Discussion:
1ARRAYS of math using Conservation Principle of Proof// There can Only Exist 5 Regular Polyhedra
Archimedes Plutonium
2017-06-14 14:53:39 UTC
Raw Message
Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY

ARRAY of MATHEMATICS, governed by Conservation of Proof Principle

Each line is a data, or fact or theorem of mathematics. Some lines are Comments

Comment_p_i::

Comment_p_j::

Fact_k::

Statement_r_u::

Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.

....,

....,

S_i

S_i+1

....,

....,

....,

S_j

S_j+1

....,

Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2

....,

....,

Proof of PT::

Picture proof

a             b
L        1                    K
c
c
2

4
c               c

I                  3            J
b                  a

then

L                 a           K

b                 b            a

I                  b            J

Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.

Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter

Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.

Proof-Statement:: Connect PB thus two triangles ABP and PBC. Since AP, PB, PC are all equal as radii, the two triangles are isosceles. The base angles are equal. Picture::

B
v  u
A  v       P       u  C

v + (v+u) + u = 180
2v + 2u = 180
v+u = 90

Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.

Data:: A history timeline goes like this::

Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.

Pythagoras of Samos 570-495 BC

Theaetetus Athens 417- 369 BC

Democritus, Abdera Thrace, 460 - 370 BC Atomic theory

Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron

Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.

Archimedes 287 - 212 BC

(sources mostly Wikipedia)

According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.

Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,

Statement-Theorem:: There can Only Exist 5 Regular Polyhedra

Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.

....,

S_k

S_k+1

....,

....,
Fundamental Theorem of Arithmetic

Fundamental Theorem of Calculus

Fundamental Theorem of Algebra

Comment:: now the order of the ARRAY is desirable to be of a history order. But, in many cases, we can lump and repeat statements. Remember, the ARRAY is going to be huge. I am talking of volumes that would fill a entire library, just on math ARRAY. And of course, in our computer age, we have the ARRAY accessible by computer.
.....,

AP

ARRAY of MATHEMATICS, governed by Conservation of Proof Principle

Each line is a data, or fact or theorem of mathematics. Some lines are Comments

Comment_p_i::

Comment_p_j::

Fact_k::

Statement_r_u::

Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.

....,

....,

S_i

S_i+1

....,

....,

....,

S_j

S_j+1

....,

Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2

....,

....,

Proof of PT::

Picture proof

a             b
L        1                    K
c
c
2

4
c               c

I                  3            J
b                  a

then

L                 a           K

b                 b            a

I                  b            J

Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.

Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter

Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.

Proof-Statement:: Connect PB thus two triangles ABP and PBC. Since AP, PB, PC are all equal as radii, the two triangles are isosceles. The base angles are equal. Picture::

B
v  u
A  v       P       u  C

v + (v+u) + u = 180
2v + 2u = 180
v+u = 90

Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.

Data:: A history timeline goes like this::

Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.

Pythagoras of Samos 570-495 BC

Theaetetus Athens 417- 369 BC

Democritus, Abdera Thrace, 460 - 370 BC Atomic theory

Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron

Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.

Archimedes 287 - 212 BC

(sources mostly Wikipedia)

According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.

Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,

Statement-Theorem:: There can Only Exist 5 Regular Polyhedra

Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.

....,

S_k

S_k+1
b***@gmail.com
2017-06-14 15:27:45 UTC
Raw Message
repast alarm, a**hole spammer AP does it again.

Post by Archimedes Plutonium
Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY
ARRAY of MATHEMATICS, governed by Conservation of Proof Principle
Each line is a data, or fact or theorem of mathematics. Some lines are Comments
Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.
....,
....,
S_i
S_i+1
....,
....,
....,
S_j
S_j+1
....,
Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2
....,
....,
Picture proof
a             b
L        1                    K
c
c
2
4
c               c
I                  3            J
b                  a
then
L                 a           K

b                 b            a
I                  b            J
Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.
Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter
Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.
B
v  u
A  v       P       u  C
v + (v+u) + u = 180
2v + 2u = 180
v+u = 90
Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.
Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.
Pythagoras of Samos 570-495 BC
Theaetetus Athens 417- 369 BC
Democritus, Abdera Thrace, 460 - 370 BC Atomic theory
Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron
Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.
Archimedes 287 - 212 BC
(sources mostly Wikipedia)
According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.
Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,
Statement-Theorem:: There can Only Exist 5 Regular Polyhedra
Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.
....,
S_k
S_k+1
....,
....,
Fundamental Theorem of Arithmetic
Fundamental Theorem of Calculus
Fundamental Theorem of Algebra
Comment:: now the order of the ARRAY is desirable to be of a history order. But, in many cases, we can lump and repeat statements. Remember, the ARRAY is going to be huge. I am talking of volumes that would fill a entire library, just on math ARRAY. And of course, in our computer age, we have the ARRAY accessible by computer.
.....,
AP
ARRAY of MATHEMATICS, governed by Conservation of Proof Principle
Each line is a data, or fact or theorem of mathematics. Some lines are Comments
Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.
....,
....,
S_i
S_i+1
....,
....,
....,
S_j
S_j+1
....,
Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2
....,
....,
Picture proof
a             b
L        1                    K
c
c
2
4
c               c
I                  3            J
b                  a
then
L                 a           K

b                 b            a
I                  b            J
Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.
Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter
Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.
B
v  u
A  v       P       u  C
v + (v+u) + u = 180
2v + 2u = 180
v+u = 90
Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.
Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.
Pythagoras of Samos 570-495 BC
Theaetetus Athens 417- 369 BC
Democritus, Abdera Thrace, 460 - 370 BC Atomic theory
Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron
Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.
Archimedes 287 - 212 BC
(sources mostly Wikipedia)
According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.
Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,
Statement-Theorem:: There can Only Exist 5 Regular Polyhedra
Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.
....,
S_k
S_k+1
Archimedes Plutonium
2017-06-14 18:32:16 UTC
Raw Message
Data:: A history timeline goes like this::

Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with the concept of arche-- water.

Pythagoras of Samos 570-495 BC

Theaetetus Athens 417- 369 BC

Democritus, Abdera Thrace, 460 - 370 BC Atomic theory

Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-aether as a dodecahedron

Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding-principle.

Archimedes 287 - 212 BC

(sources mostly Wikipedia)

According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.

Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,

....,

Statement-Theorem:: There can Only Exist 5 Regular Polyhedra-- tetrahedron, octahedron, cube, icosahedron, dodecahedron. A Regular-Polyhedron is a 3rd dimensional object with surfaces of regular-polygons. All of its surfaces, edges and vertices are the same. Tetrahedron has 4 equilateral triangles, where 3 triangles meet at each vertex. The octahedron has 8 equilateral triangles, where 4 triangles meet at each vertex. The cube or hexahedron has 6 squares, where each vertex has 3 squares meeting.
The icosahedron has 20 surfaces of equilateral triangles and each vertex has 5 triangles meeting. And finally the dodecahedron has 12 surfaces of pentagons where each vertex has 3 pentagons meeting.

Proof-Statement ;; total angles at each vertex must be less than 360 degrees. If 360 or more the surfaces would either lie flat or disassemble. So you need a gap of space to fold from 2nd dimension into 3rd dimension, and you need at least three regular polygons to fold to make solid geometry and not a flap in 2nd dimension.
For Surfaces:
3 triangles at each vertex: Tetrahedron
4 triangles at each vertex: Octahedron
5 triangles at each vertex: Icosahedron
6 triangles at each vertex would be 360 degrees, no figure possible
3 squares at each vertex: Cube
4 squares at each vertex would be again 360 degrees, no figure possible
3 pentagons at each vertex is 3*108 degrees: Dodecahedron
4 pentagons at each vertex is 4*108 degrees and no way possible.

Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.

Fact:: Theaetetus proved Only 5 Regular Polyhedra Exist and Plato was so enamored of the regular polyhedra they are often called the Platonic Solids. Plato thought the 4 of the solids were the atoms of Democritus Atomic theory where tetrahedron was fire, cube was Earth, octahedron was air, and icosahedron was water. The dodecahedron was so special to Plato he made it the Universe.

Comment:: It would be fitting, if Plato made the Universe the dodecahedron, in other words God as atom as Dodecahedron, fitting that the final words in the Euclid Elements is words of God, the 5 Regular Polyhedra are the Only ones to exist.

.....,

.....,

...,

Theorem-statement:: the sqrt2 is irrational. By irrational is meant sqrt2 is never the ratio of two integers A and B, as A/B, where A and B are reduced in lowest form-- they share no common factor.

Proving Statement:: the existence of A/B = sqrt2 implies A^2/B^2 = 2. This means A^2 = 2B^2 means both are even and share a common factor, hence no A/B.

Comment:: AP would show that the fundamental meaning or irrational number is that it is two numbers involved and playing as one. So the number sqrt2 is the two numbers of 1.414 along with 1.415 in 1000 Grid.

Comment:: a useful contrast of proving No odd perfect, except 1, exists, alongside the proof of sqrt2 is irrational, is instructive since both use the same method of proof.

Theorem Statement:: Only 1 can be Odd Perfect and no odd number larger than 1 can be perfect. By perfect is meant all factors of the number in question except the number itself is added up and if equal to the number is thus perfect. A special note is that 1 is the only singleton factor while all other factors are cofactors. For example, 9 is 1+3+3, not 1+3. And we group the factors of given odd number k. We group the factors into two groups of a m/k and p/k where k is the odd number. So that if m+p = k then k is odd perfect. A deficient odd number is one in which m+p falls short of being k. The first odd deficient is 3 with 1/3 + 0/3, the later comes 9 with 3/9 + 4/9, later comes 15 with 5/15 + 4/15.

Proof Statement:: We ask what is preventing odd deficient numbers from becoming perfect, from reaching perfect condition. We look to see if there is a pattern that prevents a odd number from attaining perfect, so we set about constructing an odd perfect number. We take any odd deficient and sum its factors and place the sum fully in the m grouping, leaving the p grouping 0, so that m falls short of k and m is odd because of singleton 1. That means for every odd number which is short of being perfect must have a p value of a even number and hence k an odd number has a even number factor. So the barrier in constructing a odd perfect number is the fact that a even number is a factor in dividing a odd number-- impossible construction.

Comment:: notice how the proof that sqrt2 is irrational has a back and forth tussle with even and odd and the same goes for odd perfect proof.
Archimedes Plutonium
2017-06-15 05:56:17 UTC
Raw Message
Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY

ARRAY of MATHEMATICS, governed by Conservation of Proof Principle

Each line is a data, or fact or theorem of mathematics. Some lines are comments. The Array tries to be historical order of math, but need not be where some items need to be mentioned several times, such as this one.

Fundamental Theorem of Math Proofs
________________________________

Statement Theorem:: Fundamental Theorem of Math Proofs, FTMP. This theorem uses the principle of Conservation, much like the same principle known in Physics, where something is conserved, even though many things change and transform, a certain quantity or certain basic aspect stays the same. The aspect of a math proof, that is Conserved is that the theorem statement to be proven true, should be the same length or about the same length as the proof statement. Meaning that each and every proof in mathematics is about the same size and length as the theorem statement. Meaning that every proof has some engine or heart of the proof, for which we do not need to repeat all the known facts that lead up to this engine or heart of the proof. But rather we can skip all that data and information and focus in on the engine of the proof. So that a proof is composed of just a quick statement of the engine of the proof, what makes the proof work, in the Proof Statement.

Proof Statement of FTMP:: the science of Logic is bigger than mathematics and gives the rules in which mathematics takes place. Logic itself is part of Physics, and the order is Physics is first, gives rise to Logic gives rise, thirdly to mathematics. When logic is done, many statements are made and compose a Logic Array. All these statements in logic are connected by the logic connector, the AND, connector, and all those statements form a Logic Array. A proof in Logic picks and sorts and uses some statements forming another proven statement. These picked and sorted and used statements are not fixed to one theorem but used in many various theorems. So what is critical in a proof of Logic, is the bare essence of what makes the theorem true, and work. So, one special statement that is the engine or heart of the proof, is all that is needed for a specific proof, while the attending associated statements form the overall background of statements used in a proof and need not be mentioned over the course of a single proof. Thus, when you look at a Proof Statement, you can look at it as if it were a Theorem Statement and use the other statements in the Array to so to speak, prove the proof.

Comments:: In the entire history of mathematics and logic, humanity was rarely blessed with someone that can think clearly and straight. For example, by 2017, it is now known that an ellipse is never a conic section but that a oval is what was thought to be an ellipse. That not until recently was it known that a sine and cosine are not sinusoid figures but are merely semicircle waves. That only recently was it known that no angle beyond 360 degrees exist for no definition can of angle can go beyond 360 degrees, unless you want to lift 361 degrees out of 2nd dimension and belong in 3rd dimension. So the fact that in math history, rarely if ever, is there someone who has a mind of logic, and can say-- look, a math proof is not where you have a paragraph theorem statement followed by 100 pages for a proof-- that is not a math proof but a joke on common sense.

....,

Comment_p_i::

....,

.....,

Comment_p_j::

Fact_k::

.....,

......,

Statement_r_u::

Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.

....,

....,

S_i

S_i+1

....,

....,

....,

S_j

S_j+1

....,

Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2

....,

....,

Proof of PT::

Picture proof

a             b
L        1                    K
c
c
2

4
c               c

I                  3            J
b                  a

then

L                 a           K

b                 b            a

I                  b            J

Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.

.....,

Theorem Statement:: Angle Sum of Triangle interior angles is 180 degrees. Triangles are three sided closed polygons with 3 interior angles.

Proof Statement:: Given any triangle ABC

1 C 2
3

A 1 2 B

Construct a line at C parallel to AB, producing transverse angles 1 with 1 and 2 with 2, and knowing a straight line is 180 degrees, we have proved the claim.

Comment:: Now that was one of the very earliest math proofs in math history and we can see a flavor of the idea that a proof statement is not much longer in size than the theorem statement.

Comment:: by the year 2017 when the full nature of irrational numbers was widely known, that an irrational angle vibrates between two different Rational numbers, that a Triangle Angle Sum can be off of 180 degrees, say 179.9 degrees vibrating with 180.1 degrees.

Comment:: Now let me see if i can offer a alternative proof to Angle Sum without using a parallel line than transversal angles as engine of proof.
Proof:: a rectangle has 4 sides, 4 right angles to sum to 360 degree. A triangle if reproduced into 2 of the same then joined forming a rectangle, hence interior angles of one sums to 180 degrees. Now there are interior angles in regular polygons and the formula that governs them is 180(n-2) degrees where n is the number of sides. So for a triangle we have 180(1) for square 180(2) for pentagon 180(3) degrees. But better we also prove the interior angles of regular polygons sum by adjoining a triangle to its given sides and arrive at the formula 180(n-2) where they get the -2 in the 180(n-2)

.....,

Comment:: the Angle Sum theorem is reportedly proven by the Pythagoreans as perhaps one of their few actual proofs given in a deductive method. I give it. And later I give my own new updated version that uses polygons rather than a parallel line argument.

.....,

Very crude dot picture of 5f6, 94TH
ELECTRON 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.

Archimedes Plutonium
konyberg
2017-06-14 20:12:33 UTC
Raw Message
Post by Archimedes Plutonium
Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY
ARRAY of MATHEMATICS, governed by Conservation of Proof Principle
Each line is a data, or fact or theorem of mathematics. Some lines are Comments
Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.
....,
....,
S_i
S_i+1
....,
....,
....,
S_j
S_j+1
....,
Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2
....,
....,
Picture proof
a             b
L        1                    K
c
c
2
4
c               c
I                  3            J
b                  a
then
L                 a           K

b                 b            a
I                  b            J
Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.
Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter
Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.
B
v  u
A  v       P       u  C
v + (v+u) + u = 180
2v + 2u = 180
v+u = 90
Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.
Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.
Pythagoras of Samos 570-495 BC
Theaetetus Athens 417- 369 BC
Democritus, Abdera Thrace, 460 - 370 BC Atomic theory
Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron
Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.
Archimedes 287 - 212 BC
(sources mostly Wikipedia)
According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.
Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,
Statement-Theorem:: There can Only Exist 5 Regular Polyhedra
Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.
....,
S_k
S_k+1
....,
....,
Fundamental Theorem of Arithmetic
Fundamental Theorem of Calculus
Fundamental Theorem of Algebra
Comment:: now the order of the ARRAY is desirable to be of a history order. But, in many cases, we can lump and repeat statements. Remember, the ARRAY is going to be huge. I am talking of volumes that would fill a entire library, just on math ARRAY. And of course, in our computer age, we have the ARRAY accessible by computer.
.....,
AP
ARRAY of MATHEMATICS, governed by Conservation of Proof Principle
Each line is a data, or fact or theorem of mathematics. Some lines are Comments
Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.
....,
....,
S_i
S_i+1
....,
....,
....,
S_j
S_j+1
....,
Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2
....,
....,
Picture proof
a             b
L        1                    K
c
c
2
4
c               c
I                  3            J
b                  a
then
L                 a           K

b                 b            a
I                  b            J
Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.
Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter
Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.
B
v  u
A  v       P       u  C
v + (v+u) + u = 180
2v + 2u = 180
v+u = 90
Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.
Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.
Pythagoras of Samos 570-495 BC
Theaetetus Athens 417- 369 BC
Democritus, Abdera Thrace, 460 - 370 BC Atomic theory
Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron
Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.
Archimedes 287 - 212 BC
(sources mostly Wikipedia)
According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.
Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.
....,
Statement-Theorem:: There can Only Exist 5 Regular Polyhedra
Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.
....,
S_k
S_k+1
Thale was a genius. However he didn't see all. He used the wrong angle for sentral. He should used the complimentary, and he could even proved the general rule.

Your struggle with the regular triangle? Try this

c^2 = a^2 + b^2 - 2*a*b*cos(C), where a, b, and c are equal.

And using this formula, all angles will have cos(angle) = 1/2

You will notice that there is no need for irrational numbers, because cos(60) = 1/2.