next question. Or, I could just leave out the 0
altogether, for
it really does not matter for the regular Fibonacci or the Factorial
Fibonacci, unless
of course I cannot get any odd numbers.

So if I define 0! = 1 then it does not alter the above of its numbers:

F_1! is 0!, 1!, 2!, 3!, 8!, 40326!, . . .

Then F_2! is 0!, 2!, 3!, . . . and same as F_1!, but then the picture
changes with
F_3! is 0!, 3!, 7!, . . .

But if 0! is not 1, then I do not see how we get any odd numbers in
the sequences.

So this Factorial Fibonacci would be a tremendously good argument in
favor of having
the value of 0! equal to 1.

Well, enough science for one night!

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
F_1! is 0, 1!, 1!, 2!, 3!, 8!, 40326!, . . .
I forgotten whether some people define 0! as equal to 1. Whether it is
1 or not
is crucial to this next question. Or, I could just leave out the 0
altogether, for
it really does not matter for the regular Fibonacci or the Factorial
Fibonacci, unless
of course I cannot get any odd numbers.
F_1! is 0!, 1!, 2!, 3!, 8!, 40326!, . . .
Then F_2! is 0!, 2!, 3!, . . . and same as F_1!, but then the picture
changes with
F_3! is 0!, 3!, 7!, . . .
But if 0! is not 1, then I do not see how we get any odd numbers in
the sequences.
So this Factorial Fibonacci would be a tremendously good argument in
favor of having
the value of 0! equal to 1.
Well, enough science for one night!

This is too important to delay.

I thought of the above as to how a log spiral accrues and I do not see
how
the curves are log curves without a crimp in every time the curve
leaves one
square to the next huge square.

So why even bother to add factorials in the regular Fibonacci
sequence?

The regular Fibonacci sequence is about additon.

Factorial is all about multiplication so I should do a Factorial
Fibonacci as
purely multiplication and that would mean only one type of Factorial
Fibonacci

Factorial Fibonacci as 0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . .

That sequence becomes 1, 1, 2, 6, 24, 120, 720, 5040, 40320, . . .

Now how to make a log spiral from those squares? The pretty thing is,
that
the crevice formed by the stacking of newer squares in a rotation is
where the
log spiral is formed. In Regular Fibonacci of addition the log spiral
is formed inside
the rectangle of whirling squares. In the Factorial Fibonacci the log
spiral is formed
outside the whirling squares in what I call the ever rotating crevice.

So I think I hit upon a inverse of geometry and algebra. That the
addition Fibonacci
yields a log spiral inside the whirling squares, whereas a
multiplication (factorial)
Fibonacci yields a log spiral in the crevice region of the whirling
squares.

Now for other Multiplicative Fibonacci Sequences as such:

M-F_1 as 1, 2, 2, 4, 8, 32, 256, . . .
M-F_2 as 2, 3, 6, 18, 108, . . .
M-F_3 as 3, 4, 12, 48, . . .

Now here again, I seem to run into the problem of not many odd
numbers, and of course
in the Factorial Fibonacci there are no odd numbers except 1. But the
big question here
again is whether when stacking those squares that a log spiral is
produced in the crevice
region? So that multiplication is the inverse of addition Fibonacci in
that the log spiral is outside the squares, and not inside the
squares. Now I wonder if the log spirals are related
to one another in the addition versus multiplication?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Now L Walk tells me that

Post by Transfer Principle
fibonacci 2954
1.00000989*10^617

And in Signal theory crytography

Post by Transfer Principle
2^2048
3.2317006*10^616

Does that have some Physics behind it?

Post by Transfer Principle
And we notice that this is within an order of magnitude of that
Fibonacci near-power of 10 we just found!
Well, elsewhere in this thread, AP asks about the numbers
19^22^2
8.2554901*10^618

Now I wondered about the Factorial Fibonacci Sequence:

0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . .

And I wondered how to construct those successive squares in order for
the whirling squares
to be able to construct a logarithmic-spiral without any crimps in the
spiral. Obviously, if a crimp is in the spiral it is no longer a log-
spiral. So is this where mathematics has a
"natural boundary between finite and infinite?" Where the log spiral
gives out and is not
able to be constructed anymore? Is the crimp showing up at 10^618? or
approx 270!

Is the crimp showing up where we have a log-spiral up to that square
at 270! but unable to
arrange all the squares so that a crimp begins to show up? And is this
somehow related to
cryptographies RSA-2048?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Now I wondered about the Factorial Fibonacci Sequence:

0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . .

That sequence looks like this:

1, 1, 2, 6, 24, 120, 720, 5040, . . .

Now each of those forms a square and we have to arrange them so as to
draw a logarithmic spiral in them just as we drew a log-spiral in the
regular
normal Fibonacci sequence which is shown in pictures as the rectangle
of whirling
squares.

Only here, in the factorial-fibonacci, since it is all multiplication
that the squares
have the log-spiral drawn in the crevasse of the squares. Let me try
to show that.

x x
XX


Now those are squares of 1 and 1 alongside each other and a square of
2 below the squares of
1 for the first three terms in factorial fibonacci.

xx
XX
__________



__________

Now those lines represents the square of 6 in the factorial fibonacci

ZZZZZZZZZZZZZZZZZZZZZZZZ







xx
XX
__________



__________|ZZZZZZZZZZZZZZZZZZZZZZZZZZZ



Now those Z's represent the 24 square in the factorial fibonacci. So
we begin
to visualize how to stack the squares and we begin to see a winding or
crevasse, an open area inside where there is empty space. It is this
empty
space that we draw the log-spiral.

And the log-spiral, I conjecture is alright up to about 270! where it
breaks down
in that it is impossible to continue without a crimp into the log-
spiral.

So this is a natural boundary of mathematics, where it is possible to
draw the
log spiral but at a specific large number, math ceases to be able to
continue
the log spiral.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

L-Walk, I looked up information theory and found RSA to mean bit keys.
Sort of like
physics application of large numbers. I guess cryptography needs to
use very large
numbers and their range is 10^500 to 10^1000. This is supporting data
for a boundary
somewhere in that region but I need a "natural math boundary".

I think I have it in the idea that the limit of whirling squares by a
Fibonacci sequence
is the factorial fibonacci and that at a specific square such as 270!
that no log-spiral
continues unless it is crimped.

Of course the regular normal Fibonacci sequences that delivers log
spirals continue
without any crimps because they generate so slowly of an increase in
squares of the
whirling squares, since those sequences are generated by addition of
two previous terms.
In the factorial fibonacci, the squares are generated very large
quickly and I speculate
that at 270! or in that vicinity, it maybe 253! where the log-spiral
that was possible to
generate up until thence, is impossible to continue for a crimp is
required.

And we can do the same generation for the microworld where the
microscopic log spiral
is impossible smaller than 1/270! without crimping. And this makes
sense also with the
idea that if absolute betweenness axiom with parallel postulate allows
to construct a microscopic triangle that has two 90 degree angles, or
two sides parallel.

So mathematics has this natural boundary between finite and infinite
and we must now
revise the Betweeness Axiom for it contradicts the Parallel Postulate.

I need David Bernier's help please. I need for him to tell me what
19^(19*19) = exactly
and what 19^(22*22) = exactly and what 22^(22*22) = exactly.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[David Bernier]

Archimedes Plutonium wrote:
[...]

Post by Archimedes Plutonium
So mathematics has this natural boundary between finite and infinite
and we must now
revise the Betweeness Axiom for it contradicts the Parallel Postulate.

I haven't heard of the Betweeness Axiom. What does it
mean for you?

Post by Archimedes Plutonium
I need David Bernier's help please. I need for him to tell me what
19^(19*19) = exactly
and what 19^(22*22) = exactly and what 22^(22*22) = exactly.

You can use the WIMS calculator at 1000 digits, base
converter module, here:
<
http://wims.auto.u-psud.fr/wims/wims.cgi?session=IQBEA5913D.3&+lang=en&+module=tool%2Fnumber%2Fbaseconv.en
David Bernier

[Archimedes Plutonium]

Post by David Bernier
[...]

Post by Archimedes Plutonium
So mathematics has this natural boundary between finite and infinite
and we must now
revise the Betweeness Axiom for it contradicts the Parallel Postulate.

I haven't heard of the Betweeness Axiom. What does it
mean for you?

Post by Archimedes Plutonium
I need David Bernier's help please. I need for him to tell me what
19^(19*19) = exactly
and what 19^(22*22) = exactly and what 22^(22*22) = exactly.

You can use the WIMS calculator at 1000 digits, base
<
http://wims.auto.u-psud.fr/wims/wims.cgi?session=IQBEA5913D.3&+lang=en&+module=tool%2Fnumber%2Fbaseconv.en
David Bernier

Hilbert's betweenness axiom:

If A and C are two points of a straight line, then there exists at
least one point B lying between A and C and at least one point D so
situated that C lies between A and D.

Which is inconsistent with the Parallel Postulate for it allows
parallel sides of a smallish triangle.

Euclidean geometry has to be refitted so that finite versus infinite
are not mixed up.

So if the boundary is 10^500 and 10^-500, there is no number between 0
and 10^-500.
So that the line segment from 0 to 1 has a hole from 0 to 10^-500.

There are only 10^500 numbers and thus, points between 0 and 1. The
number line is not
a continuum but a discrete ordering of 10^500 points or numbers
between 0 and 1.

Mathematics is confined to Finite and does not extend into the
Infinite region for mathematics
is not trustworthy that the operations stand up or breakdown in the
infinite region where
the logic is duality, no longer Aristotelian logic.

So the Kepler Packing should have been stated for 10^500 as the
boundary and what is the maximum packing for this. It is not a pure
hexagonal closed pack for it involves some
shuffling at the boundary of 10^500.

That is why Kepler Packing was never proveable.

Poincare conjecture was never proveable in old math for they had
absolute betweeness axiom
of Hilbert. There has been recent flurry of news that PC is proven.
That is a false alarm. When
there are holes smaller than 10^500 then PC is no longer a true
conjecture. And the reason
PC was difficult is because, again, Betweenness does not hold when
Finite versus Infinite
is well defined at a boundary of 10^500.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

This page shows the golden log spiral in a rectangle of whirling
squares and it
shows the spiral intersecting at the two diagonal corners of each
square.

http://en.wikipedia.org/wiki/Fibonacci_sequence

Now in an earlier post I talked about the log spiral in the Factorial
Fibonacci as
in the crevice or crevasse or the canyon region of the whirling
squares. I take that back,
for it maybe the case where we can have two log-spirals, one in the
canyon region--
outside the squares-- and one in the inside of the squares.

So tonight, I spent some time in paper cutouts of squares of 120, 24
and 6 and using
a ruler for the 720 square. And layed them out in a whirling pattern.

Now I maybe able to get two log spirals, one in the canyon region and
one inside the squares.

The important feature that this experiment and exercise is all about,
is the speculation or
conjecture that the log-spiral/s cease to be constructed at about 253!
or thereabouts.

That is the reason for the cutouts so that I can visualize what is
going on. So that I may
perceive why the log spiral can be constructed and work for 1! through
253! but then stops
the ability to go further unless the log spiral is crimped.

In other words, the Factorial Fibonacci imposes the utmost test to
being able to produce a log-spiral
from this sequence:

0!, 1!, 2!, 3!, 4!, 5!, 6!, 7!, . . .

So that if this conjecture is true and that the ability to create a
log spiral from Fibonacci (factorial) sequence is halted at 253! or in
that region, would be the boundary, a natural
boundary in mathematics that is the boundary between what we define as
finite versus
infinite.

Now from looking at the situation from my cutouts, I can perceive that
the log spiral
begins to have this characteristic of a backslide and not being able
to "center" at the
corner intersection and this tiny bit of backslide takes a toll at
253! where we can no
longer continue with the log spiral.

Now maybe, someone in mathematics has noticed this about constructing
log-spirals
with multiplication in Fibonacci sequence.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
This page shows the golden log spiral in a rectangle of whirling
squares and it
shows the spiral intersecting at the two diagonal corners of each
square.
http://en.wikipedia.org/wiki/Fibonacci_sequence
Now in an earlier post I talked about the log spiral in the Factorial
Fibonacci as
in the crevice or crevasse or the canyon region of the whirling
squares. I take that back,
for it maybe the case where we can have two log-spirals, one in the
canyon region--
outside the squares-- and one in the inside of the squares.
So tonight, I spent some time in paper cutouts of squares of 120, 24
and 6 and using
a ruler for the 720 square. And layed them out in a whirling pattern.
Now I maybe able to get two log spirals, one in the canyon region and
one inside the squares.
The important feature that this experiment and exercise is all about,
is the speculation or
conjecture that the log-spiral/s cease to be constructed at about 253!
or thereabouts.
That is the reason for the cutouts so that I can visualize what is
going on. So that I may
perceive why the log spiral can be constructed and work for 1! through
253! but then stops
the ability to go further unless the log spiral is crimped.
In other words, the Factorial Fibonacci imposes the utmost test to
being able to produce a log-spiral
0!, 1!, 2!, 3!, 4!, 5!, 6!, 7!, . . .
So that if this conjecture is true and that the ability to create a
log spiral from Fibonacci (factorial) sequence is halted at 253! or in
that region, would be the boundary, a natural
boundary in mathematics that is the boundary between what we define as
finite versus
infinite.
Now from looking at the situation from my cutouts, I can perceive that
the log spiral
begins to have this characteristic of a backslide and not being able
to "center" at the
corner intersection and this tiny bit of backslide takes a toll at
253! where we can no
longer continue with the log spiral.
Now maybe, someone in mathematics has noticed this about constructing
log-spirals
with multiplication in Fibonacci sequence.

Alright, I have tried every way possible to retain a log-spiral inside
those whirling
squares of the factorial-fibonacci and find no way, no tricks or
anything to keep
the inside of the squares as a habitat for the construction of a log-
spiral.
Maybe someone else knows of a trick in which to keep the inside of the
squares
as an environment able to produce a log spiral there. If I could have
stipulated one point of intersection inside each square, but then
again that is impossible since the curve has to enter and thus exit
each square.

However, I do find a log-spiral conducive to live in the "canyon
region" formed by
the whirling squares of the factorial-fibonacci 0!, 1!, 2!, 3!, 4!,
5!, . . . And the way
I can guarantee the existence of a log spiral in the "canyon
region" (the whirling squares
leave a gap as they go around and in this gap which I call the
canyon), the way I guarantee
a log spiral in this canyon is that the stipulation is the log-spiral
needs to intersect each
square only at one point.

In the addition-fibonacci, the regular normal fibonacci 0, 1, 1, 2, 3,
5, 8, . . . the log
spiral intersects each square at two points-- two diagonal corners. So
what allows
a log spiral in the factorial-fibonacci canyon region is the relaxed
requirement of only
one point of intersection per each square.

And now, the conjecture or speculation on my part is that since these
squares grow
enormously large very quickly, there is a backslide of the
intersection point and at about
253!, this backslide of the intersection halts the construction of the
log-spiral curve
completely.

Now I wonder if there is an easy proof or disproof of this
speculation? Whether the golden-ratio
phi can indicate the rate of backslide and calculus to indicate where
the halting of
construction occurs? L.Walk, any suggestions?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, some people are not convinced this is a viable method of
having the log-spiral outside
the whirling squares. But this is easy to demonstrate, by looking at
the classical example
of the log spiral inside the whirling squares of the regular normal
Fibonacci sequence as
shown in the picture of Wikipedia:

http://en.wikipedia.org/wiki/Fibonacci_sequence


In that picture is a golden-log-spiral of squares out to the 21 square
and it shows the two
points of intersection inside each square of its diagonal corners.

But now consider just the Golden-Log Spiral and remove the inside
squares, and for the
21 square imagine the 34 square outside with its singular point of
intersection and then
imagine for the 13 inside square the 21 outside square with its
singular point of intersection.
and so in this fashion we can start to see the "outside whirling
squares" with the inner canyon
being built by these outside whirling squares.

But in the Golden log spiral we can only do the canyon in one
revolution because the
stacking leaves no gaps. But in the Factorial-Fibonacci, the stacking
leaves gaps and
so we can have a continual canyon built.

In the Golden log spiral, the intersection at one point of the outside
squares is at the phi golden ratio of those squares but in the
factorial-fibonacci the intersection is not at the phi ratio. And so
my conjecture or speculation is that-- at the 253/4 winding or
revolution of this
log spiral of factorial, that the 253! square no longer is able to fit
the log spiral and the log
spiral comes to a halt.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Now there is another possibility that I should not dismiss. The
possibility of the log
spiral intersecting at two points of the whirling squares of the
factorial-fibonacci.
It maybe the case that by allowing only one point of intersection
barrs all log spirals
from being created in the canyons of the factorial-fibonacci. So if I
ease up on the
restriction of 1 point of intersection and allow 2 points, maybe
better.

And the speculation still remains that at the 253/4 revolutions of the
whirling squares
of the factorial-fibonacci, that the log spiral is unable to continue
at square 253! (or
in that neighborhood).

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, let sharpen this conjecture up. But first let me scold myself
for thinking that I could
have found this in 3Dimensional space with the pseudosphere versus the
sphere. That feat
may still be viable to accomplish but orders of magnitude far more
difficult. The lesson to
be learned, by me anyway, was that in difficult circumstances, never
stick to 3D, but immediately go to 2D, and later, jump back to 3D.

I am confident of this speculation or conjecture that the natural
boundary between finite
number or finite anything versus infinite number or infinite anything
is the boundary of
where log-spiral versus squares no longer are able to cooperate to
generate the log-spiral
any further than a given large square. I speculate where this
breakdown occurs is around
253! neighborhood. It may go to 270! or it may go to 22^(22*22).

By breakdown I mean several things. That at a large square, the log-
spiral misses it completely, or that the log spiral fails to intersect
with the large square.

Now by log-spiral, I mean all family classes and types of log-spirals.
Where the golden
ratio log-spiral is but one fancy example of a log spiral. So by log-
spiral, I mean the entire
gamut of all types of equiangular spirals. And at this conjectured
large square, say 253!, that
all the types of log-spirals fail to continue. That all the log
spirals were alright, and doing fine
and dandy and fanciful up to 253! square but all broke down at 253!.

Now the log-spirals can intersect with the whirling squares of the
factorial-fibonacci sequence:

0!, 1!, 2!, 3!, 4!, 5!, 6!, 7!, . . .

at either one point of intersection or at two points and if at two
points, means the log-spiral has
entered inside the square. Now in the factorial-fibonacci we notice
that the whirling squares
construct a canyon-channel that the normal regular fibonacci sequence
does not construct.
Now this feature of a canyon channel is unique to multiplicative
fibonacci sequences versus
the normal regular fibonacci sequence of 0, 1, 1, 2, 3, 5, 8, . . .

Now how much of a role the canyon-channel plays in this conjecture, I
am unsure of as yet.
Perhaps a vital role.

The important issue is to prove whether the factorial-fibonacci allows
at least one log-spiral to
reach in construction up to 253! or that neighborhood (whether it is
270! or thereabouts).
Anyway the Conjecture or Speculation is that in the factorial-
fibonacci, that we can construct at least one log spiral of all the
possible log-spirals to wind and revolve in 253/4 or 270/4, but
at those huge numbers, there no longer exists any log-spiral that can
continue beyond that
large number. That is the conjecture.

And the implications are huge. It means that mathematics, pure math,
has a built-in boundary
of what is finite versus what is infinite.

If my conjecture is proven true, meaning that there is no log spiral
that can be built after a specific Euclidean large square, means that
a line in Hyperbolic Geometry is the end boundary
of finite in Hyperbolic geometry and correspondingly, Euclidean
Geometry is a bounded finite from infinite at that same large number.

So that in geometry or algebra or numbers, whenever we say "goes to
infinity" simply means
it goes higher than 253! or whereever this special large number is
located. That our concept of
infinity means simply going beyond this special large number.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, let sharpen this conjecture up. But first let me scold myself
for thinking that I could
have found this in 3Dimensional space with the pseudosphere versus the
sphere. That feat
may still be viable to accomplish but orders of magnitude far more
difficult. The lesson to
be learned, by me anyway, was that in difficult circumstances, never
stick to 3D, but immediately go to 2D, and later, jump back to 3D.
I am confident of this speculation or conjecture that the natural
boundary between finite
number or finite anything versus infinite number or infinite anything
is the boundary of
where log-spiral versus squares no longer are able to cooperate to
generate the log-spiral
any further than a given large square. I speculate where this
breakdown occurs is around
253! neighborhood. It may go to 270! or it may go to 22^(22*22).
By breakdown I mean several things. That at a large square, the log-
spiral misses it completely, or that the log spiral fails to intersect
with the large square.
Now by log-spiral, I mean all family classes and types of log-spirals.
Where the golden
ratio log-spiral is but one fancy example of a log spiral. So by log-
spiral, I mean the entire
gamut of all types of equiangular spirals. And at this conjectured
large square, say 253!, that
all the types of log-spirals fail to continue. That all the log
spirals were alright, and doing fine
and dandy and fanciful up to 253! square but all broke down at 253!.
Now the log-spirals can intersect with the whirling squares of the
0!, 1!, 2!, 3!, 4!, 5!, 6!, 7!, . . .
at either one point of intersection or at two points and if at two
points, means the log-spiral has
entered inside the square. Now in the factorial-fibonacci we notice
that the whirling squares
construct a canyon-channel that the normal regular fibonacci sequence
does not construct.
Now this feature of a canyon channel is unique to multiplicative
fibonacci sequences versus
the normal regular fibonacci sequence of 0, 1, 1, 2, 3, 5, 8, . . .
Now how much of a role the canyon-channel plays in this conjecture, I
am unsure of as yet.
Perhaps a vital role.
The important issue is to prove whether the factorial-fibonacci allows
at least one log-spiral to
reach in construction up to 253! or that neighborhood (whether it is
270! or thereabouts).
Anyway the Conjecture or Speculation is that in the factorial-
fibonacci, that we can construct at least one log spiral of all the
possible log-spirals to wind and revolve in 253/4 or 270/4, but
at those huge numbers, there no longer exists any log-spiral that can
continue beyond that
large number. That is the conjecture.
And the implications are huge. It means that mathematics, pure math,
has a built-in boundary
of what is finite versus what is infinite.
If my conjecture is proven true, meaning that there is no log spiral
that can be built after a specific Euclidean large square, means that
a line in Hyperbolic Geometry is the end boundary
of finite in Hyperbolic geometry and correspondingly, Euclidean
Geometry is a bounded finite from infinite at that same large number.
So that in geometry or algebra or numbers, whenever we say "goes to
infinity" simply means
it goes higher than 253! or whereever this special large number is
located. That our concept of
infinity means simply going beyond this special large number.

Now I ruled out the golden-ratio-log-spiral, but since it is allowed
to intersect
at two points in each revolving square of the factorial-fibonacci, it
maybe the
key log spiral in proving the conjecture.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Now here is a funny thought, and it only goes to show that the human
mind is not
really equipped to think in terms of these very large numbers. The
idea crossed
my mind that the breakdown does not occurr but rather a specialness
occurrs in that
the golden ratio log-spiral intersects at 253! with the same number of
intersections in
that huge square as the number of intersections equal to 253!.

So the boundary of finite to infinite is not a breakdown, but rather
where the number is
special.

P.S. this also has to relate to the starting squares of 0!, 1!, 2! for
that predetermines
the windings of the log-spiral. If we can start the windings at 5! we
would have a different
outcome.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Now this gets back to what I called the family-class of log-spirals.
And this is thought
of as the angle of winding of the log spiral. So that in a sense, we
have log-spirals
that can vary from 0 degrees to 90 degrees. The log spiral cannot be
equal to
0 nor 90 degrees but has to be a value between 0 and 90. So that the
log spiral of
near 0 degrees is going to make a whole lot of revolutions or windings
whereas the
log spiral at near 90 degrees is going to revolve only a few times in
a large distance.

Now the conjecture is about this factorial-fibonacci, so that the
conjecture is pinned
down to that condition. And the conjecture is that something funny
occurs at a large
number like 253! or in that neighborhood. And that funny or strange
thing is what I
conjecture no log-spiral of the 90 degree limit can intersect at 253!
square. Alternatively
if we check the 0 degree limit log spiral that it intersects the 253!
square in precisely 253!
intersections. But these are such large numbers that our perception
and imagination plays
tricks on us.



Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Now this gets back to what I called the family-class of log-spirals.
And this is thought
of as the angle of winding of the log spiral. So that in a sense, we
have log-spirals
that can vary from 0 degrees to 90 degrees. The log spiral cannot be
equal to
0 nor 90 degrees but has to be a value between 0 and 90. So that the
log spiral of
near 0 degrees is going to make a whole lot of revolutions or windings
whereas the
log spiral at near 90 degrees is going to revolve only a few times in
a large distance.
Now the conjecture is about this factorial-fibonacci, so that the
conjecture is pinned
down to that condition. And the conjecture is that something funny
occurs at a large
number like 253! or in that neighborhood. And that funny or strange
thing is what I
conjecture no log-spiral of the 90 degree limit can intersect at 253!
square. Alternatively
if we check the 0 degree limit log spiral that it intersects the 253!
square in precisely 253!
intersections. But these are such large numbers that our perception
and imagination plays
tricks on us.

Alright, I made a mistake above in having the family class of log-
spirals as varying from
0 degrees to 90 degrees turned around.

Reading this website clarifies the family-class of log-spirals:

http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html

What I was calling close to 0 degrees as alot of revolutions is what
is truly
close to 90 degrees according to that website where 90 degrees itself
is a circle.

So the log spiral that I am truly interested in for the Factorial-
Fibonacci is close
to 0 degrees so that the log-spiral is close to looking alike a
straight line, or a curve
that seems to curve ever so slowly.

So now, what I am picturing is this Factorial Fibonacci of whirling
squares:

0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . .
And we allow the log spiral to be of all the class-families, but
concerned mostly with
those approacing 0 degrees. And place only one demand on the log
spiral that it has
at least one intersection with every whirling square. It can have more
than one intersection
but must have at least one.

So the conjecture I am raising is the idea that the Factorial
Fibonacci is alright for the
class family of log-spirals except for a large number at around 253!
or that neighborhood
where the log spirals approaching 0 degrees fail to meet the
requirement of at least
one intersection with the 253! square.

And this conjecture, if true, would mean that math has fundamentally a
natural boundary
where infinity starts and where finiteness ended.

This is tough to picture, but I think what happens is that at that
large number or in that
neighborhood, the curving of the near 0 degree log spiral does not
curve enough to capture
the huge square at around 253! (possibly 270!).

Of course the log spirals near 90 degrees are going to hit every one
of those factorial squares
since they curve so much, but even those, if we go to say 270!^(270!)
that the spacing between the revolutions may become so huge of
spacings that like a sieve it misses a factorial square completely.

These things are hard to picture, and that is why I ask for L. Walker
to have a website for
reference that shows pictures of the factorial-fibonacci of whirling
squares and a starter
log spiral near 0 degress winding through the squares. Or possibly the
Indian friend, Narashima
(excuse the spelling) who a while ago aided the conversation on
tractrix with his posting of
a website of pictures.

So words are awfully bad, alone, but with the aid of pictures would
increase this a 1000fold
in understanding.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Archimedes Plutonium wrote:

(snipped)

Post by Archimedes Plutonium
Alright, I made a mistake above in having the family class of log-
spirals as varying from
0 degrees to 90 degrees turned around.

http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html

--- quoting from that website ---
A special case of equiangular spiral is the circle, where the constant
angle is 90 Degrees.


Equiangular spiral with 40, 50, 60, 70, 80 and 85 degrees. (left to
right, top to bottom)

--- end quoting ---
 

Post by Archimedes Plutonium
What I was calling close to 0 degrees as alot of revolutions is what
is truly
close to 90 degrees according to that website where 90 degrees itself
is a circle.
So the log spiral that I am truly interested in for the Factorial-
Fibonacci is close
to 0 degrees so that the log-spiral is close to looking alike a
straight line, or a curve
that seems to curve ever so slowly.
So now, what I am picturing is this Factorial Fibonacci of whirling
0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . .
And we allow the log spiral to be of all the class-families, but
concerned mostly with
those approacing 0 degrees. And place only one demand on the log
spiral that it has
at least one intersection with every whirling square. It can have more
than one intersection
but must have at least one.
So the conjecture I am raising is the idea that the Factorial
Fibonacci is alright for the
class family of log-spirals except for a large number at around 253!
or that neighborhood
where the log spirals approaching 0 degrees fail to meet the
requirement of at least
one intersection with the 253! square.
And this conjecture, if true, would mean that math has fundamentally a
natural boundary
where infinity starts and where finiteness ended.
This is tough to picture, but I think what happens is that at that
large number or in that
neighborhood, the curving of the near 0 degree log spiral does not
curve enough to capture
the huge square at around 253! (possibly 270!).
Of course the log spirals near 90 degrees are going to hit every one
of those factorial squares
since they curve so much, but even those, if we go to say 270!^(270!)
that the spacing between the revolutions may become so huge of
spacings that like a sieve it misses a factorial square completely.
These things are hard to picture, and that is why I ask for L. Walker
to have a website for
reference that shows pictures of the factorial-fibonacci of whirling
squares and a starter
log spiral near 0 degress winding through the squares. Or possibly the
Indian friend, Narashima
(excuse the spelling) who a while ago aided the conversation on
tractrix with his posting of
a website of pictures.
So words are awfully bad, alone, but with the aid of pictures would
increase this a 1000fold
in understanding.

Maybe I need a picture of a whirling squares of that Factorial
Fibonacci or maybe I do not.
I was thinking that the canyon channel defines what angles of the log
spiral can
be used for the canyon channel is itself a closed curve and would put
a limit range on the
log spiral angle.

And the arrangement of the first few numbers 0!, 1!, 2! would also
restrict the type of angle
of the log-spiral.

So that the first few terms and the canyon channel in the whirling
squares limits the range
of log spiral angles, eliminating those near 90 degrees and those less
than 40 degrees.


But now I wonder if I am making this exercise more difficult than what
it really is. Suppose
instead of a arrangement of whirling squares of the factorial
fibonacci, instead we just simply
lined them up in a straight line, and against a straightedge wall. And
we start with either the
near 90 degree log spiral and ask at what square are the spacings
between winds too large to
intercept the factorial square? Or we start with the near 0 degree log
spiral and ask at what
square will this nearly straight line curve, curve so much with
distance that it fails to intersect
a square at a large distance?

And, I suppose this is rather neat in the fact that the near 0 degree
or the near 90 degree
is also dependent on a finite boundary mark so that whatever the large
number is that fails
for the log-spiral immediately determines the exact value of the "near
0 or near 90 value".

So in the proof, it would be far easier to determine where the
factorial number fails to have
an intercept point with the log spiral when arranged in a long row-
line rather than in a
whirling pattern.

And if a proof is easily come by, I guess the most difficult feature
of this mathematics is
the explaining of why those numbers are an end of finiteness and the
beginning of infinity?
I started to explain that in several other posts but I guess the best
way is to say that
the log spiral is a counting machine in Hyperbolic geometry and the
squares are counters
in Euclidean geometry, and because there is a breakdown of the log
spiral missing a square,
means the infinity of both Euclidean and Hyperbolic geometry stops
counting at that breakdown.

Now as time goes by, I should be able to improve on the explanation,
but I sense the most
difficult aspect of this entire exercise and experiment is to explain
to those who come running
up and saying "well, how does a log spiral and squares connect to a
infinity boundary?"

So, forget about the intuition, the conjecture, the geometry layout
experiment and the proof,
all of which were obtained in due course, but the most challenging
part of this will be the
novices who ask how are they connected to infinity?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Archimedes Plutonium wrote:

http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html

--- quoting from that website ---
A special case of equiangular spiral is the circle, where the constant
angle is 90 Degrees.


Equiangular spiral with 40, 50, 60, 70, 80 and 85 degrees. (left to
right, top to bottom)

--- end quoting ---

This is turning out to be really really pretty. The Factorial-
Fibonacci Sequence:

0!, 1!, 2!, 3!, 4!, 5!, 6!, 7!, . . .

is about "all possible arrangements of n things" So already we have a
upper limit
ingrained into the Experiment.

The use of log-spiral is an upper limit for Hyperbolic Geometry. The
use of
factorial is an upper limit for Euclidean geometry.

The fact that the log spiral has limits of approaching 0 degrees and
90 degrees
for the family-class of log spirals are limits and these limits are
defined by
what that number is in the neighborhood of 253!. So if the number is
itself 253!,
then the limit of log-spirals is 1/253! near 0 and (90 - 1/253!) near
90 degrees.

But I had a dream last night of the set-up of the experiment to
streamline it.

I spoke of two set-ups of the canyon-channel stacking in a whirling
pattern and
the up-against a straightedge wall stacking of 0!, 1!, 2!, . . .

In the dream, I saw that instead of allowing all types of log spirals
making all sorts
of intersections in the canyon-channel pattern, that there is probably
one unique
log-spiral that goes the furthest in the canyon-channel until it hits
around the
neighborhood of 253! squares where it is forced to intersect with a
square.
So instead of looking where all the log spirals happen to have one
that misses a
square at around 253!, here we do the reverse and construct the unique
log spiral
that stays within the canyon channel and happens to miss every
factorial square
up until about 253! where it cannot miss.

Now as for the lined-up-row pattern, my dreaming offered no
streamlining here. It
is probably already streamlined. Here the idea is to line them up in a
row and to
send the two most wildest, very wild log spirals from each extreme of
the closest
to 0 degrees and the closest to 90 degrees. The near 0 degree log
spiral is going
to look like a needle straight line with a slight curve in it. The
near 90 degree log spiral
is going to look like a winding up of a near circle. So in this phase
of the experiment,
the near 0 is going to puncture hole each square, enter it and exit it
except when it
gets to a distance of about 253! in that neighborhood the curve of the
near 0 degree log
spiral is enough of a curve that it misses the square and from thence
onwards misses all
the squares thereafter. And for the near 90 degree log spiral, it
intercepts every square
until it reaches that neighborhood of 253! where the windings of the
circle like log-spiral
are so huge of winding spaces that the huge square of 253! is
completely inside the spacing of the log-spiral and no intercept or
puncture intersect.

As I noted in the previous post, that the most difficult
characteristic of this entire Experiment
and proof, is not the proof itself, but the understanding that what
was accomplished here
is the locating of the natural boundary of pure math that it has a
boundary between finite versus infinite numbers. And that this number,
whatever it turns out to be, whether 253! or
270! or 22^(22*22), whatever the unique large number turns out to be
that log spiral square,
the most difficult item in this entire experiment is to be able to
explain to the complete novice
of a nonmathematician and a professional mathematician, that what was
discovered here is
the boundary of where infinity begins. I say this, because, why, what
the heck, mathematics
as a science could not do a proper valid correct Euclid Infinitude of
Primes proof, indirect method until 1991, and the community of
mathematics at large still does not accept the fact
that no-one except four people now can do a valid indirect of Euclid.
So one should never underestimate nor overestimate the state of
condition of a science.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

I am not sure at the moment. I kind of think I need triangulation of
the boundary between
finite and infinite. I need both stacking methods of the whirling
squares and of the single
row straightedge stacking of progressively larger squares

All done on the Factorial-fibonacci sequence: 0!, 1!, 2!, 3!, 4!,
5!, . . .

I need the canyon channel unique log spiral that exhausts itself at
the upper bound.

I need the near 0 degree log spiral that misses to intersect with a
square.

I need the near 90 degree log spiral that misses to intersect with a
square.

So it maybe a sort of triangulation setup. If I used only two of those
setups,
I maybe able to use a different sequence, say 3x the factorial
fibonacci, but that the canyon channel setup would reduce it back to
1/3.

So that triangulation is probably a necessary feature so as to realize
a "unique boundary",
not a boundary to satisfy whatever sequence is chosen; and thus
independent of sequence
chosen.

But correct me if wrong, I believe the factorial sequence of itself
makes the boundary unique
because only this sequence has **unique meaning of all possible
arrangement of N things**.

So that the pick of the factorial sequence makes it sequence
independent. No other sequence
holds so much meaning in mathematics as the "N! is all possible
arrangement of N things".

So what the triangulation helps to solve, where the canyon channel is
the third triangulatory.
What it solves is the fact that the other two setups of the near 0
degree and near 90 degree are just vague about "how near". So the
unique canyon channel log spiral cuts away that
vagueness. There is only one unique log spiral that can survive the
furthest distance by not
intersecting any of the whirling squares. And I speculate it is
survivable to the neighborhood
of 253! to 270!, possibly even as far as 10^650, where this uniqe
spiral can no longer avoid
from intersecting the square.

So in effect, this is a triangulation since the near 0 and near 90
degree log spirals are vague, and where the canyon channel log spiral
fixes that vagueness and pinpoints the square of large number where
all three setups converge.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

My goodness, this is turning into a 1,000 page book, but the most
famous math book
since Euclid's. Do not worry, it is already published for it is
published first on the Internet,
which all important future books will be published first on the
Internet, which saves the
author time expediency, saves the author of date time group
validation, and saves the author the less likelihood of thievery.

Now in my recent posts I was worried more about the explanation of why
this 2D Factorial
Fibonacci and the associated log-spirals will come to a breakdown, as
to how to explain
that this is the boundary of finite versus infinite numbers or any
other finite to infinity definition.

Well, I am happy to report that not only is the proof easier than I
expected, for I expected to
toil on this for months and even into 2011, but here I am close to a
proof already with the
general outline already in full view. But the explanation is also far
easier. It is just a step of
a dimension away.

I said, previously that my big mistake in this adventure was trying to
prove it in 3D of the
pseudosphere and sphere and cube. Well, the proof is easy in 2D. But
the Explanation
that, this is truly the boundary and a natural boundary in pure math
comes from 3D.

And the explanation occurrs all in this paragraph as written by
Wikipedia with reference to
Mathworld:

--- quoting Wikipedia with reference to MathWorld ---

http://en.wikipedia.org/wiki/Pseudosphere
Both its surface area and volume are finite, despite the infinite
extent of the shape along the axis of rotation. For a given edge
radius R, the area is 4πR2 just as it is for the sphere, while the
volume is 2πR3/3 and therefore half that of a sphere of that radius.
[2]

2. ^ Weisstein, Eric W., "Pseudosphere" from MathWorld.


--- end quoting Wikipedia with reference to MathWorld ---

You see the explanation? There are many people in and inside of
mathematics who actually
believe those words above that a object in math can have infinite
stretch and still be finite in
area or finite in volume. These are very silly people, to be sure, but
they happen to be
almost everyone in mathematics except of course Archimedes Plutonium
and a few others
with a dash of better commonsense than most.

The explanation of why and how I am sure that Mathematics has a
natural boundary in the
neighborhood of 253! to about 270!, possibly as high as 301!, the
reason I am so sure of this
is because, the only way to resolve the fact that the pseudosphere is
a finite surface area and
a finite volume is that the infinity of the pseudosphere is a cutaway
or cutoff at this number
270! (if that is the 2D proof boundary).

So the explanation is easier than the proof because the explanation is
the exercise of
instilling that boundary number into the equation of the surface area
of the pseudospere and the volume of the pseudosphere.

So the true surface area of pseudosphere if the boundary is 270! would
be
Pseudosphere Area = 4(pi)R^2 - (270! factor)

And the true volume of the Pseudosphere is
Pseudosphere Volume = (2/3)(pi)R^3 - (270! factor)

There is nothing of the Pseudosphere exactly equal to a sphere except
for individual
aspects such as a radius or a distance, but as for complex features as
area or
volume, you have to be really silly to think those complex features
are the same
for sphere and pseudosphere, for that would be like saying the same
projective triangle
of pseudosphere on the surface area is equal to the triangle on the
sphere, which is
patently absurd.

So if I had started this adventure all over today, and if I had been
smarter, I would have
realized that always drop to the lowest dimension possible for the
proof and only later
rise to the higher dimension. But also, I would have in this "smarter
condition" realized
that starting such an adventure would be to look for the visible
example in geometry of
where there is a "stretch to infinity" yet the poor souls thinking
that the area or volume
would still be silly pegged as finite. There maybe other places in
math where they claim
a infinite stretch, yet simultaneously think they have a finite area
or finite volume.

P.S. now the boundary of finite to infinity does not affect the truths
of Calculus as a tool
but the Calculus needs refurbishing in large parts such as its silly
notion of a continuum.
For Physics has no continuum but is discrete or quantized and thus,
mathematics
cannot have a continuum and must be discrete and quantized. There is
no more geometry
or numbers between 0 and 1/270! unless you add the phrase "you are in
infinity territory."

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright the stacking of the straightedge against the wall of these
squares


1
1
2
6
24
120
720
5040
40320
362880
3628800
39916800

And the near 0 degree and near 90 degree log spirals will traverse the
full distance
through each of the squares.

But now, here is a problem of uniformity of experiment, so we want
uniformity of the
two stackings. The key spiral is the canyon-channel log-spiral since
it is unique and
has no messy "nearness" to define. The canyon-channel spiral is
exacting and will
define what "near 0 and near 90 degrees means".

But in the whirling square stacking is a problem of uniformity that
must be addressed
so as to keep all stackings uniform. In the golden-ratio log-spiral
all the squares are
touching with no gaps anywhere and in that stacking is an inset of one
square with the
next square. It is this inset that must be eliminated to have
uniformity.

So to accomplish that uniformity of stacking between the lined-up-in-a-
row stacking
we stack the whirling squares so that the corner point is the only
point contiguous with
successive squares. So two successive squares in the whirling pattern
have a point
as intersection and not a line-segment as intersection. That gives
uniformity to both
stacking patterns.

Now the canyon channel log-spiral is the most important one to fetch
because it
determines what "near 0 and near 90 degrees" means. Once we fetch the
canyon
channel log-spiral and find it is nonintersecting out to 270! then
this is the boundary
of finite to infinity, and then we look to see if 1/270! degree and 90
- (1/270!) degrees
also fail to intersect at 270! square in the row pattern.

In the whirling pattern the unique log spiral fails to intersect any
square until 270! where
it is forced to intersect and where no other log spiral goes out that
far with the property
of nonintersection. With the near 0 and 90 degree row pattern log-
spirals, we want them
to intersect every square along the way, however at 270! (if this is
the number) neither
of the near 0 or near 90 are able to intersect. So in one pattern we
want nonintersection
and in the other pattern we want intersection. And my conjecture is
that a natural pure
math boundary occurrs around 253! to 270! and may go as high as
22^(22x22).

Now if we break these into four squares, bypassing the first 1, so we
have


1
1
2
6
24

then we have the next four
120
720
5,040
40,320

then we have the next four
362,880
3,628,800
39 916 800
12! = where my calculator gave out

So now I am wondering if there is a nice nifty trick to play to try to
fetch out this
key important canyon channel log-spiral that is going to travel the
furthest of all
log spirals in that winding of squares and where it fails to intersect
with any square until
it reaches that neighborhood of 253! and beyond.

So is there some nice trick, such as using the corner points of the
whirling squares
to fetch out this unique log-spiral?

P.S. in drawing the golden ratio log spiral to the normal fibonacci
sequence it is usually done
with a big blackened in first few squares since they are rather off
the mark of phi number,
and only after about the 4th or 5th square is the true log spiral come
into form. So let us
not get flustered with the first few squares of 1,1,2 and can blacken
them in (maybe even the
6) so that this unique log spiral takes shape with starting at 24
square.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright the stacking of the straightedge against the wall of these
squares
1
1
2
6
24
120
720
5040
40320
362880
3628800
39916800
And the near 0 degree and near 90 degree log spirals will traverse the
full distance
through each of the squares.
But now, here is a problem of uniformity of experiment, so we want
uniformity of the
two stackings. The key spiral is the canyon-channel log-spiral since
it is unique and
has no messy "nearness" to define. The canyon-channel spiral is
exacting and will
define what "near 0 and near 90 degrees means".
But in the whirling square stacking is a problem of uniformity that
must be addressed
so as to keep all stackings uniform. In the golden-ratio log-spiral
all the squares are
touching with no gaps anywhere and in that stacking is an inset of one
square with the
next square. It is this inset that must be eliminated to have
uniformity.
So to accomplish that uniformity of stacking between the lined-up-in-a-
row stacking
we stack the whirling squares so that the corner point is the only
point contiguous with
successive squares. So two successive squares in the whirling pattern
have a point
as intersection and not a line-segment as intersection. That gives
uniformity to both
stacking patterns.
Now the canyon channel log-spiral is the most important one to fetch
because it
determines what "near 0 and near 90 degrees" means. Once we fetch the
canyon
channel log-spiral and find it is nonintersecting out to 270! then
this is the boundary
of finite to infinity, and then we look to see if 1/270! degree and 90
- (1/270!) degrees
also fail to intersect at 270! square in the row pattern.
In the whirling pattern the unique log spiral fails to intersect any
square until 270! where
it is forced to intersect and where no other log spiral goes out that
far with the property
of nonintersection. With the near 0 and 90 degree row pattern log-
spirals, we want them
to intersect every square along the way, however at 270! (if this is
the number) neither
of the near 0 or near 90 are able to intersect. So in one pattern we
want nonintersection
and in the other pattern we want intersection. And my conjecture is
that a natural pure
math boundary occurrs around 253! to 270! and may go as high as
22^(22x22).
Now if we break these into four squares, bypassing the first 1, so we
have
1
1
2
6
24
then we have the next four
120
720
5,040
40,320
then we have the next four
362,880
3,628,800
39 916 800
12! = where my calculator gave out
So now I am wondering if there is a nice nifty trick to play to try to
fetch out this
key important canyon channel log-spiral that is going to travel the
furthest of all
log spirals in that winding of squares and where it fails to intersect
with any square until
it reaches that neighborhood of 253! and beyond.
So is there some nice trick, such as using the corner points of the
whirling squares
to fetch out this unique log-spiral?
P.S. in drawing the golden ratio log spiral to the normal fibonacci
sequence it is usually done
with a big blackened in first few squares since they are rather off
the mark of phi number,
and only after about the 4th or 5th square is the true log spiral come
into form. So let us
not get flustered with the first few squares of 1,1,2 and can blacken
them in (maybe even the
6) so that this unique log spiral takes shape with starting at 24
square.

Now also I have to mention a recalibration once the breakdown occurs.
What I mean is that in the line row stacking I have huge squares
progressively outward from a straight line starting
at 0 so the first square if 1, the second is 1, the third is 2, the
fourth is 6, the fifth is 24 and so on. So that the distance covered
by these first five squares is their summation which is a distance of
34. Now the log-spiral that is near 0 degrees and near 90 degrees will
have different distances of 34 metric in their windings. Now when
those two log-spirals get out to
253! the distance out there from 0 is less than the number 253!
itself, just as the distance to the seventh square 720 is less than
the number 720. So let us say or
suppose that the breakdown is at 266!, meaning that the 0 and 90
degree log spirals fail to
intersect any of the squares from 266! onwards.

By recalibration what I mean is that in the canyon-channel pattern, a
breakdown is where the
log-spiral actually has no choice but to intersect a square in the
neighborhood of 253! and the actual distance covered by the log-spiral
is close to the number 253! of metric distance. But
in the row pattern the metric distance to 266! square is recalibrated
to be 253!, whilst the log-spiral metric distance is close to 253!
itself.

So here we need a recalibration once the breakdown occurrs for the row
pattern.

Once the recalibration is applied, and suppose the breakdown occurs at
22^(22x22)
for the canyon channel log spiral, then it is also the case that the
breakdown for the
near 0 and near 90 degree log spirals in the row pattern match or
agree with 22^(22x22).
Maybe the row pattern can be altered so as no recalibration is
required, such as "planting
the backend corner of the square on the number of the number line
which corresponds to
the square side. So that for the square 5! = 120 the backend corner is
planted on the number
120 of the number line and that these squares then have holes or gaps
between them.

So maybe it is best we have gaps in between successive squares except
for the three starting
out squares of 1,1,2.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, I still do not have flexibility enough to even create a log-
spiral within that canyon channel. Whether there is a proof of that
impossibility given the setup as of yet.

So I need more flexibility for the canyon channel pattern. And since I
am lacking in
pictures, I am going to refer to this picture of four sided irregular
polygons forming
a log-spiral from Wikipedia:




--- quoting from Wikipedia ---
http://en.wikipedia.org/wiki/Logarithmic_spiral

polygonal subunits
approximate logarithmic spiral
--- end quoting ---

Instead of irregular polygons of 4 sides, let me substitute squares
from the factorial sequence:



1
1
2
6
24
120
720
5040
40320
362880
3628800
39916800

And instead of those squares meeting and intersecting at a line
segment, let me make the
intersection the corner point and allow those squares at this
intersection to change or swivel
as a hinge at this intersection. So that three huge squares could
almost make a revolution, whereas in the old setup that was
inflexible, required 4 successive squares to make a revolution. Now,
given those flexibilities I believe I have the canyon channel pattern
setup
to find a unique log-spiral that circumnavigates through this canyon-
channel of the furthest
reaches but is eventually stopped or halted at about the neighborhood
of 253! to about 22^(22x22). And this square where this unique log
spiral that reaches the furthest, is the
boundary between finite and infinite.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, after a few sputterings I am beginning to nail down this
canyon channel log spiral
that is so key to proving the boundary of finite to infinite. There is
a picture of whirling
triangles on this website:

--- quoting ---
http://jwilson.coe.uga.edu/emt669/student.folders/frietag.mark/homepage/goldenratio/goldenratio.html

Out of these Whirling Triangles, we are able to draw a logarithmic
spiral that will converge at the intersection of the the two blue
lines in Figure 3.

--- end quoting ---

Pictures are worth more than a thousand words in my estimation.

Anyway, it is this sequence I am using to prove this boundary, the
factorial sequence:

1
 1
 2
 6
 24
 120
 720
 5040
 40320
 362880
 3628800
 39916800
.
.
.



Now I have the squares of 6, 24, and 120. And to achieve that canyon-
channel pattern
I place those three squares so that they form a canyon-channel that is
shaped like a triangle
and where the 6-square corner intersects and acts as a hinge to the 24-
square and where
the 120-square just misses by a tiny distance the opposite corner of
the 6-square so as to
leave a triangle shaped canyon channel in the whirling of 6, 24, 120
squares.

Now the 720-square is connected to the corner of the 120-square as a
hinge and swivels
so that it almost touches the opposite corner of the 24-square forming
a canyon channel
in the 24, 120, 720 squares.

So the pattern is emerging of whirling squares that forms triangular
shaped canyon-channels
and they are open enough to allow a log-spiral to pass through the
channels.

Now it is this conjecture or speculation on my part that the log
spiral of these whirling squares
is unique to the Factorial sequence and that at some large number in
the neighborhood of
253! to 22^(22x22) the canyon channel log spiral breaksdown in that it
intersects with the
square in this neighborhood. Prior to this square, the log spiral
missed intersecting any of
the squares but at this square it is impossible to miss the square. It
is a unique log-spiral in that it went the furthest distance without
intersection of any of the squares.
And it matters not how wide the channel gap is for there is only one
unique log spiral that
goes the furthest distance without intersection. It is truly a great
and fantastic uniqueness
feature in all of mathematics, and it should be, for it defines where
finiteness ends and where
infinity begins.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, after a few sputterings I am beginning to nail down this
canyon channel log spiral
that is so key to proving the boundary of finite to infinite. There is
a picture of whirling
triangles on this website:

--- quoting ---
http://jwilson.coe.uga.edu/emt669/student.folders/frietag.mark/homepage/goldenratio/goldenratio.html

Out of these Whirling Triangles, we are able to draw a logarithmic
spiral that will converge at the intersection of the the two blue
lines in Figure 3.

--- end quoting ---

Pictures are worth more than a thousand words in my estimation.

Anyway, it is this sequence I am using to prove this boundary, the
factorial sequence:

1
 1
 2
 6
 24
 120
 720
 5040
 40320
 362880
 3628800
 39916800
.
.
.



Now I have the squares of 6, 24, and 120. And to achieve that canyon-
channel pattern
I place those three squares so that they form a canyon-channel that is
shaped like a triangle
and where the 6-square corner intersects and acts as a hinge to the 24-
square and where
the 120-square just misses by a tiny distance the opposite corner of
the 6-square so as to
leave a triangle shaped canyon channel in the whirling of 6, 24, 120
squares.

Now the 720-square is connected to the corner of the 120-square as a
hinge and swivels
so that it almost touches the opposite corner of the 24-square forming
a canyon channel
in the 24, 120, 720 squares.

So the pattern is emerging of whirling squares that forms triangular
shaped canyon-channels
and they are open enough to allow a log-spiral to pass through the
channels.

Now it is this conjecture or speculation on my part that the log
spiral of these whirling squares
is unique to the Factorial sequence and that at some large number in
the neighborhood of
253! to 22^(22x22) the canyon channel log spiral breaksdown in that it
intersects with the
square in this neighborhood. Prior to this square, the log spiral
missed intersecting any of
the squares but at this square it is impossible to miss the square. It
is a unique log-spiral in that it went the furthest distance without
intersection of any of the squares.
And it matters not how wide the channel gap is for there is only one
unique log spiral that
goes the furthest distance without intersection. It is truly a great
and fantastic uniqueness
feature in all of mathematics, and it should be, for it defines where
finiteness ends and where
infinity begins.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Archimedes Plutonium wrote:
(snipped)

Post by Archimedes Plutonium
1
 1
 2
 6
 24
 120
 720
 5040
 40320
 362880
 3628800
 39916800
.
.
.

Alright, I have cutouts of paper for the squares up to 720. Which is
enough to see the
pattern. Where each square starting with 6 has two opposite corners,
one a hinge corner
and the opposite diagonal corner serves as the opening of the channel.
So the figure
inside the canyon channel is not a true triangle but shaped like one,
for it actually has
one side the half of the smallest square of the three squares. I do
not know if mathematics
gives a proper name to this sort of a figure, and of course it is
connected at the hinge
vertex but the other side is unconnected since it is the channel. One
way to describe the
triangle as a unformed isosceles triangle with its smallest side that
of 1/2 of a square that
is pointed in the same direction as the vertex formed from the two
larger squares.

Now the log-spiral that spirals through this channel must have its
"bend" where the channel
opening is the smallest, otherwise it is going to intersect shortly
thereafter with that
largest square of the three.

To picture what I am saying is cut out a to scale 6 square, 24 square
and 120 square of
three different colored paper and arrange them as to form this
whirling squares with corners
as hinges and a channel.

So now we have the immediate question. Is there a log-spiral existing
that can navigate this
sequence?

0!
1!
2!
3!
4!
5!
.
.
.

Is there such a log spiral, and if there is, can it escape
intersecting the squares thus
arranged into whirling squares forming these triangle shapes?

It is my conjecture that such a log spiral exists and is unique to
this pattern and at about the neighborhood of 253! this log-spiral
breaks down and is forced to intersect with the side of
the 253! square if indeed that is the square.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
(snipped)

Post by Archimedes Plutonium
1
 1
 2
 6
 24
 120
 720
 5040
 40320
 362880
 3628800
 39916800
.
.
.

Alright, I have cutouts of paper for the squares up to 720. Which is
enough to see the
pattern. Where each square starting with 6 has two opposite corners,
one a hinge corner
and the opposite diagonal corner serves as the opening of the channel.
So the figure
inside the canyon channel is not a true triangle but shaped like one,
for it actually has
one side the half of the smallest square of the three squares. I do
not know if mathematics
gives a proper name to this sort of a figure, and of course it is
connected at the hinge
vertex but the other side is unconnected since it is the channel. One
way to describe the
triangle as a unformed isosceles triangle with its smallest side that
of 1/2 of a square that
is pointed in the same direction as the vertex formed from the two
larger squares.
Now the log-spiral that spirals through this channel must have its
"bend" where the channel
opening is the smallest, otherwise it is going to intersect shortly
thereafter with that
largest square of the three.
To picture what I am saying is cut out a to scale 6 square, 24 square
and 120 square of
three different colored paper and arrange them as to form this
whirling squares with corners
as hinges and a channel.
So now we have the immediate question. Is there a log-spiral existing
that can navigate this
sequence?
0!
1!
2!
3!
4!
5!
.
.
.
Is there such a log spiral, and if there is, can it escape
intersecting the squares thus
arranged into whirling squares forming these triangle shapes?
It is my conjecture that such a log spiral exists and is unique to
this pattern and at about the neighborhood of 253! this log-spiral
breaks down and is forced to intersect with the side of
the 253! square if indeed that is the square.

Alright, let me invert the problem to make it analytically easier.
Instead of the
unique log-spiral that runs through the canyon channel, missing all
the squares
of factorial, missing to intersect any of them until in the 253!
neighborhood, let
me invert that to saying the unique log-spiral that intercepts or
intersects at
one point of each square, starting with the 6, 24, and 120 squares,
and intersects
at one point throughout all the squares until some breakdown square in
that
253! neighborhood where it fails to intersect with a particular large
square.

What this inversion does is pinpoint along the track of the unique log-
spiral with the
use of analysis where that point is. So for the 6, 24 and 120 squares,
focusing on the
6 square where one corner is the hinge connected to the 24 square and
the opposite diagonal corner of the 6 square is the swivel opening (a
gate) to have the channel flow alongside the 120 square.

Now we can use the diagonal of the 6 square to define a angle of
swivel of the 6 square relative to the 120 square. We have not to
worry about the 24 square, since the 6 square
is hinged on the corner with the 24 square.

What we have to worry about is where the unique log spiral intersects
the 120 square
near the swivelling 6 square so that we can place the corner of the 6
square near that
point of intersection, or pointed at the direction of the intersect.
We have to remember also, that the 120 square swivels at the hinge of
the 24 square.

So my conjecture looks different in this inverted set up where we have
a unique log
spiral that goes out fairly far but in the neighborhood of 253! fails
to be able to have
an intersection with that large square, and no matter what sort of re-
adjustments are
made down the line of the hinge or swivel corner (gate corner). So
this log-spiral is
unique, for it goes the furthest out and obeys the demands the
furthest out.

Now that is a proof that a log spiral exists in that all the triangles
(those morphed
triangles that have 4 sides where the smallest side is 1/2 of a square
perimeter.
The proof that such a log-spiral exists is that every one of those
triangles formed
has the same shape as all the previous ones down the line. So the log
spiral exists,
but the problem is whether there is a breakdown or not a breakdown.
Old math
assumed there was no breakdown. I am speculating that the morphed
triangles
are the same all the way up to a breakdown in the 253! region.

New mathematics speculates that a breakdown must occur because the
squares of
factorial sequence of all possible arrangements of N things cannot
accomodate the
log-spiral family classes without a breakdown. And this breakdown is
the boundary between
finite and infinite.

So that is a key feature of my speculation of this unique log-spiral
with a breakdown in
the 253! region. That the Factorial sequence rises too fast to have
the family-class of
all log-spirals obey it with unlimited stretch, the infinity of old-
math. That the Factorial
of math is special because it is All Possible Arrangements of N things
by N!. That factorial
is thus a special sequence, because of its probability feature of
arranging things. And so the
log-spiral, the spiral that stays the same form, the same shape, no
matter how far it stretches, must have a
_breakdown_ when pitted against All Possible Arrangements. For it to
not have a breakdown
is like saying that a curve fits into a straight-line.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
So that is a key feature of my speculation of this unique log-spiral
with a breakdown in
the 253! region. That the Factorial sequence rises too fast to have
the family-class of
all log-spirals obey it with unlimited stretch, the infinity of old-
math. That the Factorial
of math is special because it is All Possible Arrangements of N things
by N!. That factorial
is thus a special sequence, because of its probability feature of
arranging things. And so the
log-spiral, the spiral that stays the same form, the same shape, no
matter how far it stretches, must have a
_breakdown_ when pitted against All Possible Arrangements. For it to
not have a breakdown
is like saying that a curve fits into a straight-line.

A better explanation is that if the log-spiral has no breakdown in All
Possible
Arrangements, then hyperbolic geometry must fit with Euclidean
geometry and
the two would be the same. A breakdown means they are distinct and
never the same.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, I think I can draw what the canyon-channel triangles look
like. They are composed of
two sides of two squares and the smallest side is 1/2 the perimeter of
the smallest square.
So for the 6,24,120 squares of the Factorial sequence 0!, 1!, 2!, 3!,
4!, etc etc
the 6, 24, 120 squares morphed-triangle looks like this, like an
arrowhead:

ascii art of the morphed triangle:
______
\
\
/
/_____

Now I only drew the side which is the 1/2 perimeter of the smallest
square such as 6
in the 6, 24, 120 trio of squares and I left the other two sides
unfinished as ascii
is hard to draw

But the message of this post is mostly of how and why the canyon
channel pattern
figures out where the log-spiral halts in the factorial sequence and
why it halts in
the neighborhood of about 253!

There are two leeways in each of the morphed-triangles. There is the
hinge leeway
where we can alter the position of the smallest square 6 at the hinge
of 6 and 24.
And we can alter the position at the hinge of the 24 and 120 squares.
Of course this
alters the gap or canyon channel that the 6 square makes with the 120
square.

Now how this all works is that every trio of squares that forms a
morphed triangle
will have a log spiral that intersects near where this smallest square
is pivoting.

So in the first one hundred factorial squares from 0! to 100! those
triangles have
enough leeway of pivoting of their squares to allow for the existence
of roughly
1 billion log spirals. But from 0! to 200! the existence of log-
spirals to satisfy winding
through the canyon channel of those 200 squares has narrowed down the
existence
of log spirals to about 1 million. Now from 200! to 220! the leeway of
the smallest
square in every trio is cut even more so that the existence of log
spirals out to
0! to 220! has narrowed the existence field to there being only
hundred thousand
log spirals up for the job. Out to 240! the field is narrowed to ten
thousand. Out
to 245! the field is narrowed to only a thousand surviving log-spirals
that fit the
description of the job. Then at about 253!, only one log spiral
remains that fits
the starting description because the leeway of the smallest square in
every trio
is at its limit of allowing a log spiral to exist. Then finally, even
this unique log-spiral
breaks down and intersects with this large square either in more than
one point
or does not intersect at all with this large square.

Try as we may in adjusting all the lower trios of squares of their
leeway, that no adjustment
can make the unique log spiral go further on under the initial
starting conditions.

I speculate the limit is a large number somewhere between 253! and
22^(22x22) = approx
5 x 10^649

This of course would be the natural and pure boundary in mathematics
between what we call
finite numbers and infinite numbers. And no longer will we have the
hypocrisy of thinking that the pseudosphere of 3D has infinite stretch
yet equal surface area as the sphere of same radius. In the New-Math
the surface area of pseudosphere has a factor of this boundary for
its area and for its volume.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, based upon the 6, 24, 120 squares I measured the pitch of the
log-spiral
to be in the vicinity of 60 degrees. Keeping in mind that the two
hinges allow for
alot of leeway.

And it is this leeway that finally gives out in the neighborhood of
253! where the
log spiral fails and breaks down.

Now I think this problem would be far easier for a computer to solve
where the breakdown
actually occurs.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, based upon the 6, 24, 120 squares I measured the pitch of the
log-spiral
to be in the vicinity of 60 degrees. Keeping in mind that the two
hinges allow for
alot of leeway.
And it is this leeway that finally gives out in the neighborhood of
253! where the
log spiral fails and breaks down.
Now I think this problem would be far easier for a computer to solve
where the breakdown
actually occurs.

I may have found a shortcut of solving where the breakdown occurs. I
noticed
the distance of the log-spiral per every trio of squares that forms a
morphed
triangle of the whirling triangles.

Given the Factorial sequence of squares:

1
  1
  2
  6
  24
  120
  720
  5040
  40320
  362880
  3628800
  39916800
.
.
.

With the first trio of 6, 24, 120 squares, I noticed the distance of
the arc of the log-spiral
is less than or equal to the distance of the semiperimeter of the
smallest square of 6, or
in this case a distance of 12. So that the arc distance of the log
spiral is less than 12 to
exit the morphed triangle.

In the case of the next trio of squares, the 24, 120, 720, the
semiperimeter distance of
the 24 square = 48 is greater than or equal to the distance covered by
the log spiral arc
in that morphed triangle exiting at the gate of the 24 square.

Same thing happens at the gate of the 120 square in the trio of 120,
720, 5040.

Somehow the arc distance is slightly, ever so slightly increasing and
at some large distance
that arc distance at the gate will be greater than the semiperimeter
of the smallest of the
trio squares.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

I always like it when math is accessible to young High School
students, or for people between
the ages of 10 to 18 to be able to follow along and do the work, not
with some fancy math
but with convincing, working with visual models math.

So I have the square cutouts of these 6, 24, 120, 720 squares from
this Factorial Sequence:



1
   1
   2
   6
   24
   120
   720
   5040
   40320
   362880
   3628800
   39916800
 .
 .
 .

And I got myself a nice fine pliable wire that is thin and easy to
work with.

I laid out the canyon-channel pattern of the morphed triangles with
square
corners as hinges or gates.

And I measured the arc distance with the wire of the trio squares 6,
24, 120
and found the arc distance of the log spiral of approx pitch of 60
degrees to be 46 and we know the semiperimeter of the 24 square is 48.
I measured the arc distance of the log-spiral
of 60 degree pitch to be 231 for the trio squares of 120, 720, 5040
and we know the
semiperimeter of 120 square is 240.

So we have these rates of change:

2/48 = 0.041

9/240 = 0.037

And according to my speculation, that arc length rate will get smaller
and smaller
as it approaches zero and then the breakdown occurs.
So it is my speculation that given the leeway in the hinges and the
gates that
eventually all the log spirals in the vicinity of 60 degree pitch will
fail to make
a point intersection with the gate of a square of large number. As we
see from
the first two gates that the intersection point is regressive and at a
large number
that intersection point will be impossible, and where no log-spiral
can continue
as per initial starting instructions. At this breakdown, either all
the log spirals
make no intersection with that large square or make more than one
point of
intersection.

Rough estimate of this breakdown square given the first two gates of
arc distance
I estimate it is somewhere closer to 300! than it is to 253!. And
closer to 19^(22x22)
rather than to 22^(22x22). Would still need the computer to make a
precise confirmation
of the breakdown. The calculations are rough because of the leeway of
the hinges and gates.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[David R Tribble]

Post by Archimedes Plutonium
I estimate it is somewhere closer to 300! than it is to 253!. And
closer to 19^(22x22) rather than to 22^(22x22).

253! = 5.1734609926400789218043308997295e+499
300! = 3.0605751221644063603537046129727e+614

19^(22x22) = 8.2554901045277384397095530071882e+618
22^(22x22) = 5.4022853245302743024619692001681e+649

All of these from the Calculator program on Windows.
Don't you have a program like that on your PC?

[Archimedes Plutonium]

Post by David R Tribble

Post by Archimedes Plutonium
I estimate it is somewhere closer to 300! than it is to 253!. And
closer to 19^(22x22) rather than to 22^(22x22).

253! = 5.1734609926400789218043308997295e+499
300! = 3.0605751221644063603537046129727e+614
19^(22x22) = 8.2554901045277384397095530071882e+618
22^(22x22) = 5.4022853245302743024619692001681e+649
All of these from the Calculator program on Windows.
Don't you have a program like that on your PC?

This is probably the world's most important computer quest. Certainly
for all of
mathematics history, the single most important use of the computers.
To tell us
exactly where finite numbers end and infinity begins in mathematics.

We need the computers because of all the variables in this setup. We
have about
600 variables. Both the hinges and the gates can move. But I guess if
one computer
program finds the limit to be 300! and another finds it at 301! and
another finds it
at 302!, well we can be sure it is in that vicinity.

Now why is this the most important task that the computers of the
world ever
were assigned with? Well, the boundary between finite and infinite is
like the first
time humanity ever knew about and discovered the Pythagorean theorem
and noted
the harmony between math and the physical world. Or like the first
time humanity realized
there is a special number known as "pi" and computed its first several
digits.

Well, this is the first time humanity realizes and knows there is a
boundary in mathematics
that marks the end of finite numbers and starts with infinite numbers.
A boundary that is
itself pure mathematics.

It changes all of mathematics for half of it was absurd falsehoods,
such as most of
Cantor's work and every subject of math is affected by this boundary.

Now I am wondering whether mathematics has only one example of
infinite stretch yet a
supposed finite surface area and finite volume? I am wondering if only
the pseudosphere
has been proclaimed with infinite stretch and yet proclaimed to have
finite surface area and
finite volume. Or did mathematics have other ludicrous propositions of
infinite stretch yet
finite something or other? I am only aware of the pseudosphere with
this silly notion. But
much of mathematics is not familar to me, and well it should not be.

I can leave it to computers to find the precise boundary.

What is most important is to have the reasoning and logic as to why a
breakdown of
all log-spirals occurs near 19^(22x22) and why that breakdown means
the start of
infinity?

And the answer is, that the pseudosphere ends at 19^(22x22). And in
the calculus equation
for the volume and surface area of pseudosphere, has in its boundary
conditions for the integral the number 19^484.

Let us ask a question of logic. The factorial for given N, means all
the possible arrangement
of N things. Does anyone honestly think there should be more log-
spirals to be arranged, and
more than all possible arrangements of log-spirals? Does it not make
more commonsense
that All Possible Arrangements is a larger set entity than all the log
spirals?

This is like asking Cantor before he was institutionalized with the
diagonal method, does anyone actually think a new number can be
retrieved from N! Once you have the entity
of All Possible Arrangements, does anyone really think some diagonal
is going to produce something new that the all possible arrangements
failed to harness? Cantor not only fooled himself but
fooled the mathematics community for over a century. If the world had
all possible numbers
as 00, 01, 10, 11, all possible arrangements of two digits of 0 and 1,
there is no way that Cantor diagonal
can work and fetch a new number out of all possible arrangements. The
Cantor diagonal fails
for the Reals, because the Reals are just like the 0 and 1
arrangements, and fails whenever all possible arrangements exist.

And the Cantor diagonal, as an analogy is the log-spiral, spiralling
through the factorial sequence. Only the Cantor diagonal is a straight
line. The log spiral breaks down, not the All Possible Arrangements.

All of math is affected by this boundary. Even the way mathematicians
do "proofs".
Because from now on, when a conjecture is offered and says "For every
Natural Number,"
what they mean is-- for all counting numbers from 0 to 301! When
Fermat penned his
last theorem, his theorem meant only from 0 to 301! and it is true for
those numbers.
When Goldbach wrote his conjecture, it was meant for all the numbers
from 0 to 301!.
When Riemann made his famous hypothesis, it was meant for all counting
numbers from
0 to 301! and because mathematics never defined finite from infinite
with precision that
none of those conjectures were ever proven nor will be proven as long
as finite is imprecisely
defined.

When Peano formalized the axioms of the Natural Numbers, he forgot
something immensely
important. He forgot to define what it means to be finite versus
infinite number. Now he
would have it by defining finite as less than 301! (if that is the
boundary).

So, is there a computer, up to speed, that can deliver where the log-
spirals breaks down
in the canyon channel pattern?

A shame that most computers around the world are doing piddly paddly
trifling stuff. Whereas
they could be doing the most important fact of mathematics since the
Pythagorean theorem
was first discovered.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
I always like it when math is accessible to young High School
students, or for people between
the ages of 10 to 18 to be able to follow along and do the work, not
with some fancy math
but with convincing, working with visual models math.
So I have the square cutouts of these 6, 24, 120, 720 squares from
1
   1
   2
   6
   24
   120
   720
   5040
   40320
   362880
   3628800
   39916800
 .
 .
 .
And I got myself a nice fine pliable wire that is thin and easy to
work with.
I laid out the canyon-channel pattern of the morphed triangles with
square
corners as hinges or gates.
And I measured the arc distance with the wire of the trio squares 6,
24, 120
and found the arc distance of the log spiral of approx pitch of 60
degrees to be 46 and we know the semiperimeter of the 24 square is 48.
I measured the arc distance of the log-spiral
of 60 degree pitch to be 231 for the trio squares of 120, 720, 5040
and we know the
semiperimeter of 120 square is 240.
2/48 = 0.041
9/240 = 0.037
And according to my speculation, that arc length rate will get smaller
and smaller
as it approaches zero and then the breakdown occurs.
So it is my speculation that given the leeway in the hinges and the
gates that
eventually all the log spirals in the vicinity of 60 degree pitch will
fail to make
a point intersection with the gate of a square of large number. As we
see from
the first two gates that the intersection point is regressive and at a
large number
that intersection point will be impossible, and where no log-spiral
can continue
as per initial starting instructions. At this breakdown, either all
the log spirals
make no intersection with that large square or make more than one
point of
intersection.

Now that I have the method squarely in mind (sorry for that pun), I
can
invert the above to asking where the log spirals never intersect the
squares through the canyon-channel, until they are forced to intersect
at some large number as 300!. So in the above, I was going after one
and
only one intersection point with the whirling morphed triangles. In
the
inversion I seek no log spiral intersection and when it does intersect
is the breakdown. So I think that model is the best model to use.

Also, let me add while on that topic, that there is a simple and easy
proof
that log-spirals exist for these canyon-channel pattern and that
reason
is that the morphed whirling triangles are similar and when you have a
similar
shape becoming increasingly larger via a sequence, you have log-
spirals existing
in that increasing shape pattern.

Post by Archimedes Plutonium
Rough estimate of this breakdown square given the first two gates of
arc distance
I estimate it is somewhere closer to 300! than it is to 253!. And
closer to 19^(22x22)
rather than to 22^(22x22). Would still need the computer to make a
precise confirmation
of the breakdown. The calculations are rough because of the leeway of
the hinges and gates.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

I have to be careful about the meaning of words like "shortcut" so
that in the future
when I pick up this post, I will have forgotten what the shortcut
stood for. The shortcut,
put bluntly is the insight that the semiperimeter of the smallest
square of a trio of squares forming a morphed triangle is larger until
the breakdown occurs at about 300!.

But here I can see why I favored the one version over the inverted
form. Since in the
one version of 1 and only 1 point of intersection as opposed by no
intersection we can
accurately keep tabs of the arc distance relative to semiperimeter.
So in the one version
we have an accurate semiperimeter tab, but in the other version we
have a better overall
understanding of the breakdown of the log spirals.

Post by Archimedes Plutonium
Alright, based upon the 6, 24, 120 squares I measured the pitch of the
log-spiral
to be in the vicinity of 60 degrees. Keeping in mind that the two
hinges allow for
alot of leeway.
And it is this leeway that finally gives out in the neighborhood of
253! where the
log spiral fails and breaks down.
Now I think this problem would be far easier for a computer to solve
where the breakdown
actually occurs.

I may have found a shortcut of solving where the breakdown occurs. I
noticed
the distance of the log-spiral per every trio of squares that forms a
morphed
triangle of the whirling triangles.


Given the Factorial sequence of squares:


1
   1
   2
   6
   24
   120
   720
   5040
   40320
   362880
   3628800
   39916800
 .
 .
 .


With the first trio of 6, 24, 120 squares, I noticed the distance of
the arc of the log-spiral
is less than or equal to the distance of the semiperimeter of the
smallest square of 6, or
in this case a distance of 12. So that the arc distance of the log
spiral is less than 12 to
exit the morphed triangle.


In the case of the next trio of squares, the 24, 120, 720, the
semiperimeter distance of
the 24 square = 48 is greater than or equal to the distance covered
by
the log spiral arc
in that morphed triangle exiting at the gate of the 24 square.


Same thing happens at the gate of the 120 square in the trio of 120,
720, 5040.


Somehow the arc distance is slightly, ever so slightly increasing and
at some large distance
that arc distance at the gate will be greater than the semiperimeter
of the smallest of the
trio squares.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
I have to be careful about the meaning of words like "shortcut" so
that in the future
when I pick up this post, I will have forgotten what the shortcut
stood for. The shortcut,
put bluntly is the insight that the semiperimeter of the smallest
square of a trio of squares forming a morphed triangle is larger until
the breakdown occurs at about 300!.

Can't even remember what I was going to say in the sentence, let alone
months
later when I pick up on this post.

I meant to say in that sentence:

semiperimeter of the smallest square of a trio of squares forming a
morphed triangle
has a larger length than the arc length of the log spiral passing
through the gate of the
smallest square in the canyon channel pattern. I have appended this to
the previous post
with a "sic" marker.

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
I have to be careful about the meaning of words like "shortcut" so
that in the future
when I pick up this post, I will have forgotten what the shortcut
stood for. The shortcut,
put bluntly is the insight that the semiperimeter of the smallest
square of a trio of squares forming a morphed triangle is larger until
the breakdown occurs at about 300!.

Can't even remember what I was going to say in the sentence, let alone
months
later when I pick up on this post.
semiperimeter of the smallest square of a trio of squares forming a
morphed triangle
has a larger length than the arc length of the log spiral passing
through the gate of the
smallest square in the canyon channel pattern. I have appended this to
the previous post
with a "sic" marker.

[Archimedes Plutonium]

Thought I would spend some time on a brief synopsis of where Physics
stands on a candidate
for where the boundary is between finite numbers and infinite numbers.
Of course, for pure
mathematics there seems to be only one candidate in play and that is
the log spiral of 2D, associated with the pseudosphere and sphere of
3D, so that where the log-spiral breaks down
is where the pseudosphere stretch ends and where the pseudosphere thus
has a finite
surface area and finite volume. I am really rather amazed of the
paltry items in mathematics
for infinite stretch yet with finite area and finite volume.

For Physics, which leads the way usually in both physics and
mathematics the largest numbers are the Coulomb Interactions and the
All Possible Arrangements of nuclides. In essence, where the Strong
Nuclear Force ends. So where finite numbers end and where
infinity begins in mathematics is a copy over from physics as to where
Physics ends because
there is no longer a Strong Nuclear Force in the world. That is akin
to mathematics if say, right
angles or 90 degree angles could no longer exist in mathematics and so
mathematics would end. Or another example is that in mathematics if
there was no more multiplicative inverse at
some large number, we would say mathematics ends there.

But let me supply this synopsis of the current progress of work in the
field of nucleosynthesis
of atomic elements:

Element 100 ranges with nuclides of from 252 to that of 257 with the
longest half lifes of
hours and days. So within this element lies my oft vaunted use of 253!
approx = 10^500.
And where we often say that the Strong Nuclear Force no longer exists,
but then again
let us continue because obviously we have produced higher atomic
elements and thus the
Strong Nuclear Force is still in existence.

Element 109: and this was the element sought for when I was in
college. The nuclide of
278 has a half life of 8 seconds. So here we reach 278! as a
meaningful number in both
physics and math.

Element 114: and this was the element sought for because it offered an
island of stability
and 289 nuclide has a half life of 2.6 seconds. So here reach 289!

Element 116: reaches out to 293 nuclides and has a half-life of 60 ms.

Element 118: and I am surprized we went past element 114 in such a
short time. Three
atoms of element 118 since 2002 have been manufactured
(nucleosynthesized) and it
has 294 nuclides with a half life of 0.89 ms. So I guess the furthest
meaningful number
in physics and math from the Elements is 294!

But wait a minute! The Atom Totality theory has some large meaningful
numbers.
Strictly speaking in "atom speak" the largest number would be 22 as
the total number
of subshells in the Plutonium Atom Totality (our current universe)
inside of 7 shells.
This delivers us the value of "pi" as 22/7 and the value of "e" as
19/7 for 19
occupied subshells in 7 shells. Now in "atom speak" the exponent
because
an energy level so that we can have 19^(22x22) and 22^(22x22). So in
"atom speak"
we can have these large numbers.

Atom Totality largest meaningful numbers: 19^(22x22) approx = 301!
approx = 10^618
and 22^(22x22) approx = 10^649


Now where will the log-spirals all breakdown in the canyon channel
pattern and what
square will the breakdown with? What square of the Factorial sequence
halts all the
log-spirals?

From my work so far, it looks to be the number 19^484 or the
19^(22x22)

So the boundary between finite and infinite comes more from Physics of
the
Atom Totality theory of the Plutonium Atom Totality, than it comes
from Physics
of where the StrongNuclear Force ends by the existence of these heavy
nuclide
elements.

So, all we can do now is wait for the computers to verify whether no
log-spiral
survives the canyon channel pattern beyond 19^(22x22), and that number
would truly be the start of infinity and below that number would still
be finite numbers
in mathematics.

So for the first time in the history of mathematics, that a precision
definition is given
for what it means to be a Finite Number and not an Infinite Number.
And for the first
time in math history, precision is given to what it means to be
infinite. It means beyond
this boundary line of 19^(22x22)

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Thought I would spend some time on a brief synopsis of where Physics
stands on a candidate
for where the boundary is between finite numbers and infinite numbers.
Of course, for pure
mathematics there seems to be only one candidate in play and that is
the log spiral of 2D, associated with the pseudosphere and sphere of
3D, so that where the log-spiral breaks down
is where the pseudosphere stretch ends and where the pseudosphere thus
has a finite
surface area and finite volume. I am really rather amazed of the
paltry items in mathematics
for infinite stretch yet with finite area and finite volume.
For Physics, which leads the way usually in both physics and
mathematics the largest numbers are the Coulomb Interactions and the
All Possible Arrangements of nuclides. In essence, where the Strong
Nuclear Force ends. So where finite numbers end and where
infinity begins in mathematics is a copy over from physics as to where
Physics ends because
there is no longer a Strong Nuclear Force in the world. That is akin
to mathematics if say, right
angles or 90 degree angles could no longer exist in mathematics and so
mathematics would end. Or another example is that in mathematics if
there was no more multiplicative inverse at
some large number, we would say mathematics ends there.
But let me supply this synopsis of the current progress of work in the
field of nucleosynthesis
Element 100 ranges with nuclides of from 252 to that of 257 with the
longest half lifes of
hours and days. So within this element lies my oft vaunted use of 253!
approx = 10^500.
And where we often say that the Strong Nuclear Force no longer exists,
but then again
let us continue because obviously we have produced higher atomic
elements and thus the
Strong Nuclear Force is still in existence.
Element 109: and this was the element sought for when I was in
college. The nuclide of
278 has a half life of 8 seconds. So here we reach 278! as a
meaningful number in both
physics and math.
Element 114: and this was the element sought for because it offered an
island of stability
and 289 nuclide has a half life of 2.6 seconds. So here reach 289!
Element 116: reaches out to 293 nuclides and has a half-life of 60 ms.
Element 118: and I am surprized we went past element 114 in such a
short time. Three
atoms of element 118 since 2002 have been manufactured
(nucleosynthesized) and it
has 294 nuclides with a half life of 0.89 ms. So I guess the furthest
meaningful number
in physics and math from the Elements is 294!

Let me add to the above list Element 120

--- quote from Wikipedia on Element 120 ---
Calculated decay characteristics


In a quantum tunneling model with mass estimates from a macroscopic-
microscopic model, the alpha-decay half-lives of several isotopes of
unbinilium (namely, 292-304Ubn) have been predicted to be around 1–20
microseconds.[8][9][10][11]

--- end quote ---

Now element 120 has never been synthesized (created). But they are
getting very close
to doing that.

And the mass number of nuclides is 302 as the most stable with the
above guess estimate
of the half-life at about 1-20 ms. If it comes to be nucleosynthesized
with a half-life of
1 ms. Then we can use that 1 ms as a time guage for the StrongNuclear
force.

But of course, here we would have a confirmation between nuclear
physics of 302! as the
end of the StrongNuclear Force and with the Atom Totality where 301!
or 302! is 19^(22x22).

Now all we need is confirmation from pure mathematics that there no
longer exists any
log spirals beyond the 301! or 302! squares in the canyon channel
pattern. I am confident
that confirmation will come sooner than later.

I am thinking of other subjects of physics which has large numbers
involved. And frankly, there are none. Except possibly, cold physics
of approaching 0 Kelvin. Now I wonder what
1/302! means in cold physics? Does it mean that the BEC condensate no
longer exists?

I don't think any physicists have ventured into thoughts of really
cold physics. And it may mean that since physics ends at 302! in the
large because no more StrongNuclear Force,
that physics also ends in the small of cold physics at 1/302!. So no
need to say absolute
zero kelvin temperature, when the temperature reaches 1/302! there is
no more physics, so
there is no more temperture drop in the interval 0 to 1/302!. So some
attention needs to be
directed at cold physics when the upper bound of numbers is 302! then
the lower bound of numbers would be 1/302!

Post by Archimedes Plutonium
But wait a minute! The Atom Totality theory has some large meaningful
numbers.
Strictly speaking in "atom speak" the largest number would be 22 as
the total number
of subshells in the Plutonium Atom Totality (our current universe)
inside of 7 shells.
This delivers us the value of "pi" as 22/7 and the value of "e" as
19/7 for 19
occupied subshells in 7 shells. Now in "atom speak" the exponent
because
an energy level so that we can have 19^(22x22) and 22^(22x22). So in
"atom speak"
we can have these large numbers.
Atom Totality largest meaningful numbers: 19^(22x22) approx = 301!
approx = 10^618
and 22^(22x22) approx = 10^649
Now where will the log-spirals all breakdown in the canyon channel
pattern and what
square will the breakdown with? What square of the Factorial sequence
halts all the
log-spirals?
From my work so far, it looks to be the number 19^484 or the
19^(22x22)
So the boundary between finite and infinite comes more from Physics of
the
Atom Totality theory of the Plutonium Atom Totality, than it comes
from Physics
of where the StrongNuclear Force ends by the existence of these heavy
nuclide
elements.
So, all we can do now is wait for the computers to verify whether no
log-spiral
survives the canyon channel pattern beyond 19^(22x22), and that number
would truly be the start of infinity and below that number would still
be finite numbers
in mathematics.
So for the first time in the history of mathematics, that a precision
definition is given
for what it means to be a Finite Number and not an Infinite Number.
And for the first
time in math history, precision is given to what it means to be
infinite. It means beyond
this boundary line of 19^(22x22)

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, I have the paper cutout setup for the canyon-channel pattern
of the Factorial
Sequence for the 6, 24, 120, and 720 squares and can visualize the
hinges and gates
of the morphed whirling triangles.

Now let me just generalize to the case of the 299!, 300!, 301!, 302!
squares by visualizing
these squares with 299! as a 1 by 1 square so that I can visualize the
other three squares
surrounding the 299! square.

Now what makes the log-spirals all possible is self similarity. That
the whirling morphed
triangles are similar, thus the log spirals still exist that can
traverse or travel in the canyon
channel, until sometime in that travel, the morphed triangles can no
longer be similar.

Now it is a wonder that since Fractal mathematics has been so
overhyped in the past decades, so overhyped, yet this sort of
information has never been realized by all those
overhyping Fractal mathematics. That the log spiral is a key
ingredient in fractal mathematics,
yet the key theorem of fractal mathematics should have been where all
log spirals breakdown
in self similarity. Do they breakdown at 253! or do they breakdown at
a smaller number?

I can report that it is highly unlikely that the breakdown of these
morphed triangles occurs at
302!. Because if we take the 299! square to be a 1 by 1 square then
the 300! would be 300
square and the 301! square would be almost a 100,000 and the 302!
would be almost
30,000,000 square. And so the morphed triangle enclosed in those three
squares would look
like a needle rather than a morphed triangle.

So the self similarity maintained is key to the survival of the log
spirals. Once that self similar
is lost in the increasing largeness of the squares, then the log
spirals breakdown.

Now in Wikipedia on log spirals they show a whirling polygon. And if
we used a whirling polygon with the Factorial sequence of areas of
polygons we also get to the point where
the increasing area of the polygons is so overwelming that like the
morphed triangles, we
are expecting the survival of a log spiral sandwiched between what can
be described as
two line segments, like a long scissors about to close together.

So I suspect that the survival of all log spirals for the whirling
morphed triangles in the
Factorial sequence or the whirling polygons as shown in Wikipedia, to
breakdown or
fall apart far earlier than 301! and that region. Whether it even
survives 253! is hard to
say. And this is where Fractal theory has been lackadaisacal and out
on vacation, for
Fractal theory should have discovered this a long time ago, long
before I discovered it.

Now do not mistake what I am saying. That a log spirals can be
generated without breakdowns of ordinary sequences of addition, where
they do not increase rapidly. But the most
important sequence in all of mathematics is not the addition sequences
but the Factoral sequence. Why is this the most important sequence?
Because the Factorial Sequence is the
cornerstone axiom of set theory in that it bespeaks of All Possible
Arrangements. It is the cornerstone of set theory and of Probability
and Statistics. For it gives you "All Possibilities
of N things arranged".

So the Factorial sequence tells us what very important features of the
log-spirals are. It tells us whether log-spirals breakdown during all
possible arrangements and where all the log spirals
breakdown.

So maybe we could not expect the mathematicians nor the hyped up
Fractal mathematicians
to have focused or concentrated on the Factorial sequence and the
limits of log-spirals. Maybe
we just could never have counted on them to have focused on Factorial
with log-spirals.

So where is the breakdown? Perhaps 253!, or, the largest Planck Units
are only in the 10^60 range, not the 10^500 range. Certainly there is
the large number of the strength of the Coulomb
force over the gravity-force as 10^40. So does infinity start at 10^40
and its inverse 10^-40.

Stranger things have come from Quantum Theory than this.

But it is fitting, grandly fitting that where the log-spirals
breakdown in the Factorial Sequence
is where one should expect that mathematics has a boundary between
finite and infinite.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, I have the paper cutout setup for the canyon-channel pattern
of the Factorial
Sequence for the 6, 24, 120, and 720 squares and can visualize the
hinges and gates
of the morphed whirling triangles.
Now let me just generalize to the case of the 299!, 300!, 301!, 302!
squares by visualizing
these squares with 299! as a 1 by 1 square so that I can visualize the
other three squares
surrounding the 299! square.
Now what makes the log-spirals all possible is self similarity. That
the whirling morphed
triangles are similar, thus the log spirals still exist that can
traverse or travel in the canyon
channel, until sometime in that travel, the morphed triangles can no
longer be similar.
Now it is a wonder that since Fractal mathematics has been so
overhyped in the past decades, so overhyped, yet this sort of
information has never been realized by all those
overhyping Fractal mathematics. That the log spiral is a key
ingredient in fractal mathematics,
yet the key theorem of fractal mathematics should have been where all
log spirals breakdown
in self similarity. Do they breakdown at 253! or do they breakdown at
a smaller number?
I can report that it is highly unlikely that the breakdown of these
morphed triangles occurs at
302!. Because if we take the 299! square to be a 1 by 1 square then
the 300! would be 300
square and the 301! square would be almost a 100,000 and the 302!
would be almost
30,000,000 square. And so the morphed triangle enclosed in those three
squares would look
like a needle rather than a morphed triangle.
So the self similarity maintained is key to the survival of the log
spirals. Once that self similar
is lost in the increasing largeness of the squares, then the log
spirals breakdown.
Now in Wikipedia on log spirals they show a whirling polygon. And if
we used a whirling polygon with the Factorial sequence of areas of
polygons we also get to the point where
the increasing area of the polygons is so overwelming that like the
morphed triangles, we
are expecting the survival of a log spiral sandwiched between what can
be described as
two line segments, like a long scissors about to close together.
So I suspect that the survival of all log spirals for the whirling
morphed triangles in the
Factorial sequence or the whirling polygons as shown in Wikipedia, to
breakdown or
fall apart far earlier than 301! and that region. Whether it even
survives 253! is hard to
say. And this is where Fractal theory has been lackadaisacal and out
on vacation, for
Fractal theory should have discovered this a long time ago, long
before I discovered it.
Now do not mistake what I am saying. That a log spirals can be
generated without breakdowns of ordinary sequences of addition, where
they do not increase rapidly. But the most
important sequence in all of mathematics is not the addition sequences
but the Factoral sequence. Why is this the most important sequence?
Because the Factorial Sequence is the
cornerstone axiom of set theory in that it bespeaks of All Possible
Arrangements. It is the cornerstone of set theory and of Probability
and Statistics. For it gives you "All Possibilities
of N things arranged".
So the Factorial sequence tells us what very important features of the
log-spirals are. It tells us whether log-spirals breakdown during all
possible arrangements and where all the log spirals
breakdown.
So maybe we could not expect the mathematicians nor the hyped up
Fractal mathematicians
to have focused or concentrated on the Factorial sequence and the
limits of log-spirals. Maybe
we just could never have counted on them to have focused on Factorial
with log-spirals.
So where is the breakdown? Perhaps 253!, or, the largest Planck Units
are only in the 10^60 range, not the 10^500 range. Certainly there is
the large number of the strength of the Coulomb
force over the gravity-force as 10^40. So does infinity start at 10^40
and its inverse 10^-40.
Stranger things have come from Quantum Theory than this.
But it is fitting, grandly fitting that where the log-spirals
breakdown in the Factorial Sequence
is where one should expect that mathematics has a boundary between
finite and infinite.

So where do all the log-spirals breakdown in the factorial sequence?
Is it 10^40 of
35! or do they manage to linger on and are only extinguished out at
about 253!

Or is there some intermediate sequence of special importance? We all
know about
the Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, . . . that delivers log
spirals with no
boundary. And where the Fibonacci sequence is one of addition. And we
all know
the Factorial sequence is a multiplication sequence that rises fastly
and is very
important because it is the "All Possible Arrangements sequence". So
the question
would then come to be, is there some intermediate sequence? Something
of terrific
importance like the Factorial that is intermediate between addition of
Fibonacci and
the immense multiplication of Factorial?

I bet there is.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
So where do all the log-spirals breakdown in the factorial sequence?
Is it 10^40 of
35! or do they manage to linger on and are only extinguished out at
about 253!

Now I do not want to give the impression that I am paniced, and given
up
on the Factorial sequence. It has not been confirmed as to where the
whirling morphed triangles pattern breaksdown for all log spirals. It
maybe
that it survives out to 301! and breaks down there.

Keep in mind that in mathematics history, noone has ventured this far
with finding a pure mathematics boundary between finite and infinite
and
since noone has ventured this far or even had a glimmer of an idea
that
a boundary exists in pure math, that we can expect there to be doubt
and hesitation of a breakdown in log spirals. That our imagination
and
intuition breakdown also as we try to picture these whirling morphed
triangles.

So it is not a case of panic on my part, but more of a backup plan
should
the Factorial Sequence not fulfill its job.

Post by Archimedes Plutonium
Or is there some intermediate sequence of special importance? We all
know about
the Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, . . . that delivers log
spirals with no
boundary. And where the Fibonacci sequence is one of addition. And we
all know
the Factorial sequence is a multiplication sequence that rises fastly
and is very
important because it is the "All Possible Arrangements sequence". So
the question
would then come to be, is there some intermediate sequence? Something
of terrific
importance like the Factorial that is intermediate between addition of
Fibonacci and
the immense multiplication of Factorial?
I bet there is.

Well there are two intermediate sequences that come to mind, readily.
There is the
Geometric Sequence which is assured to give a log-spiral setup
pattern. The Geometric
Sequence such as 0, 1, 5, 25, 125, . . . which is a multiplying of 5
of each successive term.

And then there is the Binomial sequence of the Binomial Probability:

0, 2, 4, 8, 16, 32, 64, . . .

It is also a geometric sequence and is sure to deliver log-spirals.

But do they deliver a breakdown of all log-spirals at a large number
out?

That question would be tantamount to asking whether there is a
breakdown
in the self-similarity with increasing size? And there is no breakdown
as far
as I can intuit.

So the difference between Factorial sequence and Binomial sequence is
a difference
of one having a variable multiplier whereas the other has a constant
multiplier.

So the question then becomes if there is a important Probability
theory sequence
that is intermediate between a Factorial sequence and a Geometric
Sequence?

Is there something in Probability theory, of importance, that fits
that description?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
So where do all the log-spirals breakdown in the factorial sequence?
Is it 10^40 of
35! or do they manage to linger on and are only extinguished out at
about 253!

Now I do not want to give the impression that I am paniced, and given
up
on the Factorial sequence. It has not been confirmed as to where the
whirling morphed triangles pattern breaksdown for all log spirals. It
maybe
that it survives out to 301! and breaks down there.
Keep in mind that in mathematics history, noone has ventured this far
with finding a pure mathematics boundary between finite and infinite
and
since noone has ventured this far or even had a glimmer of an idea
that
a boundary exists in pure math, that we can expect there to be doubt
and hesitation of a breakdown in log spirals. That our imagination
and
intuition breakdown also as we try to picture these whirling morphed
triangles.
So it is not a case of panic on my part, but more of a backup plan
should
the Factorial Sequence not fulfill its job.

Post by Archimedes Plutonium
Or is there some intermediate sequence of special importance? We all
know about
the Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, . . . that delivers log
spirals with no
boundary. And where the Fibonacci sequence is one of addition. And we
all know
the Factorial sequence is a multiplication sequence that rises fastly
and is very
important because it is the "All Possible Arrangements sequence". So
the question
would then come to be, is there some intermediate sequence? Something
of terrific
importance like the Factorial that is intermediate between addition of
Fibonacci and
the immense multiplication of Factorial?
I bet there is.

Well there are two intermediate sequences that come to mind, readily.
There is the
Geometric Sequence which is assured to give a log-spiral setup
pattern. The Geometric
Sequence such as 0, 1, 5, 25, 125, . . . which is a multiplying of 5
of each successive term.
0, 2, 4, 8, 16, 32, 64, . . .
It is also a geometric sequence and is sure to deliver log-spirals.
But do they deliver a breakdown of all log-spirals at a large number
out?
That question would be tantamount to asking whether there is a
breakdown
in the self-similarity with increasing size? And there is no breakdown
as far
as I can intuit.
So the difference between Factorial sequence and Binomial sequence is
a difference
of one having a variable multiplier whereas the other has a constant
multiplier.
So the question then becomes if there is a important Probability
theory sequence
that is intermediate between a Factorial sequence and a Geometric
Sequence?
Is there something in Probability theory, of importance, that fits
that description?

Well, let me try to engineer something on the spot, but only briefly,
since I am
expected to do some business elsewhere.

Seems to me that we can engineer all sorts of sequences like the
Factorial of its
variable multiplier by inserting a fraction multiplier in each
successive term that is thence subtracted.

For example the Factorial sequence is 1, 1, 2, 6, 24, 120, 720, . . .

Now let me engineer a sequence from the Factorial that subtracts a
fraction off of each
successive term such as subtraction of 90% from each successive term
leaving this
sequence:

0.1, 0.1, 0.2, 0.6, 2.4, . . .

So, in some manner, have I tamed the Factorial sequence into becoming
a geometric sequence?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, I am waiting for the confirmation that the Factorial Sequence
with its whirling morphed
triangles of squares and the canyon channel pattern allows log-spirals
to exist in that condition until about 301! or that vicinity of
19^(22x22).

In case no log spiral can exist in that Factorial Sequence of morphed
triangles which have
hinges and gates for the factorial squares, and is broken down not at
301! region but breaks down at the mere paltry site of around 35! or
10^40.

So let me just take a sneek preview of the 35! region. We already know
about the 6! region of
6, 24, 120, 720 but what about the 35! region? Here again let us look
at 32! and treat it as a
1 by 1 square. Then we have 33!, 34!, 35! and those would be squares
that are 33 times bigger, 1122 times bigger and 39,270 times bigger.

So can any log spiral survive and exist by winding its way through 35
gates of those whirling
squares?

I am hoping that one log spiral survives all the way to 301! when it
finally is destroyed by the
conditions. But if that is too much to hope for, then I hope one log
spiral survives the gates
up to 35! when it is come to a breakdown.

Now if it comes to pass that there does exist one log spiral that
survives the conditions to 10^40, then this is obviously not where
finiteness ends and infinity begins since in physics
the StrongNuclear Force prospers in the conditions of 253! all the way
to 302! or 19^(22x22)

So if it comes to pass that a log spiral in the canyon channel pattern
only survives to 35!, then I suspect we have to find a intermediate
sequence of the Factorial Sequence that
tempers the Factorial Sequence by taking a constant percentage out of
each term of the
Factorial Sequence so that this new sequence allows the existence of
log-spirals all the way
up to 19^(22x22) and then ends there or breaks down there.

Now many are going to shout and cry "foul foul, you "ad hoc". And I
would agree with them
except that perhaps this percentage term that tempers the Factorial
Sequence is a special
number itself. If it is a special number, then it is not ad hoc.

Now I am going to return to the Atom Totality theory book solely, and
let the above confirmation work its way out.

The above is the most important new work in all of mathematics, since
the time of
Euclid when geometry was first systematized. Important because it
affects every subject
of mathematics by specifically detailing what it means to be finite
versus infinite. Until now,
mathematics worked on that every person had their own belief system as
to what finite meant
and what infinity meant. And thus we had conjectures never able to be
proven such as Riemann Hypothesis, Goldbach, Fermat's Last Theorem,
Kepler Packing, Poincare Conjecture, and thousands of others.

Once it is confirmed that math itself has a boundary of finite versus
infinite in the Factorial Sequence where the pseudosphere surface area
and volume have boundary conditions, is a
magnificently huge and important moment of progress for mathematics.
In the meantime
math resides in dirty and murky imprecision.

So while I wait for confirmation, I move on to Atom Totality.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, I am waiting for the confirmation that the Factorial Sequence
with its whirling morphed
triangles of squares and the canyon channel pattern allows log-spirals
to exist in that condition until about 301! or that vicinity of
19^(22x22).
In case no log spiral can exist in that Factorial Sequence of morphed
triangles which have
hinges and gates for the factorial squares, and is broken down not at
301! region but breaks down at the mere paltry site of around 35! or
10^40.
So let me just take a sneek preview of the 35! region. We already know
about the 6! region of
6, 24, 120, 720 but what about the 35! region? Here again let us look
at 32! and treat it as a
1 by 1 square. Then we have 33!, 34!, 35! and those would be squares
that are 33 times bigger, 1122 times bigger and 39,270 times bigger.
So can any log spiral survive and exist by winding its way through 35
gates of those whirling
squares?
I am hoping that one log spiral survives all the way to 301! when it
finally is destroyed by the
conditions. But if that is too much to hope for, then I hope one log
spiral survives the gates
up to 35! when it is come to a breakdown.
Now if it comes to pass that there does exist one log spiral that
survives the conditions to 10^40, then this is obviously not where
finiteness ends and infinity begins since in physics
the StrongNuclear Force prospers in the conditions of 253! all the way
to 302! or 19^(22x22)
So if it comes to pass that a log spiral in the canyon channel pattern
only survives to 35!, then I suspect we have to find a intermediate
sequence of the Factorial Sequence that
tempers the Factorial Sequence by taking a constant percentage out of
each term of the
Factorial Sequence so that this new sequence allows the existence of
log-spirals all the way
up to 19^(22x22) and then ends there or breaks down there.
Now many are going to shout and cry "foul foul, you "ad hoc". And I
would agree with them
except that perhaps this percentage term that tempers the Factorial
Sequence is a special
number itself. If it is a special number, then it is not ad hoc.

Now we probably all learned the binomial probability and the factorial
probability via the examples of arranging N things is N! and the
example of
taking a test of multiply choice true or false, with N questions is
2^N.

Now in mathematics probability theory they usually split up these two
permutations but I like to think of them as both the "outcome space"
the all possible outcome events.

I call the one the Factorial outcome space and the other the Binomial
outcome space.

So by a special number cited in my prior post, what if I subtract the
Binomial outcome
space from the Factorial outcome space and have a sequence as thus:

0! - 0, 1! - 2^1, 2! - 2^2, 3! - 2^3, 4! - 2^4, 5! - 2^5, . . .

Now let me pause and examine the meaning of say 5! - 2^5. From what we
learned
in school that means all possible arrangement in order of 5 items as
5!, and the
number 2^5 is the all possible outcomes of 5 decisions with two
choices.

So what is the meaning of say the subtraction of the binomial from the
factorial?

Is it a special sequence? And does it have a Physics physical meaning?

What is the number 301! - 2^301

And will this sequence provide a whirling figure of squares or
triangles or quadrilaterals
that generates log-spirals?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Archimedes Plutonium wrote:
(snipped)

Post by Archimedes Plutonium
Now we probably all learned the binomial probability and the factorial
probability via the examples of arranging N things is N! and the
example of
taking a test of multiply choice true or false, with N questions is
2^N.
Now in mathematics probability theory they usually split up these two
permutations but I like to think of them as both the "outcome space"
the all possible outcome events.
I call the one the Factorial outcome space and the other the Binomial
outcome space.
So by a special number cited in my prior post, what if I subtract the
Binomial outcome
0! - 0, 1! - 2^1, 2! - 2^2, 3! - 2^3, 4! - 2^4, 5! - 2^5, . . .
Now let me pause and examine the meaning of say 5! - 2^5. From what we
learned
in school that means all possible arrangement in order of 5 items as
5!, and the
number 2^5 is the all possible outcomes of 5 decisions with two
choices.
So what is the meaning of say the subtraction of the binomial from the
factorial?
Is it a special sequence? And does it have a Physics physical meaning?
What is the number 301! - 2^301
And will this sequence provide a whirling figure of squares or
triangles or quadrilaterals
that generates log-spirals?

Alright, if you joined this thread late, what I am doing now is taming
the Factorial Sequence.
Taming it in case it is too wild for having log-spirals exist in the
sequence because the numbers build too large, too rapidly. The whole
exercise is to find where in mathematics is
the Natural Pure Boundary of where finite ends and infinity begins.
This excercise is prompted
by the ludicrous situation of mathematics where old and young, stupid
and silly mathematicians actually believe a pseudosphere stretches to
infinity yet retains a finite
surface area and finite volume. We really must laugh at those kind of
so called mathematicians who earned their degree from a bubble gum
machine.

So to fix where the pseudosphere ends as a boundary between finite and
infinite, and moreover, for all of mathematics, to have for the first
time in history a precision definition
of what it means to be Finite versus Infinite. Because in any college
and university across
the world, you walk in and ask the local math professor for a
precision definition of finite
versus infinite and all you get is some goobledygook crap.

What this thread is going to end up with is that precision definition
of finite versus infinite.
It will be some number such as 301! or 19^(22x22) or possibly even
22^(22x22).

So once this exercise/experiment is over with and the number is found
that is pure math
and is a pure boundary between finite and infinite, so that we can
safely say that the area and
volume of a pseudosphere has a extra term of subtraction compared to
area and volume of the
sphere of identical radius, without laughing out loud at modern day
mathematicians who actually do believe infinite stretch is still
finite area and finite volume.

Last night I offered this new hybrid sequence of the Factorial
probability space of outcomes
subtracting the Binomial probability space of outcomes:

Factorial subtracting Binomial: 0! - 0, 1! - 2^1, 2! - 2^2, 3! - 2^3,
4! - 2^4, 5! - 2^5, . . .

One thing is certain for me about that new sequence. It delivers more
log-spirals that
can survive the windings of those squares. But does it have much
meaning? When
we subtract the outcome space from another outcome space, does it have
a Physics
physical meaning?

So let me offer today a division of the Factorial probability by the
Binomial probability
and this sequence offers us a physical meaning in that some of the
events of the outcome
space had a choice of being either particle or wave nature in physics.

Factorial/ Binomial sequence: 0! /0, 1! /2^1, 2! /2^2, 3! /2^3, 4! /
2^4, 5! /2^5, . . .

Now I suspect this sequence will have that natural pure math boundary
between Finite
versus Infinity and will allow the existence of log-spirals all the
way up to 19^(22x22)
or possibly 22^(22x22) but then extinguish the existence of any log-
spirals continuing
thereafter.

I suspect this to be true because if the pure Factorial sequence can
survive with the existence of log-spirals out to 35! that this
Factorial divided by Binomial sequence would survive out
to 19^(22x22) or even maybe 22^(22x22)

P.S. I still hold out hope that the plain old Factorial sequence alone
can tell the boundary of
finite vs. infinity.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
Factorial subtracting Binomial: 0! - 0, 1! - 2^1, 2! - 2^2, 3! - 2^3,
4! - 2^4, 5! - 2^5, . . .

Now I probably made a mistake on that first binomial term, or maybe
not, since
we have 0 or nothing if we have no decisions to make. So the question
is
whether we have 2^0 as the first Binomial term or whether we have 0.
Maybe in
addition or subtraction we start the Binomial probability with 0 and
not 2^0 and use
that for division or multiplication.

Post by Archimedes Plutonium
Factorial/ Binomial sequence: 0! /0, 1! /2^1, 2! /2^2, 3! /2^3, 4! /
2^4, 5! /2^5, . . .

Now in that division sequence the first term should likely be 0!/ 2^0
which is 1
rather than "undefined by zero division"

Anyway, all log spirals start out quirky with there first few terms,
even in the Fibonacci
sequence which is the native homeland of log-spirals start off quirky
where we usually
heavily blacken in the first curve windings of the log-spiral.

Now the question on my mind this night is whether that Binomial
probability division tames
the wildness of the Factoral probability outcome space. And checking
on some spots of those
terms, I see that 35! which is approx 10^40 is divided by 2^35 which
is about 10^10 so we have a square 10^30 rather than 10^40. And about
253! which is 10^500 we are softened by the binomial division of
2^253 = 10^76, and further yet that 301! of about 10^600 is softened
by 2^301 of about 10^90.

So the question is, if the all the families of log spirals can only
survive out to 35! for the pure
Factorial sequence and then all of them are extinguished at 35!, the
question is whether this
binomial softening of the too large and too fast of the factorial, is
enough of a softening that
at least one log spiral of say (guessing) of pitch 60 degrees in the
canyon channel pattern
(the morphed triangle pattern), that at least one log spiral survives
out to 302! before it is rendered extinguished from existence.

But I want to comment also, that the division of Factorial by Binomial
makes alot of sense
in terms of Physics and the nuclides in a atom nucleus. In an atom of
253 nuclides we have
the factorial 253! which is all the possible arrangements in order of
those 253 nuclides. And
now when we divide 253! by 2^253 we arrive at a number which tells us
the total outcome space of whether a nuclide in particular is a
"particle or a wave". So that has physics significance.

But maybe there is even a better Physics significance of the Factorial
divided by something else.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

This is sort of a eureka moment for me, either a half-hearted eureka
or a doubly
eureka, or both.

Has anyone in math ever wondered why probability spaces or outcomes
come
in only two varieties? Either the factorial or binomial (generalized
binomial)?

For example we have a true false test of two questions and the outcome
space
is TT , TF, FT, FF for binomial of 2^2. Now for factorial we have all
possible arrangement
in order of two things as AB, BA for 2!. For three in binomial of
questions and outcome
space we have TTT, TTF, TFF, TFT, FTT, FTF, FFT, FFF and for three
with factorial
we have ABC, BCA, CAB, ACB, BAC, CBA, giving us 2^3 for binomial and
3! for factorial.
At the start, the binomial is far larger than the factorial, but a
short ways out, the factorial
quickly overtakes the binomial. But in physics, with quantum duality,
we would not
see these two sequences as different from one another. We would see
them as a duality
in which they are substitutable for one another, just as in one
experiment we focus on
the wave nature and in another on the particle nature. So this is
extremely difficult for the
old generations of mathematicians to understand, that their logic of
Aristotelian does not
apply to all of mathematics and is only good for a small region of
mathematics which is
the finite region. The whole of mathematics is quantum logic and so
the factorial sequence
and binomial are not different and distinct in the whole of
mathematics, only different and
distinct when doing small finite numbers in mathematics. There are
only two of these sequences for probability outcome spaces, just as in
physics there are only two dualities
for that of particle and wave.

So in the past weeks I have been batting myself over the head with
these log-spirals, and
coming down to this showdown of the factorial and binomial. And the
answer was there all
along, and it was simple. I can use the Binomial to do the geometry of
whirling squares or
whirling morphed triangles and when I am finished, I simply convert
over to factorial. So the
expectation is that the log-spirals, all of them, cease to exist
around 2^2048, which is 302!.

Now the question is, which to use for atomic nucleus nuclides such as
element 120 with
302 nuclides? Do I use 302! or do I use the equivalent binomial
probability with the question as
to whether particle or wave? This becomes the binomial of approx
2^2048

In the previous posts, I was trying to soften down or dampen down the
rapid speed of size
of the factorial squares to accommodate the existence of log spirals.
I attempted subtraction and then division on the factorial with the
binomial. This is foolish on my part, except for that
it leads me to understand that Factorial is a quantum dual of
Binomial, just as particle is dual
to wave or that geometry is dual to arithmetic.

So instead of whirling morphed triangles with the Factorial sequence:
0!, 1!, 2!, 3!, 4!, 5!, . . .

I can use the Binomial sequence:
0, 2, 4, 8, 16, 32, 64, . . .

And where the whirling morphed triangles cease to allow the existence
of any
log-spirals at around the 2^2048 vicinity is the equivalent to the
factorial 302!.

In physics, what the factorial and binomial have to do with the
nuclides and nucleus
is ask two questions which are quantum dualities. We ask what total
arrangement in
order can we place 302 nuclides and thus asking a factorial question
that is a question
of the interactions of those 302 particles.

Or, alternatively, we can ask the question of what total outcomes when
each of the 302
nuclides is given a choice of being either a particle or a wave. And
here the answer is
where the 2^N binomial equals the outcome space of the 302!

So tonight, what I need to do is cut out squares of 2, 4, 8, 16, 32,
64 and then see how
well I can construct the pattern of whirling morphed triangles. The
binomial sequence is not
called a geometric sequence for nothing.

So the question about the binomial sequence as per log-spirals, is
what makes them extinguished at 2^2048 square? And the likely answer
is that at that square, there is no
longer any room of flexibility of the canyon-channel of its hinges and
gates to allow any
log-spiral to exist from thereafter.

P.S. I should note that the mention of cypher codes with the RSA
2^2048 did turn out to
be quite handy, in that it redirected my attention to the binomial
which I probably would have
otherwise skipped out completely. What the RSA gave me, is the idea
that we can look at
any Universal Space Outcome as either a factorial space or a binomial
space. And naturally
in computers, the yes, no or gate open or closed is ideal for
binomial, but has anyone in electronics or computers switched over to
the factorial space? Probably not, and not because
it is impossible, but only because it is not as practical or easy as
binomial.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Enrico]

On Sep 10, 12:01 pm, Archimedes Plutonium

Post by Archimedes Plutonium
This is sort of a eureka moment for me, either a half-hearted eureka
or a doubly
eureka, or both.
Has anyone in math ever wondered why probability spaces or outcomes
come
in only two varieties? Either the factorial or binomial (generalized
binomial)?
For example we have a true false test of two questions and the outcome
space
is TT , TF, FT, FF for binomial of 2^2. Now for factorial we have all
possible arrangement
in order of two things as AB, BA for 2!. For three in binomial of
questions and outcome
space we have TTT, TTF, TFF, TFT, FTT, FTF, FFT, FFF and for three
with factorial
we have ABC, BCA, CAB, ACB, BAC, CBA, giving us 2^3 for binomial and
3! for factorial.
At the start, the binomial is far larger than the factorial, but a
short ways out, the factorial
quickly overtakes the binomial. But in physics, with quantum duality,
we would not
see these two sequences as different from one another. We would see
them as a duality
in which they are substitutable for one another, just as in one
experiment we focus on
the wave nature and in another on the particle nature. So this is
extremely difficult for the
old generations of mathematicians to understand, that their logic of
Aristotelian does not
apply to all of mathematics and is only good for a small region of
mathematics which is
the finite region. The whole of mathematics is quantum logic and so
the factorial sequence
and binomial are not different and distinct in the whole of
mathematics, only different and
distinct when doing small finite numbers in mathematics. There are
only two of these sequences for probability outcome spaces, just as in
physics there are only two dualities
for that of particle and wave.
So in the past weeks I have been batting myself over the head with
these log-spirals, and
coming down to this showdown of the factorial and binomial. And the
answer was there all
along, and it was simple. I can use the Binomial to do the geometry of
whirling squares or
whirling morphed triangles and when I am finished, I simply convert
over to factorial. So the
expectation is that the log-spirals, all of them, cease to exist
around 2^2048, which is 302!.
Now the question is, which to use for atomic nucleus nuclides such as
element 120 with
302 nuclides? Do I use 302! or do I use the equivalent binomial
probability with the question as
to whether particle or wave? This becomes the binomial of approx
2^2048
In the previous posts, I was trying to soften down or dampen down the
rapid speed of size
of the factorial squares to accommodate the existence of log spirals.
I attempted subtraction and then division on the factorial with the
binomial. This is foolish on my part, except for that
it leads me to understand that Factorial is a quantum dual of
Binomial, just as particle is dual
to wave or that geometry is dual to arithmetic.
0!, 1!, 2!, 3!, 4!, 5!, . . .
0, 2, 4, 8, 16, 32, 64, . . .
And where the whirling morphed triangles cease to allow the existence
of any
log-spirals at around the 2^2048 vicinity is the equivalent to the
factorial 302!.
In physics, what the factorial and binomial have to do with the
nuclides and nucleus
is ask two questions which are quantum dualities. We ask what total
arrangement in
order can we place 302 nuclides and thus asking a factorial question
that is a question
of the interactions of those 302 particles.
Or, alternatively, we can ask the question of what total outcomes when
each of the 302
nuclides is given a choice of being either a particle or a wave. And
here the answer is
where the 2^N binomial equals the outcome space of the 302!
So tonight, what I need to do is cut out squares of 2, 4, 8, 16, 32,
64 and then see how
well I can construct the pattern of whirling morphed triangles. The
binomial sequence is not
called a geometric sequence for nothing.
So the question about the binomial sequence as per log-spirals, is
what makes them extinguished at 2^2048 square? And the likely answer
is that at that square, there is no
longer any room of flexibility of the canyon-channel of its hinges and
gates to allow any
log-spiral to exist from thereafter.
P.S. I should note that the mention of cypher codes with the RSA
2^2048 did turn out to
be quite handy, in that it redirected my attention to the binomial
which I probably would have
otherwise skipped out completely. What the RSA gave me, is the idea
that we can look at
any Universal Space Outcome as either a factorial space or a binomial
space. And naturally
in computers, the yes, no or gate open or closed is ideal for
binomial, but has anyone in electronics or computers switched over to
the factorial space? Probably not, and not because
it is impossible, but only because it is not as practical or easy as
binomial.
Archimedes Plutoniumhttp://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

==========================================================

Post by Archimedes Plutonium
Has anyone in math ever wondered why probability spaces or outcomes
come
in only two varieties? Either the factorial or binomial (generalized
binomial)?

You may be needlessly limiting yourself.

There are more than 2 varieties.

With 3 state logic, a 2-variable function has
3^2 = 9 combinations of inputs possible and
3^9 total possible different functions.

Physically, the 3 states could be represented as
(1) Zero volts
(2) +5 volts
(3) -5 volts


Enrico

[Archimedes Plutonium]

Post by Enrico
On Sep 10, 12:01 pm, Archimedes Plutonium

Post by Archimedes Plutonium
Has anyone in math ever wondered why probability spaces or outcomes
come
in only two varieties? Either the factorial or binomial (generalized
binomial)?

You may be needlessly limiting yourself.
There are more than 2 varieties.
With 3 state logic, a 2-variable function has
3^2 = 9 combinations of inputs possible and
3^9 total possible different functions.
Physically, the 3 states could be represented as
(1) Zero volts
(2) +5 volts
(3) -5 volts
Enrico

Thanks for the suggestion. After I read your post, I made the cutouts
for the
2, 4, 8, 16, 32, 64, . . .

sequence, and found them to my stark surprize too cramped to do the
morphed triangle pattern. The squares bump into one another.

So then I went to the next sequence of 3^n

3, 9, 27, 81, . . .

and found this sequence able to hold the morphed triangle for the
canyon channel pattern.

But then I was in a bit of a puzzle predicament, for why should the
Factorial not deliver
and the 2^n not deliver but why should the 3^n deliver channels for
log-spirals to
flourish?

Then I realized I was making it overly difficult. That all three
sequences should work as
to delivery of the boundary between Finite and Infinity.

Scratch all that about the morphed triangle pattern with its gates and
hinges.

Instead, take the Factorial Sequence:

1, 1, 2, 6, 24, 120, 720, 5040, . . .

and just whirl around the squares where one corner side fastens to the
corner
side of the successor square and where every four squares make a
revolution or winding.

Do the same for the 2^n binomial sequence or the 3^n sequence.

I conjecture that log-spirals exist throughout the windings or
revolutions until the square of
side 301! comes up, or in that neighborhood:

--- various contributors wrote ---

Post by Enrico
253! = 5.1734609926400789218043308997295e+499
300! = 3.0605751221644063603537046129727e+614

(sic) >RSA 2^2048 result 3.2317006*10^616
(sic) > 301! Result: 9.2123311...*10^616
(sic) > fibonacci 2954 Result: 1.00000989*10^617

Post by Enrico
19^(22x22) = 8.2554901045277384397095530071882e+618
22^(22x22) = 5.4022853245302743024619692001681e+649

--- end quote ---

So why are all log-spirals extinguished in those channels of whirling
squares, no matter
if the factorial channels or the 2^n channels? The reason is that at
large numbers
such as 10^618 or at 10^649, the curvature needed to be a log spiral
cannot travel that
large distance and stay confined within that channel.

In the factorial sequence the pitch of the log spirals in order to
flourish must be
radically different from the pitch of the binomial or geometric
sequences. But no matter
what the pitch, once those huge squares of 301! and 302!, it
extinguishes all the log-spirals
for they can no longer be log-spirals due to the vast distance that
needs to be traversed
yet still have a curvature to stop from intercepting any square.

I need to get the boundary first, Enrico, then I can focus on
generalization.

One generalization I should ponder is that although I have 302! from
302 protons plus
neutrons in element 120 of 302 nuclides, I should consider the
electrons involved. It
is my theory that the StrongNuclear force is just the normal electron
becoming a
"nuclear electron" which has extra Coulomb force power since it has
traded off its immense
space in return for more Coulomb sticking power of nearby protons. And
a neutron is
really composed of three particles inside itself of the proton,
nuclear-electron, neutrino.

So for an element like 120, with its 302 nuclides of which 120 are
protons and 182 are
neutrons, we have an accounting of the nucleus as 120 + 182 protons,
182 nuclear
electrons, and 182 neutrinos. Which gives me 666 nuclear particles.
Enrico, what I am
trying to explain is, if the boundary comes in as 302! which is about
2^2048, that I am
going to have to explain why 2048 comes into the picture? So if the
element 302 has a
total of 666 + normal electrons 120 = 786. Finally, counting the
neutrons of 182 plus
786 we have 968. And if we say that the StrongNuclear force
is the 3 particles of neutron, proton, electron, then is 3^968 equal
to 2^2048 equal to
302!

I don't know. I need to focus on finding that boundary, first, before
I care to try to
explain the number 2048.

This is a thrilling adventure, and no-one else before has ever
entertained the idea
that pure math has a natural boundary between finite and infinity. So
it is thrilling and
exciting and all new.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

We all know about the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, . . .
and how it is the native habitat for log-spirals in a whirling square
pattern.

And according to my conjecture that the boundary between finite and
infinity is somewhere
in the vicinity of 301! where no log spiral can exist in whirling
squares beyond 301! for they
all intercept or intersect a square. So that the Factorial sequence
delivers this intersection
as well as the binomial sequence of 2, 4, 8, 16, 32, . . . altough at
different log spiral pitches.

So the question occurred to me, since the Fibonacci sequence has log-
spirals without ever
any halt their windings, yet the binomial sequence halts all log-
spirals at 301!, the question occurred to me whether the Fibonacci
sequence sort of comes close to a
succession of numbers where A, B, C such that B is almost 2xA and
where C is almost
4xA

So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

But I was wondering if the Fibonacci sequence gave some sort of hint
that geometric sequences would fall apart in maintaining the existence
of log-spirals around 301!

So does the Fibonacci sequence lend itself as a prediction or hint of
the breakdown of
log-spirals in any geometric sequence >2, other than a "addition
sequence"?

So the question is, does the Fibonacci sequence provide a hint of
breakdowns of log-spirals
in the factorial or binomial sequences?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[quasi]

On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<***@gmail.com> wrote:

<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.

Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.

Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".

quasi

[Archimedes Plutonium]

Post by quasi
On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.
Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.
Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".
quasi

What I am worried about is the leeway of where the Fibonacci sequence
is "additive" and seeks to be "geometric as per the phi number
1.6180339887498948482......."

Current mathematics uses expressions as "limit, as n approaches
infinity"
without ever there being a precision definition of infinity. This is
what this
exercise aims to correct, the use of concepts-- infinity that was
never well
defined.

The entire excercise is to define infinity with precision, such as
where infinity starts
at 302!.

So, now, is there something strange that goes on with the Fibonacci
Sequence at about
302! where the term F_n , F_n+1, F_n+2 where a strangeness occurs in
that F_n+1
is equal to or exceeds the phi value and where F_n+2 equals or exceeds
the phi value
so that a kink in the Fibonacci sequence occurs at this spot where it
actually equals or
exceeds in a few terms.

For example we have terms 89, 144, 233, 377 and we have 144/89 =
1.617.. and
233/144 = 1.618.. which exceeds phi, and then 377/233 = 1.61802...
which is less
than phi.

So the question becomes whether at the huge number of around 302!
whether something
funny or strange happens to the Fibonacci sequence that is a portend
or predictor that
log-spirals all breakdown at that large number no matter what
geometric sequence is
engineered.

LWalker  > fibonacci 2954  Result: 1.00000989*10^617

So I am looking at the neighborhood of Fibonacci 2954, perhaps 2952,
then 2953, then 2954, then 2955
and asking is there something pecular happening here with the additive
Fibonacci in that
perhaps we have a string of four terms where all four of them come to
divide successors
and deliver a phi value that exceeds phi.

So that **if all log-spirals breakdown in all geometric sequences >phi
in the vicinity of 302! **
then the question is does the Fibonacci sequence give us any warning
of that breakdown?
It may or may not but worth a peek.

Now we can sharpen up the search for the boundary between finite and
infinity, and let
me define the Old Math infinity. Let me call it that as Old-Math-
Infinity.

Old-Math-Infinity was ill-defined and left it to every reader to
conjure or imagine "endless"
as being infinity. The trouble with this ill-defined Old-Math-Infinity
is that it does not allow
us to define what Finite means. Is the number 0000123456..... finite
or infinite? In Old Math
that would be finite since it has four zeroes to the leftward and that
is the guide rule Old Math
had on finite numbers. If it has zeroes in the leftward string then it
is finite. But we recognize
000012345678.... as an infinite number, not a finite number.

So the question of the boundary between finite and infinite, in the
end analysis, must straighten out the pseudosphere surface area and
volume as compared to the sphere of equal
radius. Not even the most hypocritical Cantor believer or continuum
hypothesis believer, really
believes that Old Math Infinity with its infinite endless stretch has
a finite volume or finite surface area. Even those hypocrites would
admit their mistakes.

So that the question of the boundary at 302! is that we have the
Geometric Phi Sequence

1.618.... , then phi^2, then phi^3, then phi^4, . . . .

And we ask, can every log spiral endure and exist out to the Old-Math-
Infinity in this geometric
sequence or does every possible type of log-spiral, extinguish and die
at around 302! You can tell what I am betting on.

P.S. So does the Fibonacci sequence have a tiny kink in it at around
the number 302! where a
string of terms in succession exceed phi? A kink that portends or
predicts that the boundary in
the Phi Geometric Sequence will reveal?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.
Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.
Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".
quasi

What I am worried about is the leeway of where the Fibonacci sequence
is "additive" and seeks to be "geometric as per the phi number
1.6180339887498948482......."
Current mathematics uses expressions as "limit, as n approaches
infinity"
without ever there being a precision definition of infinity. This is
what this
exercise aims to correct, the use of concepts-- infinity that was
never well
defined.
The entire excercise is to define infinity with precision, such as
where infinity starts
at 302!.
So, now, is there something strange that goes on with the Fibonacci
Sequence at about
302! where the term F_n , F_n+1, F_n+2 where a strangeness occurs in
that F_n+1
is equal to or exceeds the phi value and where F_n+2 equals or exceeds
the phi value

I am typing too fast for then I make these minor mistakes. I should
not bother to
say "equal to" because none of the terms via a division are going to
be equal to
phi, for all of them are going to either exceed or be less.

I corrected on original with a "sic" sign and deleted all those "equal
to"

Post by Archimedes Plutonium
so that a kink in the Fibonacci sequence occurs at this spot where it
actually equals or
exceeds in a few terms.
For example we have terms 89, 144, 233, 377 and we have 144/89 =
1.617.. and
233/144 = 1.618.. which exceeds phi, and then 377/233 = 1.61802...
which is less
than phi.
So the question becomes whether at the huge number of around 302!
whether something
funny or strange happens to the Fibonacci sequence that is a portend
or predictor that
log-spirals all breakdown at that large number no matter what
geometric sequence is
engineered.
LWalker  > fibonacci 2954  Result: 1.00000989*10^617
So I am looking at the neighborhood of Fibonacci 2954, perhaps 2952,
then 2953, then 2954, then 2955
and asking is there something pecular happening here with the additive
Fibonacci in that
perhaps we have a string of four terms where all four of them come to
divide successors
and deliver a phi value that exceeds phi.
So that **if all log-spirals breakdown in all geometric sequences >phi
in the vicinity of 302! **
then the question is does the Fibonacci sequence give us any warning
of that breakdown?
It may or may not but worth a peek.
Now we can sharpen up the search for the boundary between finite and
infinity, and let
me define the Old Math infinity. Let me call it that as Old-Math-
Infinity.
Old-Math-Infinity was ill-defined and left it to every reader to
conjure or imagine "endless"
as being infinity. The trouble with this ill-defined Old-Math-Infinity
is that it does not allow
us to define what Finite means. Is the number 0000123456..... finite
or infinite? In Old Math
that would be finite since it has four zeroes to the leftward and that
is the guide rule Old Math
had on finite numbers. If it has zeroes in the leftward string then it
is finite. But we recognize
000012345678.... as an infinite number, not a finite number.
So the question of the boundary between finite and infinite, in the
end analysis, must straighten out the pseudosphere surface area and
volume as compared to the sphere of equal
radius. Not even the most hypocritical Cantor believer or continuum
hypothesis believer, really
believes that Old Math Infinity with its infinite endless stretch has
a finite volume or finite surface area. Even those hypocrites would
admit their mistakes.
So that the question of the boundary at 302! is that we have the
Geometric Phi Sequence
1.618.... , then phi^2, then phi^3, then phi^4, . . . .
And we ask, can every log spiral endure and exist out to the Old-Math-
Infinity in this geometric
sequence or does every possible type of log-spiral, extinguish and die
at around 302! You can tell what I am betting on.
P.S. So does the Fibonacci sequence have a tiny kink in it at around
the number 302! where a
string of terms in succession exceed phi? A kink that portends or
predicts that the boundary in
the Phi Geometric Sequence will reveal?
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[quasi]

On Sat, 11 Sep 2010 12:41:52 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.
Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.
Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".
quasi

What I am worried about is the leeway of where the Fibonacci sequence
is "additive" and seeks to be "geometric as per the phi number
1.6180339887498948482......."
Current mathematics uses expressions as "limit, as n approaches
infinity"
without ever there being a precision definition of infinity. This is
what this
exercise aims to correct, the use of concepts-- infinity that was
never well
defined.
The entire excercise is to define infinity with precision, such as
where infinity starts
at 302!.
So, now, is there something strange that goes on with the Fibonacci
Sequence at about
302! where the term F_n , F_n+1, F_n+2 where a strangeness occurs in
that F_n+1
is equal to or exceeds the phi value and where F_n+2 equals or exceeds
the phi value
so that a kink in the Fibonacci sequence occurs at this spot where it
actually equals or
exceeds in a few terms.
For example we have terms 89, 144, 233, 377 and we have 144/89 =
1.617.. and
233/144 = 1.618.. which exceeds phi, and then 377/233 = 1.61802...
which is less
than phi.

It can be proved that for even n, the ratio F_(n+1) / F_n always
exceeds phi, and for odd n, it's always less than phi.

Post by Archimedes Plutonium
So the question becomes whether at the huge number of around 302!
whether something
funny or strange happens to the Fibonacci sequence that is a portend
or predictor that
log-spirals all breakdown at that large number no matter what
geometric sequence is
engineered.

I strongly doubt it.

For one thing, the absolute value of the difference between the ratio
F_(n+1) / F_n and phi is strictly decreasing (and approaches 0 very
rapidly).

My feeling is that, in general, it's more natural to expect uniformity
rather than a one-time break.

quasi

[Archimedes Plutonium]

Post by quasi
On Sat, 11 Sep 2010 12:41:52 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.
Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.
Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".
quasi

What I am worried about is the leeway of where the Fibonacci sequence
is "additive" and seeks to be "geometric as per the phi number
1.6180339887498948482......."
Current mathematics uses expressions as "limit, as n approaches
infinity"
without ever there being a precision definition of infinity. This is
what this
exercise aims to correct, the use of concepts-- infinity that was
never well
defined.
The entire excercise is to define infinity with precision, such as
where infinity starts
at 302!.
So, now, is there something strange that goes on with the Fibonacci
Sequence at about
302! where the term F_n , F_n+1, F_n+2 where a strangeness occurs in
that F_n+1
is equal to or exceeds the phi value and where F_n+2 equals or exceeds
the phi value
so that a kink in the Fibonacci sequence occurs at this spot where it
actually equals or
exceeds in a few terms.
For example we have terms 89, 144, 233, 377 and we have 144/89 =
1.617.. and
233/144 = 1.618.. which exceeds phi, and then 377/233 = 1.61802...
which is less
than phi.

It can be proved that for even n, the ratio F_(n+1) / F_n always
exceeds phi, and for odd n, it's always less than phi.

Thanks, does that theorem have a name, and when proven?

Post by quasi

Post by Archimedes Plutonium
So the question becomes whether at the huge number of around 302!
whether something
funny or strange happens to the Fibonacci sequence that is a portend
or predictor that
log-spirals all breakdown at that large number no matter what
geometric sequence is
engineered.

I strongly doubt it.
For one thing, the absolute value of the difference between the ratio
F_(n+1) / F_n and phi is strictly decreasing (and approaches 0 very
rapidly).

Fine, I am not sure of the logic here of a boundary. I am not sure
whether
the Fibonacci sequence has to have something strange happening at the
number which is the boundary for finite and infinity. What happens to
all the
log-spirals is that they extinguish at this boundary, but whether that
shows
up in signs of the Fibonacci Sequence, is not clear in my mind as of
yet.

Perhaps, as you say, nothing happens, and if we knew about every
region
of terms of the sequence, we find no strangeness.

But on the other hand, perhaps the proof that every even is exceeding
phi
and every odd is less than phi maybe in error and is not a theorem.
Maybe
the error is that such is true except when the log spiral reaches the
large
numbers of the vicinity of 302!.

Do you know how the proof of the theorem was conducted? Perhaps there
was
a error in the proof and it really was not a theorem.

Post by quasi
My feeling is that, in general, it's more natural to expect uniformity
rather than a one-time break.
quasi

Again, I am not certain of the logic of this, if all log spirals in
geometric sequences > = phi
breakdown at 302!, then we have to question whether that has some
affect on the Fibonacci
Sequence and the theorem cited that all even exceed. I would have to
guess it must affect
the Fibonacci if a pure math boundary exists.

And perhaps the affect relates to what you said that the difference
approaches zero to
phi, rapidly. So, maybe, just maybe, at 302! that theorem is broken
and that the even
exceeds while odd is less, is broken at 302!

So how was that theorem proven? Was it a solid proof? I doubt any
other related fields
of math rely on that theorem and is a "one off theorem". Am I right
about that opinion?

Now maybe I should be focused not just on the golden ratio log spiral
but log spirals
in general. And we should be able to examine the Phi Geometric
Sequence out at the
numbers 300!, 301!, 302!, 303!. And examine the Fibonacci Sequence out
at those
numbers as well as the Factorial Sequence.

In a Whirling Pattern of an Identical Shaped Object that only
increases in size, and if all those
exercises or experiments come to a grinding halt at a number such as
302! would that
not be affecting the golden ratio log-spiral at 302!

What Quasi mentions the fact of difference of ratio rapidly
approaching zero, is supporting
evidence in my favor that something strange happens in the vicinity of
302!, but since it
approaches zero so rapidly, no-one would bother to review what happens
out at 302!

So if all the log spirals cease to exist in the channels of these
whirling squares of geometric
sequences >= phi, and that they cease to exist because the imposed
conditions of the walls
of the squares yet the turning curve of the spiral, then perhaps that
theorem is also broken and
not true at this boundary line between finite and infinity of 302!

So anyone know of how the proof of that theorem went? Or did it have a
hidden assumption of
"a precision definition of infinity"? Now would that not be a cake and
eat it too? That here I am
trying to show the boundary of finite and infinity and I am thrown a
theorem that presumes even exceeds and odd is less, but to prove the
theorem, a hidden assumption of "what infinity
means" was used. Would that not be a laughable, eat your own cake.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 12:41:52 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.
Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.
Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".
quasi

What I am worried about is the leeway of where the Fibonacci sequence
is "additive" and seeks to be "geometric as per the phi number
1.6180339887498948482......."
Current mathematics uses expressions as "limit, as n approaches
infinity"
without ever there being a precision definition of infinity. This is
what this
exercise aims to correct, the use of concepts-- infinity that was
never well
defined.
The entire excercise is to define infinity with precision, such as
where infinity starts
at 302!.
So, now, is there something strange that goes on with the Fibonacci
Sequence at about
302! where the term F_n , F_n+1, F_n+2 where a strangeness occurs in
that F_n+1
is equal to or exceeds the phi value and where F_n+2 equals or exceeds
the phi value
so that a kink in the Fibonacci sequence occurs at this spot where it
actually equals or
exceeds in a few terms.
For example we have terms 89, 144, 233, 377 and we have 144/89 =
1.617.. and
233/144 = 1.618.. which exceeds phi, and then 377/233 = 1.61802...
which is less
than phi.

It can be proved that for even n, the ratio F_(n+1) / F_n always
exceeds phi, and for odd n, it's always less than phi.

Thanks, does that theorem have a name, and when proven?

Post by quasi

Post by Archimedes Plutonium
So the question becomes whether at the huge number of around 302!
whether something
funny or strange happens to the Fibonacci sequence that is a portend
or predictor that
log-spirals all breakdown at that large number no matter what
geometric sequence is
engineered.

I strongly doubt it.
For one thing, the absolute value of the difference between the ratio
F_(n+1) / F_n and phi is strictly decreasing (and approaches 0 very
rapidly).

Fine, I am not sure of the logic here of a boundary. I am not sure
whether
the Fibonacci sequence has to have something strange happening at the
number which is the boundary for finite and infinity. What happens to
all the
log-spirals is that they extinguish at this boundary, but whether that
shows
up in signs of the Fibonacci Sequence, is not clear in my mind as of
yet.
Perhaps, as you say, nothing happens, and if we knew about every
region
of terms of the sequence, we find no strangeness.
But on the other hand, perhaps the proof that every even is exceeding
phi
and every odd is less than phi maybe in error and is not a theorem.
Maybe
the error is that such is true except when the log spiral reaches the
large
numbers of the vicinity of 302!.
Do you know how the proof of the theorem was conducted? Perhaps there
was
a error in the proof and it really was not a theorem.

Post by quasi
My feeling is that, in general, it's more natural to expect uniformity
rather than a one-time break.
quasi

Again, I am not certain of the logic of this, if all log spirals in
geometric sequences > = phi
breakdown at 302!, then we have to question whether that has some
affect on the Fibonacci
Sequence and the theorem cited that all even exceed. I would have to
guess it must affect
the Fibonacci if a pure math boundary exists.
And perhaps the affect relates to what you said that the difference
approaches zero to
phi, rapidly. So, maybe, just maybe, at 302! that theorem is broken
and that the even
exceeds while odd is less, is broken at 302!
So how was that theorem proven? Was it a solid proof? I doubt any
other related fields
of math rely on that theorem and is a "one off theorem". Am I right
about that opinion?
Now maybe I should be focused not just on the golden ratio log spiral
but log spirals
in general. And we should be able to examine the Phi Geometric
Sequence out at the
numbers 300!, 301!, 302!, 303!. And examine the Fibonacci Sequence out
at those
numbers as well as the Factorial Sequence.
In a Whirling Pattern of an Identical Shaped Object that only
increases in size, and if all those
exercises or experiments come to a grinding halt at a number such as
302! would that
not be affecting the golden ratio log-spiral at 302!
What Quasi mentions the fact of difference of ratio rapidly
approaching zero, is supporting
evidence in my favor that something strange happens in the vicinity of
302!, but since it
approaches zero so rapidly, no-one would bother to review what happens
out at 302!
So if all the log spirals cease to exist in the channels of these
whirling squares of geometric
sequences >= phi, and that they cease to exist because the imposed
conditions of the walls
of the squares yet the turning curve of the spiral, then perhaps that
theorem is also broken and
not true at this boundary line between finite and infinity of 302!
So anyone know of how the proof of that theorem went? Or did it have a
hidden assumption of
"a precision definition of infinity"? Now would that not be a cake and
eat it too? That here I am
trying to show the boundary of finite and infinity and I am thrown a
theorem that presumes even exceeds and odd is less, but to prove the
theorem, a hidden assumption of "what infinity
means" was used. Would that not be a laughable, eat your own cake.

I suppose I should not say "eat your own cake" if the theorem
marshalled against
me is tainted because it used a definition of infinity that was
assumed to be accurate
when it was not. So I guess the appropriate metaphor is that I buy a
cake to demonstrate
something good and new, only to have the cake thrown into my face as a
bad outcome
overall. Because this whole thread is after the location of the
boundary between finite
and infinite, and a theorem that has a assumed ill-defined definition
of infinity.

But let me add on another point. Suppose I am correct in that 302! is
the boundary
between finite and infinity. That would mean that the number phi, for
the use of
mathematics is only reliable as a mathematics number when it has only
618 digit
place value. Mathematics is only a Aristotelian logic subject when
confined to the
finite portion of mathematics. Anything beyond 302! is not
trustworthy, but has
quantum logic as its underpinning.

So the question of whether the golden ratio log spiral using the
Fibonacci Sequence
with phi, the question would be is phi out to 618 digits place value a
natural boundary
to stop with phi? As Quasi wrote that the ratio differences rapidly
converge to zero, so
does a phi with 618 digits place value, as well as a "pi and e" out to
618 digit place value.
Is there harmony in the special numbers of mathematics if they all
halted at 618 digits
in the decimal system?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Archimedes Plutonium wrote:
(snipped)

Post by Archimedes Plutonium
So the question of whether the golden ratio log spiral using the
Fibonacci Sequence
with phi, the question would be is phi out to 618 digits place value a
natural boundary
to stop with phi? As Quasi wrote that the ratio differences rapidly
converge to zero, so
does a phi with 618 digits place value, as well as a "pi and e" out to
618 digit place value.
Is there harmony in the special numbers of mathematics if they all
halted at 618 digits
in the decimal system?

Now I used the word "harmony" but perhaps I can throw in the words
concord,
or validate or, I guess harmonize is the best word to use because it
was
the Pythagorean theorem that started the "harmony of the spheres" of
bringing
together mathematics with physics in Ancient Greek times.

But a harmony would be very eerie, and strange. What I mean is.
Suppose it is
found that the log-spirals all die out at 302! in those canyon
channels of geometric
sequences. So that 302! is the missing factor in the volume and
surface area
boundary for the pseudosphere of identical radius to a sphere.

But the eerie thing would be that if we look at the place value of
"pi, e, phi" at the
618 digit place value, and should all three of them come in as "0"
digit in that
place value, then that would be eerie and strange, and pretty. It is
likely not to
happen. And whether "pi, e, and phi" all three simultaneously ever
have the same digit in any of its place values in decimal? I would not
be surprized if in binary that they all three have
the same digit in a place value, but in decimal, I would be surprized.
So is the boundary
line between finite and infinite also marked by a concordance or
harmony of its last digit
in the three numbers listed simultaneously?

Anyone have a proof that pi, e, phi, simultaneously do or do not have
the same digits in the
same place value in binary? Let me see:

1.618
2.718
3.141

Well the 3rd and 4th digits of 1 and 8 came awfully close to matching
but pi threw then off.
So I ask the question, if at 618 decimal place value, do all three
simultaneously have a 0
digit?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[quasi]

On Sat, 11 Sep 2010 22:45:58 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 12:41:52 -0700 (PDT), Archimedes Plutonium

Post by Archimedes Plutonium

Post by quasi
On Sat, 11 Sep 2010 00:10:02 -0700 (PDT), Archimedes Plutonium
<snip>

Post by Archimedes Plutonium
So I was wondering if at Fibonacci sequence where it reaches 301! that
we see a closeness
of A, B, C by a factor of almost 2 and 4? I know the Fibonacci
sequence is bounded above
by the geometric sequence of 2, 4, 8, 16, 32, . . .

Let F_n denote the n'th Fibonacci number, where F_1 = 1 and F_2 = 1.
Then the limit, as n approaches infinity, of the ratio F_(n+1) / F_n
is equal to (1 + sqrt(5)) / 2.
Thus, in the above sense, the Fibonacci sequence is "approximately
geometric".
quasi

What I am worried about is the leeway of where the Fibonacci sequence
is "additive" and seeks to be "geometric as per the phi number
1.6180339887498948482......."
Current mathematics uses expressions as "limit, as n approaches
infinity"
without ever there being a precision definition of infinity. This is
what this
exercise aims to correct, the use of concepts-- infinity that was
never well
defined.
The entire excercise is to define infinity with precision, such as
where infinity starts
at 302!.
So, now, is there something strange that goes on with the Fibonacci
Sequence at about
302! where the term F_n , F_n+1, F_n+2 where a strangeness occurs in
that F_n+1
is equal to or exceeds the phi value and where F_n+2 equals or exceeds
the phi value
so that a kink in the Fibonacci sequence occurs at this spot where it
actually equals or
exceeds in a few terms.
For example we have terms 89, 144, 233, 377 and we have 144/89 =
1.617.. and
233/144 = 1.618.. which exceeds phi, and then 377/233 = 1.61802...
which is less
than phi.

It can be proved that for even n, the ratio F_(n+1) / F_n always
exceeds phi, and for odd n, it's always less than phi.

Thanks, does that theorem have a name, and when proven?

It's more of an exercise.

One approach is to use the closed form expression for F_n, sometimes
called "Binet's Formula", given by

F_n = (phi^n - (-1/phi)^n) / sqrt(5)

See:


<http://en.wikipedia.org/wiki/Binet%27s_formula#Closed_form_expression>

or

<http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html>

Simplifying the ratio F_(n+1) / F_n (using Binet's formula) yields
easy proofs of the claims I previously mentioned.

quasi

[Transfer Principle]

On Aug 29, 2:26 am, Archimedes Plutonium

M-F_1  as 1, 2, 2, 4, 8, 32, 256, . . .
M-F_2 as 2, 3, 6, 18, 108, . . .
M-F_3 as 3, 4, 12, 48, . . .
Now here again, I seem to run into the problem of not many odd
numbers

The product of two odd numbers is odd. The product of an even
number and any integer is even. Thus, as soon as an even number
appears, all subsequent values are even.

In order to avoid even numbers, we must start with two odds:

3, 5, 15, 75, ...

Since AP is trying to find odd numbers, he might be interested
in the double factorial function. The double factorial function
n!! multiplies only the numbers of the same parity as , so that
we have:

1!! = 1
3!! = 3*1 = 3
5!! = 5*3*1 = 15
7!! = 7*5*3*1 = 105
9!! = 9*7*5*3*1 = 945

Wolfram Alpha Input:
253!!

Result:
2.563629865...*10^250

In full, the number is:

25636298652726149519498162316800050160690516553691185885978470937405189330789095527872524496004383713964303517031849017184365418389038192598536836267652918506994672777677156197866953108524381733356230463150028909812609980688452101312577724456787109375

which is evidently odd.

Naturally, 253!! is much less than 253! since we're multiplying
only the odds up to 253, not all the naturals up to 253.

[Transfer Principle]

On Aug 29, 12:46 am, Archimedes Plutonium

Post by Archimedes Plutonium
I forgotten whether some people define 0! as equal to 1.

They do:

Wolfram Alpha Input:
0!

Result:
1

[Archimedes Plutonium]

Post by Transfer Principle
On Aug 29, 12:46 am, Archimedes Plutonium

Post by Archimedes Plutonium
I forgotten whether some people define 0! as equal to 1.

0!
1

L. Walk, can you please, set up a website where you draw the stacking
of the factorial-fibonacci so that it has the canyon channel visable
of the sequence:

0!, 1!, 2!, 3!, 4!, 5!, 6!, . . .

And so that the website can begin to draw log-spirals that would
relate to the whirling
squares.

The conjecture is that some log-spirals fit into these whirling
squares up to 253!, but
at 253!, no log-spiral can continue.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Transfer Principle]

On Aug 28, 10:39 pm, Archimedes Plutonium

Post by Archimedes Plutonium

Post by Transfer Principle
Not only that, but let's recall that some posters have
criticized AP's value 10^500 as the finite-infinite boundary
because the RSA-2048 values exceed this boundary. Well, let's

L. Walker, please talk about this RSA and what it means.

RSA numbers are large semiprimes that are difficult to factor. In
cryptography, one can encode a message using an RSA semiprime
such that one needs to know the factors of the semiprime in order
to decode it.

This Wikipedia link can explain it in full:

http://en.wikipedia.org/wiki/RSA

The important part is that small semiprimes can be factored in a
reasonable amount of time, thus breaking the code. So we need to
use large semiprimes that take a long time to factor. And right
now, the size of these semiprimes exceeds 10^500.

Post by Archimedes Plutonium
LWalk, please tell what 22^(22)(22) is?

Wolfram Alpha Input:
22^22^2

Result:
5.40228532...*10^649

Post by Archimedes Plutonium
Can someone tell what 266! is equal to?

Wolfram Alpha Input:
266!

Result:
1.28188766...*10^531

Post by Archimedes Plutonium
And can someone tell what factorial is
close to 10^617

Wolfram Alpha Input:
Solve log(x!)=617 for x (click on base 10 log)

Result:
301.014

This is always tricky since we have to use the base 10 log
in order for the number to be small enough for Wolfram
Alpha to handle, but log is a multivalued function with
complex values. We either need to ignore the imaginary
part or look at the plot to find the real-valued solution.

We can check the solution be entering in:

Wolfram Alpha Input:
301!

Result:
9.2123311...*10^616

By the way, does AP know that Wolfram Alpha is a site that
anyone can click on and use?

http://www.wolframalpha.com

Post by Archimedes Plutonium
But now, here is a new twist on all of this. Instead of just those
plain numbers what about
a Fibonacci sequence of the factorial so we have F_1! as equal to 0,
1!, 1!, 2!, 3!, 8!, . . .

What happened to 5! might I ask?

[Archimedes Plutonium]

Post by Transfer Principle
On Aug 28, 10:39 pm, Archimedes Plutonium

Post by Archimedes Plutonium

Post by Transfer Principle
Not only that, but let's recall that some posters have
criticized AP's value 10^500 as the finite-infinite boundary
because the RSA-2048 values exceed this boundary. Well, let's

L. Walker, please talk about this RSA and what it means.

RSA numbers are large semiprimes that are difficult to factor. In
cryptography, one can encode a message using an RSA semiprime
such that one needs to know the factors of the semiprime in order
to decode it.
http://en.wikipedia.org/wiki/RSA
The important part is that small semiprimes can be factored in a
reasonable amount of time, thus breaking the code. So we need to
use large semiprimes that take a long time to factor. And right
now, the size of these semiprimes exceeds 10^500.

Post by Archimedes Plutonium
LWalk, please tell what 22^(22)(22) is?

22^22^2
5.40228532...*10^649

Post by Archimedes Plutonium
Can someone tell what 266! is equal to?

266!
1.28188766...*10^531

Post by Archimedes Plutonium
And can someone tell what factorial is
close to 10^617

Solve log(x!)=617 for x (click on base 10 log)
301.014
This is always tricky since we have to use the base 10 log
in order for the number to be small enough for Wolfram
Alpha to handle, but log is a multivalued function with
complex values. We either need to ignore the imaginary
part or look at the plot to find the real-valued solution.
301!
9.2123311...*10^616
By the way, does AP know that Wolfram Alpha is a site that
anyone can click on and use?
http://www.wolframalpha.com

Post by Archimedes Plutonium
But now, here is a new twist on all of this. Instead of just those
plain numbers what about
a Fibonacci sequence of the factorial so we have F_1! as equal to 0,
1!, 1!, 2!, 3!, 8!, . . .

What happened to 5! might I ask?

Thanks for the large numbers above and I need to archive them for
quick reference.

As for the 5!, I made a mistake when I entered into the idea of
multiplication with Fibonacci
sequences thinking that the factorial fibonacci had that same addition
of prior two terms so that there would not be a 5!.

But later, I realized it was silly to mix multiplication with addition
and I opted for pure multiplication whereas Fibonacci opted for pure
addition.

So the Factorial Fibonacci is this:

0!, 1!, 2!, 3!, 4! . . .

where it is simply the factorial of all the Naturals in order.

As for being "bothered" that there are no odd numbers in factorial, I
am not bothered for when
it is taken as meaning all possible arrangements of those n-things,
then we expect the universe to be symmetrical and not have a "one off
extra alone item".


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Transfer Principle]

On Aug 26, 4:34 pm, Archimedes Plutonium

Post by Archimedes Plutonium
So is there a quick proof that no golden numbers is of the form a^2 =
b where a and b are golden numbers?

I assume, of course, that AP is excluding the trivial solutions
a=b=0 and a=b=1 (note that both 0 and 1 are golden numbers). Also
we note that a=-1, b=1 is also a solution since -1 is the -2nd
Fibonacci number (according to the Wikipedia page listed above).

Now looking at the Wikipedia page again, we scroll down to
"Primes and Divisibility," where it reads:

"144 is the only nontrivial square Fibonacci number."

and then as sqrt(144) = 12, which isn't a golden number, we
conclude that a^2 = b has only the three trivial solutions.

This fact appears to be a corollary of Carmichael's Theorem,
which has its own Wikipedia page:

http://en.wikipedia.org/wiki/Carmichael%27s_theorem

"Carmichael's theorem, named after the American mathematician
R.D. Carmichael, states that for n greater than 12, the nth
Fibonacci number F(n) has at least one prime factor that is
not a factor of any earlier Fibonacci number."

Since F(n) has at least one prime factor that isn't a factor
of any earlier golden number, it certainly can't be the
_square_ of that golden number (which would have the same
prime factors as that golden number).

The Wikipedia page gives a link to the following page, which
gives the proof that 144 is the largest Fibonacci square:

http://math.asu.edu/~checkman/SquareFibonacci.html

Theorem 3 is what we seek here, but its proof depends on those
of Theorems 1 and 2.

[Archimedes Plutonium]

Post by Transfer Principle
On Aug 26, 4:34 pm, Archimedes Plutonium

Post by Archimedes Plutonium
So is there a quick proof that no golden numbers is of the form a^2 =
b where a and b are golden numbers?

I assume, of course, that AP is excluding the trivial solutions
a=b=0 and a=b=1 (note that both 0 and 1 are golden numbers). Also
we note that a=-1, b=1 is also a solution since -1 is the -2nd
Fibonacci number (according to the Wikipedia page listed above).
Now looking at the Wikipedia page again, we scroll down to
"144 is the only nontrivial square Fibonacci number."
and then as sqrt(144) = 12, which isn't a golden number, we
conclude that a^2 = b has only the three trivial solutions.
This fact appears to be a corollary of Carmichael's Theorem,
http://en.wikipedia.org/wiki/Carmichael%27s_theorem
"Carmichael's theorem, named after the American mathematician
R.D. Carmichael, states that for n greater than 12, the nth
Fibonacci number F(n) has at least one prime factor that is
not a factor of any earlier Fibonacci number."
Since F(n) has at least one prime factor that isn't a factor
of any earlier golden number, it certainly can't be the
_square_ of that golden number (which would have the same
prime factors as that golden number).
The Wikipedia page gives a link to the following page, which
http://math.asu.edu/~checkman/SquareFibonacci.html
Theorem 3 is what we seek here, but its proof depends on those
of Theorems 1 and 2.

Thanks for the references above. And I need to look up what is known
about
multiplication or division of "pi and or "e"" with respect to phi.
Seems like these
three numbers do not interface much.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Transfer Principle
On Aug 26, 4:34 pm, Archimedes Plutonium

Post by Archimedes Plutonium
So is there a quick proof that no golden numbers is of the form a^2 =
b where a and b are golden numbers?

I assume, of course, that AP is excluding the trivial solutions
a=b=0 and a=b=1 (note that both 0 and 1 are golden numbers). Also
we note that a=-1, b=1 is also a solution since -1 is the -2nd
Fibonacci number (according to the Wikipedia page listed above).
Now looking at the Wikipedia page again, we scroll down to
"144 is the only nontrivial square Fibonacci number."
and then as sqrt(144) = 12, which isn't a golden number, we
conclude that a^2 = b has only the three trivial solutions.
This fact appears to be a corollary of Carmichael's Theorem,
http://en.wikipedia.org/wiki/Carmichael%27s_theorem
"Carmichael's theorem, named after the American mathematician
R.D. Carmichael, states that for n greater than 12, the nth
Fibonacci number F(n) has at least one prime factor that is
not a factor of any earlier Fibonacci number."
Since F(n) has at least one prime factor that isn't a factor
of any earlier golden number, it certainly can't be the
_square_ of that golden number (which would have the same
prime factors as that golden number).
The Wikipedia page gives a link to the following page, which
http://math.asu.edu/~checkman/SquareFibonacci.html
Theorem 3 is what we seek here, but its proof depends on those
of Theorems 1 and 2.

Well, we have to revise all the work done on Fibonacci Sequence
and Fibonacci Numbers, because the old-math only focused on
a narrow stretch of Fibonacci Numbers and ignored the general.

In this Fibonacci sequence of F_98 is a Fibonacci Sequence

F_98 = 0, 98, 98, 196, 294, . . .

And obviously it has the perfect-square of 196 or 14^2

So the problem is that Fibonacci Sequences taken in whole have an
infinitude of perfect squares since all perfect squares will appear in
a Fibonacci generalized
Sequence.

I looked at that website:

--- quoting from http://math.asu.edu/~checkman/SquareFibonacci.html
---

JOHN H. E. COHN
Bedford College, University of London, London, N.W.1.

INTRODUCTION
An old conjecture about Fibonacci numbers is that 0, 1 and 144 are the
only perfect squares. Recently there appeared a report that
computation had revealed that among the first million numbers in the
sequence there are no further squares [1]. This is not surprising, as
I have managed to prove the truth of the conjecture, and this short
note is written by invitation of the editors to report my proof. The
original proof will appear shortly in [2] and the reader is referred
there for details. However, the proof given there is fairly long, and
although the same method gives similar results for the Lucas numbers,
I have recently discovered a rather neater method, which starts with
the Lucas numbers, and it is of this method that an account appears
below. It is hoped that the full proof together with its consequences
for Diophantine equations will appear later this year. I might add
that the same method seems to work for more general sequences of
integers, thus enabling equations like

--- end quoting from that website ---


L, Walker, a question I need an answer for. Has no-one in mathematics
ever realized that
this sequence

0, 1, 1, 2, 3, 5, 8, . . .

is only a narrow slice of what should be all the Fibonacci Sequences?

I call that the F_1

and then there is F_2 = 0, 2, 2, 4, 6, 10, . . .

Question L, Walker, so is this the first time anyone realized that
Fibonacci is not just
F_1 but the entire collection?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[David R Tribble]

Post by Archimedes Plutonium

Post by Transfer Principle
"144 is the only nontrivial square Fibonacci number."

Well, we have to revise all the work done on Fibonacci Sequence
and Fibonacci Numbers, because the old-math only focused on
a narrow stretch of Fibonacci Numbers and ignored the general.

Seriously? What makes you think that?

Post by Archimedes Plutonium
In this Fibonacci sequence of F_98 is a Fibonacci Sequence
F_98 = 0, 98, 98, 196, 294, . . .
And obviously it has the perfect-square of 196 or 14^2
So the problem is that Fibonacci Sequences taken in whole have an
infinitude of perfect squares since all perfect squares will appear in
a Fibonacci generalized Sequence.
L, Walker, a question I need an answer for. Has no-one in mathematics
ever realized that this sequence
0, 1, 1, 2, 3, 5, 8, . . .
is only a narrow slice of what should be all the Fibonacci Sequences?
I call that the F_1
and then there is F_2 = 0, 2, 2, 4, 6, 10, . . .
Question L, Walker, so is this the first time anyone realized that
Fibonacci is not just F_1 but the entire collection?

So your "generalized" Fibonacci sequence F_n is simply the
sequence you get when you multiply the elements of F_1 by n.
That does not seem to be very interesting or useful.

A better "generalized" sequence would be:
F(a,b) = a, b, a+b, b+(a+b), (a+b)+(b+a+b), ...
or:
F(a,b)(0) = a
F(a,b)(1) = b
F(a,b)(n) = F(a,b)(n-2) + F(a,b)(n-1)

In other words, a Fibonacci-like sequence in which the first
two starting numbers are specified as parameters.

I'm sure this has already been given a name by someone.

And lo, a quick Google search turns up these:
http://mathworld.wolfram.com/GeneralizedFibonacciNumber.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibGen.html
http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers

-drt

[quasi]

On Sat, 28 Aug 2010 19:29:25 -0700 (PDT), David R Tribble

Post by David R Tribble

Post by Archimedes Plutonium

Post by Transfer Principle
"144 is the only nontrivial square Fibonacci number."

Well, we have to revise all the work done on Fibonacci Sequence
and Fibonacci Numbers, because the old-math only focused on
a narrow stretch of Fibonacci Numbers and ignored the general.

Seriously? What makes you think that?

Post by Archimedes Plutonium
In this Fibonacci sequence of F_98 is a Fibonacci Sequence
F_98 = 0, 98, 98, 196, 294, . . .
And obviously it has the perfect-square of 196 or 14^2
So the problem is that Fibonacci Sequences taken in whole have an
infinitude of perfect squares since all perfect squares will appear in
a Fibonacci generalized Sequence.
L, Walker, a question I need an answer for. Has no-one in mathematics
ever realized that this sequence
0, 1, 1, 2, 3, 5, 8, . . .
is only a narrow slice of what should be all the Fibonacci Sequences?
I call that the F_1
and then there is F_2 = 0, 2, 2, 4, 6, 10, . . .
Question L, Walker, so is this the first time anyone realized that
Fibonacci is not just F_1 but the entire collection?

So your "generalized" Fibonacci sequence F_n is simply the
sequence you get when you multiply the elements of F_1 by n.
That does not seem to be very interesting or useful.
F(a,b) = a, b, a+b, b+(a+b), (a+b)+(b+a+b), ...
F(a,b)(0) = a
F(a,b)(1) = b
F(a,b)(n) = F(a,b)(n-2) + F(a,b)(n-1)
In other words, a Fibonacci-like sequence in which the first
two starting numbers are specified as parameters.
I'm sure this has already been given a name by someone.
http://mathworld.wolfram.com/GeneralizedFibonacciNumber.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibGen.html
http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers
-drt

[quasi]

Post by quasi
On Sat, 28 Aug 2010 19:29:25 -0700 (PDT), David R Tribble

Post by David R Tribble

Post by Archimedes Plutonium

Post by Transfer Principle
"144 is the only nontrivial square Fibonacci number."

Well, we have to revise all the work done on Fibonacci Sequence
and Fibonacci Numbers, because the old-math only focused on
a narrow stretch of Fibonacci Numbers and ignored the general.

Seriously? What makes you think that?

Post by Archimedes Plutonium
In this Fibonacci sequence of F_98 is a Fibonacci Sequence
F_98 = 0, 98, 98, 196, 294, . . .
And obviously it has the perfect-square of 196 or 14^2
So the problem is that Fibonacci Sequences taken in whole have an
infinitude of perfect squares since all perfect squares will appear in
a Fibonacci generalized Sequence.
L, Walker, a question I need an answer for. Has no-one in mathematics
ever realized that this sequence
0, 1, 1, 2, 3, 5, 8, . . .
is only a narrow slice of what should be all the Fibonacci Sequences?
I call that the F_1
and then there is F_2 = 0, 2, 2, 4, 6, 10, . . .
Question L, Walker, so is this the first time anyone realized that
Fibonacci is not just F_1 but the entire collection?

So your "generalized" Fibonacci sequence F_n is simply the
sequence you get when you multiply the elements of F_1 by n.
That does not seem to be very interesting or useful.
F(a,b) = a, b, a+b, b+(a+b), (a+b)+(b+a+b), ...
F(a,b)(0) = a
F(a,b)(1) = b
F(a,b)(n) = F(a,b)(n-2) + F(a,b)(n-1)
In other words, a Fibonacci-like sequence in which the first
two starting numbers are specified as parameters.
I'm sure this has already been given a name by someone.
http://mathworld.wolfram.com/GeneralizedFibonacciNumber.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibGen.html
http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers

Sorry for the accidental previous blank reply.

I intended to mention that at least one author uses the name
"Gibonacci" sequence (G for "generalized") for a Fibonacci-like
sequence with arbitrary starting values a,b.

Personally, I prefer to call them "Fibonacci-like" sequences.

quasi

[David R Tribble]

Post by David R Tribble
F(a,b) = a, b, a+b, b+(a+b), (a+b)+(b+a+b), ...
F(a,b)(0) = a
F(a,b)(1) = b
F(a,b)(n) = F(a,b)(n-2) + F(a,b)(n-1)
In other words, a Fibonacci-like sequence in which the first
two starting numbers are specified as parameters.
I'm sure this has already been given a name by someone.
http://mathworld.wolfram.com/GeneralizedFibonacciNumber.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibGen.html
http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers

At least one author uses the name
"Gibonacci" sequence (G for "generalized") for a Fibonacci-like
sequence with arbitrary starting values a,b.
Personally, I prefer to call them "Fibonacci-like" sequences.

There's also the generalization of adding F(i)+F(i-c) to get the
next term F(i+1), where c > 0.

Then there's the generalization of adding more than two previous
elements of the sequence, which have names like "tribonacci
numbers", "tetranacci", "heptanacci", and so on.

-drt

[Transfer Principle]

On Aug 26, 4:34 pm, Archimedes Plutonium

Post by Archimedes Plutonium
Now obviously we have a square of a golden number that is shy of equal
by 1 unit in that
of 2^2 = 4 is shy by 1 of 5 and we have 3^2 = 9 is shy by 1 unit of 8.

As it turns out, there are no larger golden numbers that are
within one of the square of a golden number either. This also
follows from Carmichael's Theorem, along with the following two
formulae that I found that deal with numbers that are one more
or less than the square of a golden number:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html

"Two formulae answer our question immediately:
F(n)^2 + 1 = F(n – 2)F(n + 2) if n is odd
F(n – 1)F(n + 1) if n is even
F(n)^2 – 1 = F(n – 1)F(n + 1) if n is odd
F(n – 2)F(n + 2) if n is even
These two formulae tell us that
the neighbours of F(n)^2 are never prime,
in fact they are always the product of two Fibonacci numbers!"

Notice that the two exceptions given by AP correspond to the
cases of n=3 and n=4. In either case, n-2 is one or two, and
since F(1) = F(2) = 1 this factor disappears, leaving us with
F(n+2) alone on the right side.

So now we can prove the following:

Theorem:
If n>4, then F(n)^2 is not the neighbor of a golden number.

Proof:
The formulae from the above link tells us that F(n)^2 is the
product of two golden numbers wih index as least 3 -- in
other words, neither number is 1, 0, or -1. So now it remains
to show that the product of two golden numbers greater than 1
can't itself be a golden number.

This is where Carmichael's Theorem comes in. Assume that some
golden number F(m) could indeed be written as the product of
two golden numbers, F(q)and F(r), greater than 1. But by
Carmichael's theorem, F(m) has a prime factor which both F(q)
and F(r) lack. This is a contradiction!

(An indirect proof involving primes? Hey, doesn't that sound
strangely familiar? As it turns out, Carmichael's Theorem
does imply Euclid's Theorem that there are infinitely many
primes, though Carmichael's proof is clearly more complex
than Euclid's. Oh, and before anyone asks, we _cannot_
conclude that F(m) is necessarily prime, so we have _not_
proved that there are infinitely many Fibonacci primes. Let's
burn that bridge before anyone tries to cross it!)

All that's left is to check to make sure that 144, the lone
exception to Carmichael's Theorem, isn't the product of two
golden numbers greater than 1. We easily check that it isn't.

Therefore F(n)^2, with n>4, can never be the neighbor of a
golden number. QED

[Archimedes Plutonium]

Post by Transfer Principle
On Aug 26, 4:34 pm, Archimedes Plutonium

Post by Archimedes Plutonium
Now obviously we have a square of a golden number that is shy of equal
by 1 unit in that
of 2^2 = 4 is shy by 1 of 5 and we have 3^2 = 9 is shy by 1 unit of 8.

As it turns out, there are no larger golden numbers that are
within one of the square of a golden number either. This also
follows from Carmichael's Theorem, along with the following two
formulae that I found that deal with numbers that are one more
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html
F(n)^2 + 1 = F(n – 2)F(n + 2) if n is odd
F(n – 1)F(n + 1) if n is even
F(n)^2 – 1 = F(n – 1)F(n + 1) if n is odd
F(n – 2)F(n + 2) if n is even
These two formulae tell us that
the neighbours of F(n)^2 are never prime,
in fact they are always the product of two Fibonacci numbers!"
Notice that the two exceptions given by AP correspond to the
cases of n=3 and n=4. In either case, n-2 is one or two, and
since F(1) = F(2) = 1 this factor disappears, leaving us with
F(n+2) alone on the right side.
If n>4, then F(n)^2 is not the neighbor of a golden number.
The formulae from the above link tells us that F(n)^2 is the
product of two golden numbers wih index as least 3 -- in
other words, neither number is 1, 0, or -1. So now it remains
to show that the product of two golden numbers greater than 1
can't itself be a golden number.
This is where Carmichael's Theorem comes in. Assume that some
golden number F(m) could indeed be written as the product of
two golden numbers, F(q)and F(r), greater than 1. But by
Carmichael's theorem, F(m) has a prime factor which both F(q)
and F(r) lack. This is a contradiction!
(An indirect proof involving primes? Hey, doesn't that sound
strangely familiar? As it turns out, Carmichael's Theorem
does imply Euclid's Theorem that there are infinitely many
primes, though Carmichael's proof is clearly more complex
than Euclid's. Oh, and before anyone asks, we _cannot_
conclude that F(m) is necessarily prime, so we have _not_
proved that there are infinitely many Fibonacci primes. Let's
burn that bridge before anyone tries to cross it!)
All that's left is to check to make sure that 144, the lone
exception to Carmichael's Theorem, isn't the product of two
golden numbers greater than 1. We easily check that it isn't.
Therefore F(n)^2, with n>4, can never be the neighbor of a
golden number. QED

Seeing this Carmichael's theorem for the first time as posted by
LWalk, I am thinking there is probably a far easier proof using
geometry.
I am guessing Carmichael did it algebraically.

But an easier proof would look at those rectangles of whirling squares
and ask
where can those rectangles have a Semiperimeter equal to that of a
square?


Now if we take the Fibonacci sequence as 0, 1, 1, 2, 3, 5, 8, 13, 21,
34, . . .

And we multiply it by 2, obtaining the new sequence of
0, 2, 2, 4, 6, 10, 16, 26, 42, 64, . . .

Now in this new sequence do we still have a rectangle of whirling
squares?

Quite obviously we do, and is it also governed by phi? Yes of course.

And now, we have boatloads of perfect squares.

So, now what if we multiply the Fibonacci Sequence by 3, calling it
Fibonacci_3 sequence, will we have a unique perfect square >1, just
like we had for Fibonacci_1 Seqeunce with 144?

Here we have 0, 3, 3, 6, 9, 15, 24, 39, 63, . . . Now the question is
in Fibonacci_3 is there a unique perfect square just as Fibonacci_1
was unique for 144? Already I see 9, but whether it is unique is
unknown as of yet.

So for odd numbered Fibonacci sequences we may have a unique perfect
square >1. Now the
number 10^500 is an even number but I can alter that with little
effort to be 10^500 subtract 1.
And here I would ask, what is the perfect square in that Fibonacci
sequence?


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Transfer Principle]

On Aug 26, 11:16 am, Archimedes Plutonium

Post by Archimedes Plutonium
So, has anyone taken the Fibonacci sequence out to the vicinity of
10^500 and can tell
me what two Fibonacci numbers surround below and above 10^500

Let's try asking Wolfram Alpha again.

The command "fibonacci n" does return the nth Fibonacci number,
but it doesn't tell us with Fibonacci number AP seeks. So let
us visit the Wikipedia page for Fibonacci number:

http://en.wikipedia.org/wiki/Fibonacci_number

"Computation by rounding:
Since abs(1-phi)^n/sqrt(5) < 1/2 for all n >= 0, the number
F(n) is the closest integer to phi^n/sqrt(5)"

Here phi is the golden ratio (1+sqrt(5))/2. So now we can ask
Wolfram Alpha the following query:

solve log(((1+sqrt(5))/2)^n/sqrt(5)) = 500 for x

and remembering to click on the base-10 log. Since the log
function is multivalued, there are infinitely many solutions,
but only one is real-valued and is given as 2394.16.

So the Fibonacci numbers that AP seeks are the 2394th and
2395th Fibonacci numbers. So now we can type in:

fibonacci 2394 (approximately 9.2667*10^499)
fibonacci 2395 (approximately 1.4994*10^500)

Post by Archimedes Plutonium
I would want to know the same answer to the question of 10^1000.

Replacing 500 with 1000 in the formula above gives the
approximate value of 4786.64. So we type in:

fibonacci 4786 (approximately 7.3343*10^999)
fibonacci 4787 (approximately 1.1867*10^1000)

Notice that 2394*2 = 4787+1. The reason that this is not
exact is the extra factor of sqrt(5) in the denominator.

[Archimedes Plutonium]

Post by Transfer Principle
On Aug 26, 11:16 am, Archimedes Plutonium

Post by Archimedes Plutonium
So, has anyone taken the Fibonacci sequence out to the vicinity of
10^500 and can tell
me what two Fibonacci numbers surround below and above 10^500

Let's try asking Wolfram Alpha again.
The command "fibonacci n" does return the nth Fibonacci number,
but it doesn't tell us with Fibonacci number AP seeks. So let
http://en.wikipedia.org/wiki/Fibonacci_number
Since abs(1-phi)^n/sqrt(5) < 1/2 for all n >= 0, the number
F(n) is the closest integer to phi^n/sqrt(5)"
Here phi is the golden ratio (1+sqrt(5))/2. So now we can ask
solve log(((1+sqrt(5))/2)^n/sqrt(5)) = 500 for x
and remembering to click on the base-10 log. Since the log
function is multivalued, there are infinitely many solutions,
but only one is real-valued and is given as 2394.16.
So the Fibonacci numbers that AP seeks are the 2394th and
fibonacci 2394 (approximately 9.2667*10^499)
fibonacci 2395 (approximately 1.4994*10^500)

Post by Archimedes Plutonium
I would want to know the same answer to the question of 10^1000.

Replacing 500 with 1000 in the formula above gives the
fibonacci 4786 (approximately 7.3343*10^999)
fibonacci 4787 (approximately 1.1867*10^1000)
Notice that 2394*2 = 4787+1. The reason that this is not
exact is the extra factor of sqrt(5) in the denominator.

Thanks alot.

I seem to be too impatient to plug into Wolfram, their requests, in
order for them to
spit out the numbers I want. And I seemed to have picked up a bad
habit over the years
of never reading a article fully, but impatiently spot reading.

Now what I am going to do is treat the fact that they are not squares
or square roots, is
to mend them or hospitalized these two numbers and doctor them up with
pi and "e"
multiples so that they are squares and square roots.

This doctoring is the equivalent to the Geometry where I talk of the
"defect of the pseudosphere cutaway of the poles".

So the Fibonacci sequence or the logarithmic spiral of "rectangle of
whirling squares"
is the algebra side of the house and the pseudosphere nesting with
sphere is the geometry
side of the house.

It would have been if fibonacci 2394 (approximately 9.2667*10^499)
were the unique perfect square rather than 144 the unique perfect
square >1. Because then the champagne corks would fly in a party of
celebration. Instead, I am faced with adversity and have to roll up
the
sleeves and dig deeper.

I have to doctor the patient fibonacci 2394 (approximately
9.2667*10^499) with pi and "e"
multiples to see if I can find anything special for Fibonacci 2394 or
2395, and 4786 or 4787.

How nice and sweet it would have been if Fibonacci 13 was not a
perfect square and only
Fibonacci 2394 was a perfect square.

But let me translate that to geometry of sphere and pseudosphere, and
what happens at
radius 12 for sphere? The cube is 12 and dodecahedron and that is the
Atom Totality geometry.

But there is also something else in Physics with 12 and 144, involving
the special number
of the Inverse Fine Structure Constant. I remember a story in physics
but may have forgotten the name. The name is perhaps Eddington. The
story goes that when the Fine Structure Constant was first discovered
and used in the 19th century of Maxwell, it initially was pegged at
144. And the story goes that Eddington was giving a geometry
explanation for why it had
to be 144. Now if Eddington was the correct person of this story, I do
not believe he was
using rectangles in whirling squares of the golden ratio for his
derivation.

And of course, as the story unfolded in modern times that the Inverse
Fine Structure Constant
is a number near 137 and not 144.

But here also, let me play around because the defect of 137 to 144 is
7 and pi is 22 subshells in 7 shells as well as "e" 19 occupied
subshells in 7 shells.

So what is the algebraic defect, if I can define an algebraic defect
that is the analogy of the
pseudosphere cutaway defect. What is the algebraic defect of Fibonacci
2394 or 2395, and 4786 or 4787 as being perfect squares of one
another ? Is it a multiplicative factor of pi and
"e". Where I multiply or divide one or the other by either pi or "e"
or both and which thus
doctor's up or fixes the wound and makes them perfect squares or
square root of one another?

And if I am extremely lucky, that these numbers are unique to being
doctored up as perfect squares?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[David Bernier]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
from 0 to 10^500 and infinity occupies 10^500 up to 10^1000.
Now I probably am making some calculation and proportion mistakes, but
that is not important.
for what I am seeking is whether there is a proportion that involves
the number 10^-500 and
10^500 that is special in mathematics.
This reminds me of the golden ratio.
So what I am wondering is that if we look for a boundary between
finite and infinite, it will definitely have to exist both in the
microworld and macroworld and it will have to have a
ratio between that of 0 and 1, and that ratio will then transfer to a
ratio of finite region for
macroworld and infinite region for macroworld. So, finally, is 10^500
a special number because
it serves as the Golden Ratio, or golden rectangle in the interval 1
to 0 and then the macroworld? Is 10^500 special because it is unique
in determining in this ratio of intervals?

Alright, maybe I found what is special about 10^500 ( or a number in
that neighborhood).
We have the golden ratio, (1+sqrt5) /2 = 1.618... And we have the
Golden Spiral and
we have the golden logarithmic spiral
The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, . . .
0, 10^-500, 10^-500, 2/10^500, 3/10^500, . . .
Now, let me multiply that backed-up Fibonacci sequence by the number,
you guessed it,
10^500.
Question is, is there a 10^500 and then later on a number 10^1000 in
that sequence?
As I said, it maybe a number in the vicinity of 10^500 but it has the
property of inverse
within the Fibonacci sequence. Whether it is a unique inverse or
whether it is just the
first inverse in the Fibonacci sequence is a different open question
at this moment.
So, has anyone taken the Fibonacci sequence out to the vicinity of
10^500 and can tell
me what two Fibonacci numbers surround below and above 10^500, then I
would want to
know the same answer to the question of 10^1000.

I've got one Fibonacci number just below 10^500.

Using the closed form expression given at Wikipedia
here:
< http://en.wikipedia.org/wiki/Fibonacci_number#Properties >

I get with the PARI-gp calculator:

F(2394) =
9266713930 6451455317 0267211300 7974015280 6612955569
2161356252 7702248286 8327850791 3248601375 2412188872
8550426752 4085972060 5595521966 1565596749 5312075833
4829145780 1748195991 8570803716 1483563558 3285007772
7014440670 6166099198 8774874808 0858804480 0110232223
1215830258 7904863484 6037567418 1261004808 0932031433
6673509514 5053530810 5355555174 6802839894 6454979881
5075779846 8476243746 6053840737 5048965612 5397875236
6811998837 3466269125 4130379481 8734916078 3492643239
0396628509 5901494232 2704632834 3235788510 9699331192

so F(2394) ~= 0.92667 e+500 , and
F(2395) > 10^500.


David Bernier

[Archimedes Plutonium]

David Bernier wrote:
(snipped)

Post by David Bernier
I've got one Fibonacci number just below 10^500.
Using the closed form expression given at Wikipedia
< http://en.wikipedia.org/wiki/Fibonacci_number#Properties >
F(2394) =
9266713930 6451455317 0267211300 7974015280 6612955569
2161356252 7702248286 8327850791 3248601375 2412188872
8550426752 4085972060 5595521966 1565596749 5312075833
4829145780 1748195991 8570803716 1483563558 3285007772
7014440670 6166099198 8774874808 0858804480 0110232223
1215830258 7904863484 6037567418 1261004808 0932031433
6673509514 5053530810 5355555174 6802839894 6454979881
5075779846 8476243746 6053840737 5048965612 5397875236
6811998837 3466269125 4130379481 8734916078 3492643239
0396628509 5901494232 2704632834 3235788510 9699331192
so F(2394) ~= 0.92667 e+500 , and
F(2395) > 10^500.
David Bernier

Thanks David, I need to keep that data for future use. To see if it
has
some "specialness".

[Archimedes Plutonium]

I am a bit tired here, but I do not give up easily. So let me recap
with all the assumptions I
am carrying forward.

I am assuming that pure math has a natural boundary between finite and
infinite numbers.

In Physics it has this natural boundary as where the StrongNuclear
force no longer exists
at element with atomic number 100 with its 253! of Coulomb
Interactions equals approx
10^500.

Now here we can have two possibilities, either math has a natural
boundary or it does not.
So what prompts me to go on looking? Well, there is supporting
evidence that such a natural
boundary exists with a special number because I was able to derive the
speed of light out of
pure mathematics. And there is supporting evidence from the Formula:

Euclid geometry = Elliptic geometry unioned with Hyperbolic geometry

It is indice of supporting evidences like this and other data that
keeps me going to searching.

In that famous formula above, we can use the sphere as Elliptic and
the pseudosphere
as Hyperbolic. So is there something special about a sphere and
pseudosphere at radius
10^500?

In my derivation out of pure math of the speed of light, you may
remember I used a sphere
and band-meridians and I used what the defect of the enclosed
pseudosphere was.

So this gives me hope that something special is with the sphere and
pseudosphere at
radius 10^500. Maybe the defect-- the polar cutaway of the
pseudosphere of radius
10^500 is a special number itself and thus rendering the number 10^500
as also being
special.

Since the speed of light hinges upon the pseudosphere defect, it is
reasonable to think that the
specialness of 10^500 would also rest upon the pseudosphere defect.




--- quoting an older post of mine that talks of the derivation of
inverse fine structure
constant out of pure math ---

Date: Apr 26, 2010 3:25 PM

(most snipped)
Postscript: Chapter 18: "pi" and "e" and "i" explained; inverse fine

structure constant, and proton to electron mass ratio, speed

of

light,
all linked and explained.

(snipped)

So looking back at my posts on this topic, where I finally
speed of light = summation of meridian strips distance / log-spiral-
radius
Definition: Log-spiral radius is the 1/4 of semicircumference.
Definition: meridian strips are strips and not lines for they have a
width
involved and in the case of Earth in kilometers the width of

the

strips
is a kilometer wide.

(i) Alright, so I have the number "pi" from pure physics as 22
subshells in 7 shells
of the 231Pu Atom Totality
(ii) I have the number "e" from pure physics as 19 occupied subshells
in 7 shells
(iii) I have the Fine-Structure Constant as 22 / (22/7)^7 from pure
physics
(iv) I have the mass ratio of proton to electron as 6 (22/7)^5 from
the fact that
231Pu has the 5f6 energy level, where the fifth energy level has
exponent 5
and the seventh energy level has exponent 7 in Fine Structure
Constant.
speed of light = summation of meridian strips distance / log-spiral-
radius
Now let me see if I can write that in terms of just pi and "e" where
time is in
1 units and distance is in 1 units and using the fifth energy

level:

So the circumference of this generalized unit distance and unit time
is that of 22


So in the equation of the Speed of Light we would have:


Speed of Light = 22 x 22/ log-spiral-radius


Now the Log-spiral-radius is going to be a tiny bit larger than the
Euclidean
radius of 7/2 = 3.5. Remember the golden-ratio-log-spiral is
approximated
by 1/4 turn circles, where sometimes it is slightly larger than the
radius of
the true 1/4 circle.


So we have 22x22/ 3.5 = 138


And the inverse fine structure constant is 137. But if we had the
tiny
bit
larger portion of the golden ratio the phi-log-spiral-radius


22x22/3.53 = 137


Now the Inverse Fine Structure Constant uses the speed of light c as
in:


hbar*c/(e^2)


But here, I have derived the Inverse Fine Structure Constant from the
speed of light itself. How is that possible? Well, I simply removed
the
hbar and the (e^2) by calling them unit distance and unit time.


So the speed of light is basically one and the same as the Fine
Structure Constant.

--- end quoting earlier post of mine to refresh my memory on fine
structure derivation ---


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Post by Archimedes Plutonium
I am a bit tired here, but I do not give up easily. So let me recap
with all the assumptions I
am carrying forward.
I am assuming that pure math has a natural boundary between finite and
infinite numbers.
In Physics it has this natural boundary as where the StrongNuclear
force no longer exists
at element with atomic number 100 with its 253! of Coulomb
Interactions equals approx
10^500.
Now here we can have two possibilities, either math has a natural
boundary or it does not.
So what prompts me to go on looking? Well, there is supporting
evidence that such a natural
boundary exists with a special number because I was able to derive the
speed of light out of
Euclid geometry = Elliptic geometry unioned with Hyperbolic geometry
It is indice of supporting evidences like this and other data that
keeps me going to searching.
In that famous formula above, we can use the sphere as Elliptic and
the pseudosphere
as Hyperbolic. So is there something special about a sphere and
pseudosphere at radius
10^500?
In my derivation out of pure math of the speed of light, you may
remember I used a sphere
and band-meridians and I used what the defect of the enclosed
pseudosphere was.
So this gives me hope that something special is with the sphere and
pseudosphere at
radius 10^500. Maybe the defect-- the polar cutaway of the
pseudosphere of radius
10^500 is a special number itself and thus rendering the number 10^500
as also being
special.
Since the speed of light hinges upon the pseudosphere defect, it is
reasonable to think that the
specialness of 10^500 would also rest upon the pseudosphere defect.
--- quoting an older post of mine that talks of the derivation of
inverse fine structure
constant out of pure math ---
Date: Apr 26, 2010 3:25 PM

(most snipped)
Postscript: Chapter 18: "pi" and "e" and "i" explained; inverse fine

structure constant, and proton to electron mass ratio, speed

of

light,
all linked and explained.

(snipped)

So looking back at my posts on this topic, where I finally
speed of light = summation of meridian strips distance / log-spiral-
radius
Definition: Log-spiral radius is the 1/4 of semicircumference.
Definition: meridian strips are strips and not lines for they have a
width
involved and in the case of Earth in kilometers the width of

the

strips
is a kilometer wide.

(i) Alright, so I have the number "pi" from pure physics as 22
subshells in 7 shells
of the 231Pu Atom Totality
(ii) I have the number "e" from pure physics as 19 occupied subshells
in 7 shells
(iii) I have the Fine-Structure Constant as 22 / (22/7)^7 from pure
physics
(iv) I have the mass ratio of proton to electron as 6 (22/7)^5 from
the fact that
231Pu has the 5f6 energy level, where the fifth energy level has
exponent 5
and the seventh energy level has exponent 7 in Fine Structure
Constant.
speed of light = summation of meridian strips distance / log-spiral-
radius
Now let me see if I can write that in terms of just pi and "e" where
time is in
1 units and distance is in 1 units and using the fifth energy

So the circumference of this generalized unit distance and unit time
is that of 22
Speed of Light = 22 x 22/ log-spiral-radius
Now the Log-spiral-radius is going to be a tiny bit larger than the
Euclidean
radius of 7/2 = 3.5. Remember the golden-ratio-log-spiral is
approximated
by 1/4 turn circles, where sometimes it is slightly larger than the
radius of
the true 1/4 circle.
So we have 22x22/ 3.5 = 138
And the inverse fine structure constant is 137. But if we had the
tiny
bit
larger portion of the golden ratio the phi-log-spiral-radius
22x22/3.53 = 137
Now the Inverse Fine Structure Constant uses the speed of light c as
hbar*c/(e^2)
But here, I have derived the Inverse Fine Structure Constant from the
speed of light itself. How is that possible? Well, I simply removed
the
hbar and the (e^2) by calling them unit distance and unit time.
So the speed of light is basically one and the same as the Fine
Structure Constant.
--- end quoting earlier post of mine to refresh my memory on fine
structure derivation ---

Alright, let me just play around with atom number speak, or talk. Here
the biggest
number is 22 subshells in 7 shells and 19 occupied subshells giving pi
as 22/7 and
e as 19/7 for the plutonium atom , element 94. But we have numbers
such as the
Coulomb Interactions so we have element 100 with say 253! coulomb
interactions and
where the StrongNuclear Force no longer exists. We have element 109
with 266 nucleons
to give 266! which is somewhere around 10^600 whereas 253! is
somewhere close to
10^500.

In atom speak, do I get near 10^500 or 10^600 with just simply numbers
from 0 to 22?

Well we all know the famous equation e^(i)(2pi) = 1 or e^(i)(pi) = -1

So let us take pi as 22/7 and e as 19/7, but truncate them for we seek
only large
numbers in "atom speak". So can I achieve a huge number of 10^500 to
10^600 from
atom speak with the number 22 or 19?

So if e truncated is 19 and pi truncated is 22 we have 19^(i)(22)

So in this "atom speak" what is "i"? Well, is not "i" the hyperbolic
geometry of pi
in euclidean or maybe elliptic geometry? So that "i" has value of
(22).

So in "atom speak" where we are never bothered or annoyed with
distractions but
focused on only what is important with numbers and atoms, we have:

19^(22)(22) = 19^484

Now how close is that to 10^500 or 10^600?

Or we could have in "atom speak" the larger number of

22^(22)(22) = 22^484

And what is that number as 10 an exponent? For surely it is in the
neighborhood
of 10^500 where the StrongNuclear force becomes nonexistent. And hence
physics
is no longer existing and hence no mathematics exists beyond those
numbers. In a
sense, physics breaks down and how that relates to mathematics is that
math no longer
has Aristotelian Logic of straightline logic but breaks down and where
the operations of
math are no longer trustworthy. Logic beyond 10^500 is duality logic
of quantum mechanics.
In mathematics we see this breakdown in that Riemann Hypothesis,
Goldbach conjecture,
Fermat's Last Theorem are no longer provable in mathematics.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Testing

Post by Archimedes Plutonium

Post by Archimedes Plutonium
I am a bit tired here, but I do not give up easily. So let me recap
with all the assumptions I
am carrying forward.
I am assuming that pure math has a natural boundary between finite and
infinite numbers.
In Physics it has this natural boundary as where the StrongNuclear
force no longer exists
at element with atomic number 100 with its 253! of Coulomb
Interactions equals approx
10^500.
Now here we can have two possibilities, either math has a natural
boundary or it does not.
So what prompts me to go on looking? Well, there is supporting
evidence that such a natural
boundary exists with a special number because I was able to derive the
speed of light out of
Euclid geometry = Elliptic geometry unioned with Hyperbolic geometry
It is indice of supporting evidences like this and other data that
keeps me going to searching.
In that famous formula above, we can use the sphere as Elliptic and
the pseudosphere
as Hyperbolic. So is there something special about a sphere and
pseudosphere at radius
10^500?
In my derivation out of pure math of the speed of light, you may
remember I used a sphere
and band-meridians and I used what the defect of the enclosed
pseudosphere was.
So this gives me hope that something special is with the sphere and
pseudosphere at
radius 10^500. Maybe the defect-- the polar cutaway of the
pseudosphere of radius
10^500 is a special number itself and thus rendering the number 10^500
as also being
special.
Since the speed of light hinges upon the pseudosphere defect, it is
reasonable to think that the
specialness of 10^500 would also rest upon the pseudosphere defect.
--- quoting an older post of mine that talks of the derivation of
inverse fine structure
constant out of pure math ---
Date: Apr 26, 2010 3:25 PM

(most snipped)
Postscript: Chapter 18: "pi" and "e" and "i" explained; inverse fine

structure constant, and proton to electron mass ratio, speed

of

light,
all linked and explained.

(snipped)

So looking back at my posts on this topic, where I finally
speed of light = summation of meridian strips distance / log-spiral-
radius
Definition: Log-spiral radius is the 1/4 of semicircumference.
Definition: meridian strips are strips and not lines for they have a
width
involved and in the case of Earth in kilometers the width of

the

strips
is a kilometer wide.

(i) Alright, so I have the number "pi" from pure physics as 22
subshells in 7 shells
of the 231Pu Atom Totality
(ii) I have the number "e" from pure physics as 19 occupied subshells
in 7 shells
(iii) I have the Fine-Structure Constant as 22 / (22/7)^7 from pure
physics
(iv) I have the mass ratio of proton to electron as 6 (22/7)^5 from
the fact that
231Pu has the 5f6 energy level, where the fifth energy level has
exponent 5
and the seventh energy level has exponent 7 in Fine Structure
Constant.
speed of light = summation of meridian strips distance / log-spiral-
radius
Now let me see if I can write that in terms of just pi and "e" where
time is in
1 units and distance is in 1 units and using the fifth energy

So the circumference of this generalized unit distance and unit time
is that of 22
Speed of Light = 22 x 22/ log-spiral-radius
Now the Log-spiral-radius is going to be a tiny bit larger than the
Euclidean
radius of 7/2 = 3.5. Remember the golden-ratio-log-spiral is
approximated
by 1/4 turn circles, where sometimes it is slightly larger than the
radius of
the true 1/4 circle.
So we have 22x22/ 3.5 = 138
And the inverse fine structure constant is 137. But if we had the
tiny
bit
larger portion of the golden ratio the phi-log-spiral-radius
22x22/3.53 = 137
Now the Inverse Fine Structure Constant uses the speed of light c as
hbar*c/(e^2)
But here, I have derived the Inverse Fine Structure Constant from the
speed of light itself. How is that possible? Well, I simply removed
the
hbar and the (e^2) by calling them unit distance and unit time.
So the speed of light is basically one and the same as the Fine
Structure Constant.
--- end quoting earlier post of mine to refresh my memory on fine
structure derivation ---

Alright, let me just play around with atom number speak, or talk. Here
the biggest
number is 22 subshells in 7 shells and 19 occupied subshells giving pi
as 22/7 and
e as 19/7 for the plutonium atom , element 94. But we have numbers
such as the
Coulomb Interactions so we have element 100 with say 253! coulomb
interactions and
where the StrongNuclear Force no longer exists. We have element 109
with 266 nucleons
to give 266! which is somewhere around 10^600 whereas 253! is
somewhere close to
10^500.
In atom speak, do I get near 10^500 or 10^600 with just simply numbers
from 0 to 22?
Well we all know the famous equation e^(i)(2pi) = 1 or e^(i)(pi) = -1
So let us take pi as 22/7 and e as 19/7, but truncate them for we seek
only large
numbers in "atom speak". So can I achieve a huge number of 10^500 to
10^600 from
atom speak with the number 22 or 19?
So if e truncated is 19 and pi truncated is 22 we have 19^(i)(22)
So in this "atom speak" what is "i"? Well, is not "i" the hyperbolic
geometry of pi
in euclidean or maybe elliptic geometry? So that "i" has value of
(22).
So in "atom speak" where we are never bothered or annoyed with
distractions but
19^(22)(22) = 19^484
Now how close is that to 10^500 or 10^600?
Or we could have in "atom speak" the larger number of
22^(22)(22) = 22^484
And what is that number as 10 an exponent? For surely it is in the
neighborhood
of 10^500 where the StrongNuclear force becomes nonexistent. And hence
physics
is no longer existing and hence no mathematics exists beyond those
numbers. In a
sense, physics breaks down and how that relates to mathematics is that
math no longer
has Aristotelian Logic of straightline logic but breaks down and where
the operations of
math are no longer trustworthy. Logic beyond 10^500 is duality logic
of quantum mechanics.
In mathematics we see this breakdown in that Riemann Hypothesis,
Goldbach conjecture,
Fermat's Last Theorem are no longer provable in mathematics.
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[David R Tribble]

Post by Archimedes Plutonium
I am assuming that pure math has a natural boundary between finite and
infinite numbers.

Of course it does. But it's not what you think it is.

[The Charity Algorithm]

Post by Archimedes Plutonium
Let me for a moment change my tactic; instead of
trying to get 10^500
as a special number, let me assume it is infinity
from the start.
Now the formulas for area and volume of the
surface area is
4*pi*R*R while its volume is
(2/3)*pi*R*R*R
So now we define pseudosphere cutaway as a euclidean
planar cut.
And we define the boundary of finite versus infinite
at
10^500 where that number is infinity itself and all
numbers higher.
Now we ask the question of the pseudosphere of radius
1 where it ends
at 10^500 of its poles as a Euclidean
planar cut.
And now we ask, how much of a tiny piece of area do
we have missing
from the formula 4*pi*R*R and again ask how much
volume is missing
from (2/3)*pi*R*R*R
So when we do the integral, the infinity is replaced
by the number
10^500.
So the formulas would no longer be equal to the above
but have a tiny
subtraction of missing area and missing volume.
Now maybe that is the specialness of the number
10^500 in that the
missing portions of area and volume, by the selection
of infinity as
10^500 is a portion that is
special elsewhere.
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

listen i am not even going to read anything but your subject line because i do not play games and i do not have to play games and there is nothing you can say or do about this and there never will be so get used it. stop playing your game now and disclose to the world what you keep secret or i suspect something that was with me a second and left might start really rocking the place we play games with because they are really under a lot of stress and playing these games this way is not fair for everyone when they are about to say go ahead my day was made and it was yesterday before you were born i got up early and stocked up on profits and aereson cans causing whooping cough for bad people.

[The Charity Algorithm]

Post by Archimedes Plutonium
Let me for a moment change my tactic; instead of
trying to get 10^500
as a special number, let me assume it is infinity
from the start.
Now the formulas for area and volume of the
surface area is
4*pi*R*R while its volume is
(2/3)*pi*R*R*R
So now we define pseudosphere cutaway as a euclidean
planar cut.
And we define the boundary of finite versus infinite
at
10^500 where that number is infinity itself and all
numbers higher.
Now we ask the question of the pseudosphere of radius
1 where it ends
at 10^500 of its poles as a Euclidean
planar cut.
And now we ask, how much of a tiny piece of area do
we have missing
from the formula 4*pi*R*R and again ask how much
volume is missing
from (2/3)*pi*R*R*R
So when we do the integral, the infinity is replaced
by the number
10^500.
So the formulas would no longer be equal to the above
but have a tiny
subtraction of missing area and missing volume.
Now maybe that is the specialness of the number
10^500 in that the
missing portions of area and volume, by the selection
of infinity as
10^500 is a portion that is
special elsewhere.
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

But I want to thank the Spanish guy for translating to the nice lady at work. I do not know who you are but you guys should run to the battle becase I am fine and got it here guys go ahead I hope.

[The Charity Algorithm]

Big man at the counter said it is myself so I am going to try this next.