Archimedes Plutonium
2018-08-16 04:34:30 UTC
5Graciously let us thank AP for proving a ellipse is a cylinder section, never a conic section
CONIC SECTION IS OVAL, never an ellipse; (Proof below)
Before I do the formal proof, let us experiment first, where High School students can perform and prove in their own minds that ellipse is never a conic.
Conics = oval, 4 Experiments
4th experiment Re: -World's first proofs that the Conic section is an
Oval, never an ellipse// yes, Apollonius and Dandelin were wrong
by Archimedes Plutonium
Easy and fast experiment, and gets the person able to make more cones
and cylinders in a rush. Only fault I have of this experiment is that
it leaves a scissors mark-- a vertex so to speak. But it is fast and
easy. The proof is in the comparison. Now the cut should be at a steep
enough angle. If you cut straight across, both will be circles, so
make a steep cut.
This experiment is the best for it immediately shows you the asymmetry
of an axis, where the upward needs little area to fill in any gap and
the downward needs an entire "crescent shaped area add-on to the
circle lid.
This experiment is cumbersome and takes much precision and good
materials. It is just a repeat of the Dandelin work on this topic, and
one can easily see how the Dandelin fake proof is constructed-- he
starts off with assuming the figure is an ellipse. Which tells us, he
never had a good-working-model if any at all. For you cannot stuff a
ellipse inside a cone. You can stuff a ellipse inside a cylinder. So
this suggests the entire Dandelin nonsense was all worked out in the
head and never in hands on actual reality. So, in this experiment, we
give a proof that Dandelin was utterly wrong and that it is a cylinder
that you can stuff a ellipse sandwiched by two identical spheres-- one
upper and one lower.
The only amazing part of the Dandelin story is how an utterly fake
proof could have survived from 1822, and not until 2017 is it
thoroughly revealed as ignorant nonsense. One would think in math,
there is no chance such a hideously flawed proof could even be
published in a math journal, and if anything is learned from Dandelin,
is that the math journal publishing system is a whole entire garbage
network. A network that is corrupt and fans fakery.
reveals to us another proof that the conic section is a oval. For the
square-pyramid section is a Isosceles Trapezoid, and what is so great
about that, is we can take a cone and place inside of the cone a
square pyramid and then place a second square pyramid over the cone,
so the cone is sandwiched in between two square pyramids.
Now the square pyramids are tangent to the cone at 4 line segments, 8
altogether for the two, and what is so intriguing about the tangents
is that it allows us to quickly develop a analytic geometry that the
cone section must be a oval in order for the two square pyramids to be
both isosceles trapezoids as sections.
Archimedes Plutonium
--------------------
Conics = oval, 2 proofs, synthetic, analytic
Synthetic Geometry & Analytical Geometry Proofs that Conic section =
Oval, never an ellipse-- World's first proofs thereof
by Archimedes Plutonium
_Synthetic Geometry proofs that Cylinder section= Ellipse// Conic section= Oval
First Synthetic Geometry proofs, later the Analytic Geometry proofs.
Alright I need to get this prepared for the MATH ARRAY of proofs, that
the Ellipse is a Cylinder section, and that the Conic section is an
oval, never an ellipse
PROOF that Cylinder Section is an Ellipse, never a Oval::
I would have proven it by Symmetry. Where I indulge the reader to
place a circle inside the cylinder and have it mounted on a swivel, a
tiny rod fastened to the circle so that you can pivot and rotate the
circle. Then my proof argument would be to say--when the circle plate
is parallel with base, it is a circle but rotate it slightly in the
cylinder and determine what figure is produced. When rotated at the
diameter, the extra area added to the upper portion equals the extra
area added to bottom portion in cylinder, symmetrical area added,
hence a ellipse. QED
Now for proof that the Conic section cannot be an ellipse but an oval,
I again would apply the same proof argument by symmetry.
Proof:: Take a cone in general, and build a circle that rotates on a
axis. Rotate the circle just a tiny bit for it is bound to get stuck
or impeded by the upward slanted walls of the cone. Rotate as far as
you possibly can. Now filling in the area upwards is far smaller than
filling in the area downwards. Hence, only 1 axis of symmetry, not 2
axes of symmetry. Define Oval as having 1 axis of symmetry. Thus a
oval, never an ellipse. QED
The above two proofs are Synthetic Geometry proofs, which means they
need no numbers, just some concepts and axioms to make the proof work.
A Synthetic geometry proof is where you need no numbers, no coordinate
points, no arithmetic, but just using concepts and axioms. A Analytic
Geometry proof is where numbers are involved, if only just coordinate
points.
Array:: Analytic Geometry proof that Cylinder section= Ellipse//Conic
section = Oval, never ellipse
Now I did 3 Experiments and 3 models of the problem, but it turns out
that one model is superior over all the other models. One model is the
best of all.
That model is where you construct a cone and a cylinder and then
implant a circle inside the cone and cylinder attached to a handle so
that you can rotate the circle inside. Mine uses a long nail that I
poked holes into the side of a cylinder and another one inside a cone
made from heavy wax paper of magazine covers. And I used a Mason or
Kerr used lid and I attached them to the nail by drilling two holes
into each lid and running a wire as fastener. All of this done so I
can rotate or pivot the circle inside the cylinder and cone. You need
a long nail, for if you make the models too small or too skinny, you
lose clarity.
ARRAY, Analytic Geometry Proof, Cylinder Section is a Ellipse::
E
__
.-' `-.
.' `.
/ \
; ;
| G c | H
; ;
\ /
`. .'
`-. _____ .-'
F
The above is a view of a ellipse with center c and is produced by the
Sectioning of a Cylinder as long as the cut is not perpendicular to
the base, and as long as the cut involves two points not larger than
the height of the cylinder walls. What we want to prove is that the
cut is always a ellipse, which is a plane figure of two axes of
symmetry with a Major Axis and Minor Axis and center at c.
Side view of Cylinder EGFH above with entry point cut at E and exit
point cut at F and where c denotes the central axis of the cylinder
and where x denotes a circle at c parallel with the base-circle of
cylinder
| |
| | E
| |
| |
|x c |x
| |
| |
| |
|F |
| |
| |
| |
So, what is the proof that figure EGFH is always an ellipse in the
cylinder section? The line segment GH is the diameter of the circle
base of cylinder and the cylinder axis cuts this diameter in half such
that Gc = cH. Now we only need to show that Fc = cE. This is done from
the right triangles cxF and cxE, for we note that by Angle-Side-Angle
these two right triangles are congruent and hence Fc = cE, our second
axis of symmetry and thus figure EGFH is always an ellipse. QED
Array proof:: Analytic Geometry proof that Conic section= Oval// never ellipse
ARRAY, Analytic Geometry Proof, Conic Section is a Oval, never an ellipse::
A
,'" "`.
/ \
C | c | D
\ /
` . ___ .'
B
The above is a view of a figure formed from the cut of a conic with
center c as the axis of the cone and is produced by the Sectioning of
a Cone as long as the cut is not perpendicular to the base, and as
long as the cut is not a hyperbola, parabola or circle (nor line).
What we want to prove is that this cut is always a oval, never an
ellipse. An oval is defined as a plane figure of just one axis of
symmetry and possessing a center, c, with a Major Diameter as the axis
of symmetry and a Minor Diameter. In our diagram above, the major
diameter is AB and minor diameter is CD.
Alright, almost the same as with Cylinder section where we proved the
center was half way between Major Axis and Minor Axis of cylinder,
only in the case of the Conic, we find that the center is half way
between CD the Minor Diameter, but the center is not halfway in
between the Major Diameter, and all of that because of the reason the
slanted walls of the cone cause the distance cA to be far smaller than
the distance cB. In the diagram below we have the circle of x centered
at c and parallel to base. The angle at cx is not 90 degrees as in
cylinder. The angle of cAx is not the same as the angle cBx, as in the
case of the cylinder, because the walls of the cone-for line segments-
are slanted versus parallel in the cylinder. Triangles cAx and cBx are
not congruent, and thus, the distance of cA is not equal to cB,
leaving only one axis of symmetry AB, not CD.
/ \A
x/ c \x
B/ \
Hence, every cut in the Cone, not a hyperbola, not a parabola, not a
circle (not a line) is a Oval, never an ellipse.
QED
--Archimedes Plutonium
Very crude dot picture of 5f6, 94TH
ELECTRON=muon DOT CLOUD of 231Pu
::\ ::|:: /::
::\::|::/::
_ _
(:Y:)
- -
::/::|::\::
::/ ::|:: \::
One of those dots is the Milky Way galaxy. And each dot represents another galaxy.
. \ . . | . /.
. . \. . .|. . /. .
..\....|.../...
::\:::|::/::
--------------- -------------
--------------- (Y) -------------
--------------- --------------
::/:::|::\::
../....|...\...
. . /. . .|. . \. .
. / . . | . \ .
http://www.iw.net/~a_plutonium/ whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of spammers, off-topic-misfits, front-page-hogs, stalking mockers, suppression-bullies, and demonizers.
Read my recent posts in peace and quiet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium
CONIC SECTION IS OVAL, never an ellipse; (Proof below)
Before I do the formal proof, let us experiment first, where High School students can perform and prove in their own minds that ellipse is never a conic.
Conics = oval, 4 Experiments
4th experiment Re: -World's first proofs that the Conic section is an
Oval, never an ellipse// yes, Apollonius and Dandelin were wrong
by Archimedes Plutonium
1st EXPERIMENT:: Fold paper into cone and cylinder, (I prefer the waxy cover of a magazine). Try to make both about the same size, so the perspective is even. Now tape the cone and cylinder so they do not come undone in the scissor or paper cutter phase. A paper cutter is best but dangerous, so be careful, be very careful with paper cutter. Make the same angle of cut in each. and the best way of insuring that is to temporary staple the two together so the angle is the same. Once cut, remove the staples. Now we inspect the finished product. Hold each in turn on a sheet of paper and with a pencil trace out the figure on the flat piece of paper. Notice the cylinder gives an ellipse with 2 Axes of Symmetry, while the conic gives a oval because it has just one, yes 1 axis of symmetry.
That was my first experiment.and cylinders in a rush. Only fault I have of this experiment is that
it leaves a scissors mark-- a vertex so to speak. But it is fast and
easy. The proof is in the comparison. Now the cut should be at a steep
enough angle. If you cut straight across, both will be circles, so
make a steep cut.
2nd EXPERIMENT:: get a Kerr or Mason canning lid and repeat the above production of a cone and cylinder out of stiff waxy paper (magazine covers). Try to make the cone and cylinder about the same size as the lid. Now either observe with the lid inside the cone and cylinder, or, punch two holes in the cone and cylinder and fasten the lid inside. What you want to observe is how much area and where the area is added to make a section. So that in the cylinder, there is equal amount of area to add upwards as to add downwards of the lid, but in the cone, the area upwards added is small, while the area added downwards is huge new area. Thus the cylinder had two axes of symmetry and is an ellipse, while cone is 1 axis of symmetry and is an oval.
of an axis, where the upward needs little area to fill in any gap and
the downward needs an entire "crescent shaped area add-on to the
circle lid.
3rd EXPERIMENT:: Basically this is a repeat of the Dandelin fake proof, only we use a cylinder. Some tennis balls or ping pong balls come in see through plastic cylinder containers. And here you need just two balls in the container and you cut out some cardboard in the shape of ellipse that fits inside the container. You will be cutting many different sizes of these ellipses and estimating their foci. Now you insert these ellipse and watch to see the balls come in contact with the foci. Now, you build several cones in which the ellipses should fit snugly. Trouble is, well, there is never a cone that any ellipse can fit inside, for only an oval fits inside the cones.
materials. It is just a repeat of the Dandelin work on this topic, and
one can easily see how the Dandelin fake proof is constructed-- he
starts off with assuming the figure is an ellipse. Which tells us, he
never had a good-working-model if any at all. For you cannot stuff a
ellipse inside a cone. You can stuff a ellipse inside a cylinder. So
this suggests the entire Dandelin nonsense was all worked out in the
head and never in hands on actual reality. So, in this experiment, we
give a proof that Dandelin was utterly wrong and that it is a cylinder
that you can stuff a ellipse sandwiched by two identical spheres-- one
upper and one lower.
The only amazing part of the Dandelin story is how an utterly fake
proof could have survived from 1822, and not until 2017 is it
thoroughly revealed as ignorant nonsense. One would think in math,
there is no chance such a hideously flawed proof could even be
published in a math journal, and if anything is learned from Dandelin,
is that the math journal publishing system is a whole entire garbage
network. A network that is corrupt and fans fakery.
4th EXPERIMENT:: this is a new one. And I have it resting on my coffee table at the moment and looking at it. It comes from a toy kit of plastic see through geometry figures, cost me about $5. And what I have is a square pyramid and a cone of about the same size. Both see through. And what I did was rest the square pyramid apex on top of the cone apex, so the cone is inside the square pyramid. Now I wish I had a rectangular box to fit a cylinder inside the box. But this toy kit did not have that, but no worries for the imagination can easily picture a cylinder inside a rectangular box. Now the experiment is real simple in that we imagine a Planar Cut into the rectangular box with cylinder inside and the cut will make a rectangle from the box and a ellipse from the cylinder. Now with the cut of the square pyramid that contains a cone inside, the square pyramid is a trapezoid section while the cone is a oval section. If the cut were parallel to the base, the square pyramid yields a square and the cone yields a circle. This experiment proves to all the dunces, the many dunces who think a conic section is an ellipse, that it cannot be an ellipse, for obviously, a cone is not the same as a cylinder.
Now this 4th Experiment is a delicious fascinating experiment, for itreveals to us another proof that the conic section is a oval. For the
square-pyramid section is a Isosceles Trapezoid, and what is so great
about that, is we can take a cone and place inside of the cone a
square pyramid and then place a second square pyramid over the cone,
so the cone is sandwiched in between two square pyramids.
Now the square pyramids are tangent to the cone at 4 line segments, 8
altogether for the two, and what is so intriguing about the tangents
is that it allows us to quickly develop a analytic geometry that the
cone section must be a oval in order for the two square pyramids to be
both isosceles trapezoids as sections.
Archimedes Plutonium
--------------------
Conics = oval, 2 proofs, synthetic, analytic
Synthetic Geometry & Analytical Geometry Proofs that Conic section =
Oval, never an ellipse-- World's first proofs thereof
by Archimedes Plutonium
_Synthetic Geometry proofs that Cylinder section= Ellipse// Conic section= Oval
First Synthetic Geometry proofs, later the Analytic Geometry proofs.
Alright I need to get this prepared for the MATH ARRAY of proofs, that
the Ellipse is a Cylinder section, and that the Conic section is an
oval, never an ellipse
PROOF that Cylinder Section is an Ellipse, never a Oval::
I would have proven it by Symmetry. Where I indulge the reader to
place a circle inside the cylinder and have it mounted on a swivel, a
tiny rod fastened to the circle so that you can pivot and rotate the
circle. Then my proof argument would be to say--when the circle plate
is parallel with base, it is a circle but rotate it slightly in the
cylinder and determine what figure is produced. When rotated at the
diameter, the extra area added to the upper portion equals the extra
area added to bottom portion in cylinder, symmetrical area added,
hence a ellipse. QED
Now for proof that the Conic section cannot be an ellipse but an oval,
I again would apply the same proof argument by symmetry.
Proof:: Take a cone in general, and build a circle that rotates on a
axis. Rotate the circle just a tiny bit for it is bound to get stuck
or impeded by the upward slanted walls of the cone. Rotate as far as
you possibly can. Now filling in the area upwards is far smaller than
filling in the area downwards. Hence, only 1 axis of symmetry, not 2
axes of symmetry. Define Oval as having 1 axis of symmetry. Thus a
oval, never an ellipse. QED
The above two proofs are Synthetic Geometry proofs, which means they
need no numbers, just some concepts and axioms to make the proof work.
A Synthetic geometry proof is where you need no numbers, no coordinate
points, no arithmetic, but just using concepts and axioms. A Analytic
Geometry proof is where numbers are involved, if only just coordinate
points.
Array:: Analytic Geometry proof that Cylinder section= Ellipse//Conic
section = Oval, never ellipse
Now I did 3 Experiments and 3 models of the problem, but it turns out
that one model is superior over all the other models. One model is the
best of all.
That model is where you construct a cone and a cylinder and then
implant a circle inside the cone and cylinder attached to a handle so
that you can rotate the circle inside. Mine uses a long nail that I
poked holes into the side of a cylinder and another one inside a cone
made from heavy wax paper of magazine covers. And I used a Mason or
Kerr used lid and I attached them to the nail by drilling two holes
into each lid and running a wire as fastener. All of this done so I
can rotate or pivot the circle inside the cylinder and cone. You need
a long nail, for if you make the models too small or too skinny, you
lose clarity.
ARRAY, Analytic Geometry Proof, Cylinder Section is a Ellipse::
E
__
.-' `-.
.' `.
/ \
; ;
| G c | H
; ;
\ /
`. .'
`-. _____ .-'
F
The above is a view of a ellipse with center c and is produced by the
Sectioning of a Cylinder as long as the cut is not perpendicular to
the base, and as long as the cut involves two points not larger than
the height of the cylinder walls. What we want to prove is that the
cut is always a ellipse, which is a plane figure of two axes of
symmetry with a Major Axis and Minor Axis and center at c.
Side view of Cylinder EGFH above with entry point cut at E and exit
point cut at F and where c denotes the central axis of the cylinder
and where x denotes a circle at c parallel with the base-circle of
cylinder
| |
| | E
| |
| |
|x c |x
| |
| |
| |
|F |
| |
| |
| |
So, what is the proof that figure EGFH is always an ellipse in the
cylinder section? The line segment GH is the diameter of the circle
base of cylinder and the cylinder axis cuts this diameter in half such
that Gc = cH. Now we only need to show that Fc = cE. This is done from
the right triangles cxF and cxE, for we note that by Angle-Side-Angle
these two right triangles are congruent and hence Fc = cE, our second
axis of symmetry and thus figure EGFH is always an ellipse. QED
Array proof:: Analytic Geometry proof that Conic section= Oval// never ellipse
ARRAY, Analytic Geometry Proof, Conic Section is a Oval, never an ellipse::
A
,'" "`.
/ \
C | c | D
\ /
` . ___ .'
B
The above is a view of a figure formed from the cut of a conic with
center c as the axis of the cone and is produced by the Sectioning of
a Cone as long as the cut is not perpendicular to the base, and as
long as the cut is not a hyperbola, parabola or circle (nor line).
What we want to prove is that this cut is always a oval, never an
ellipse. An oval is defined as a plane figure of just one axis of
symmetry and possessing a center, c, with a Major Diameter as the axis
of symmetry and a Minor Diameter. In our diagram above, the major
diameter is AB and minor diameter is CD.
Alright, almost the same as with Cylinder section where we proved the
center was half way between Major Axis and Minor Axis of cylinder,
only in the case of the Conic, we find that the center is half way
between CD the Minor Diameter, but the center is not halfway in
between the Major Diameter, and all of that because of the reason the
slanted walls of the cone cause the distance cA to be far smaller than
the distance cB. In the diagram below we have the circle of x centered
at c and parallel to base. The angle at cx is not 90 degrees as in
cylinder. The angle of cAx is not the same as the angle cBx, as in the
case of the cylinder, because the walls of the cone-for line segments-
are slanted versus parallel in the cylinder. Triangles cAx and cBx are
not congruent, and thus, the distance of cA is not equal to cB,
leaving only one axis of symmetry AB, not CD.
/ \A
x/ c \x
B/ \
Hence, every cut in the Cone, not a hyperbola, not a parabola, not a
circle (not a line) is a Oval, never an ellipse.
QED
--Archimedes Plutonium
Very crude dot picture of 5f6, 94TH
ELECTRON=muon DOT CLOUD of 231Pu
::\ ::|:: /::
::\::|::/::
_ _
(:Y:)
- -
::/::|::\::
::/ ::|:: \::
One of those dots is the Milky Way galaxy. And each dot represents another galaxy.
. \ . . | . /.
. . \. . .|. . /. .
..\....|.../...
::\:::|::/::
--------------- -------------
--------------- (Y) -------------
--------------- --------------
::/:::|::\::
../....|...\...
. . /. . .|. . \. .
. / . . | . \ .
http://www.iw.net/~a_plutonium/ whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies
I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of spammers, off-topic-misfits, front-page-hogs, stalking mockers, suppression-bullies, and demonizers.
Read my recent posts in peace and quiet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium