Archimedes Plutonium
2017-06-14 14:59:47 UTC
Need to include modern way of proving a set is infinite.
....,
....,
Fundamental Theorem of Arithmetic
Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.
Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.
....,
Comment:: Here we have the fine example of where the proof statement is smaller in length than the theorem-statement. But it only goes to show, that if you have a valid proof, you cut away the fat, and that is ironic since FTArithmetic is UPFAT, unique prime factorization theorem.
....,
Fundamental Theorem of Calculus
...,
Statement of Fundamental Theorem of Calculus: The integral of calculus is the area under the function graph, and the derivative of calculus is the dy/dx of a point (x_1, y_1) to the next successor point (x_2, y_2). The integral is area of the rectangle involved with (x_1, y_1) and (x_2, y_2), and the derivative is the dy/dx of y_2 - y_1 / x_2 - x_1. Prove that the integral and derivative are inverses of one another, meaning that given one, you can get the other, they are reversible.
....,
.....,
Proof of FTC::
From this:
/|
/ |
/----|
/ |
/ |
_____
To this:
______
| |
| |
| |
----------
You can always go from a trapezoid with slanted roof to being a rectangle, by merely a midpoint that etches out a right triangle tip which when folded down becomes a rectangle.
....,
Comment::
Here we see that a geometry diagram is ample proof of Fundamental Theorem of Calculus. And a geometry diagram is preferred for its simplicity and brevity, as the Pythagorean theorem started to do with mathematics in Ancient times. Other pieces and parts of the proof are scattered among the 10^60 facts and data of the Array of math. Here we show that a Statement is proven by a kernel of math knowledge-- you take a rectangle and procure a right triangle from the midpoint of the width and form a trapezoid for derivative and then reform back to a rectangle for integral. The derivative and integral pivot back and forth on a hinge at the midpoint of the rectangle width. The two operatives of integration and differentiation are reversible operators.
.....,
.....,
Fundamental Theorem of Algebra
...,
Old Math statement of FTA:: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number A is a complex number with an imaginary part equal to zero.
....,
Comment:: New Math statement theorem of FTA need not be so long, because in New Math we recognize that sqrt-1 is simply A/0 where A is a real-number (New Real Number).
Theorem-Statement of FTA:: given any polynomial with New Real Numbers, that there always exists at least one New Real Number, call it A, as a solution.
Proof of FTA: x^2 +1 = 0, goes to 1/x^2 = -1, with solution x=0 as 1/0 = sqrt-1, thus any polynomial is 1/polynomial = +-k has at least, one solution for A.
Comment:: Notice the beauty of the Statement of Theorem then Proof are almost identical in length of words. This is what happens when you have a true proof of a statement in mathematics.
....,
S_m
S_m+1
....,
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.
.....,
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.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium
On Monday, June 5, 2017 at 1:42:51 AM UTC-5, Archimedes Plutonium wrote:
Correction to Only 5 Regular Polyhedra proof
....,
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. The vertex angle summation has to be always less than 360 degrees and always have more than two regular polygons participating.
Otherwise, the surfaces would lie flat or disassemble or not form a solid in third dimension.
AP
Newsgroups: sci.math
Date: Mon, 5 Jun 2017 01:46:17 -0700 (PDT)
Subject: Correction to Only 5 Regular Polyhedra proof
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Mon, 05 Jun 2017 08:46:18 +0000
Correction to Only 5 Regular Polyhedra proof
...,
...,
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.
Proof Statement:: 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. We ask what is preventing odd deficient numbers from becoming perfect. We take any odd deficient and sum its factors and place the sum in the m grouping 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.
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.
AP
2, ARRAY of Math, governed by Conservation Principle
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.
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.
....,
....,
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
Democritus, Abdera Thrace, 460 - 370 BC Atomic theory
Theaetetus Athens 417- 369 BC
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.
.....,
Theorem Statement:: Angle Sum of Triangle interior angles is 180 degrees.
.....,
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.
.....,
......,
Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.
Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.
....,
...,
...,
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.
....,
....,
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.
....,
S_k
S_k+1
....,
....,
Comments: The Fundamental theorems were entered in earlier in the Array in historical timeline. But here we repeat them as bunched together for it may spur new ideas.
Fundamental Theorem of Arithmetic
Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.
Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.
....,
Comment:: Here we have the fine example of where the proof statement is smaller in length than the theorem-statement. But it only goes to show, that if you have a valid proof, you cut away the fat, and that is ironic since FTArithmetic is UPFAT, unique prime factorization theorem.
....,
Fundamental Theorem of Calculus
...,
Statement of Fundamental Theorem of Calculus: The integral of calculus is the area under the function graph, and the derivative of calculus is the dy/dx of a point (x_1, y_1) to the next successor point (x_2, y_2). The integral is area of the rectangle involved with (x_1, y_1) and (x_2, y_2), and the derivative is the dy/dx of y_2 - y_1 / x_2 - x_1. Prove that the integral and derivative are inverses of one another, meaning that given one, you can get the other, they are reversible.
....,
.....,
Proof of FTC::
From this:
/|
/ |
/----|
/ |
/ |
_____
To this:
______
| |
| |
| |
----------
You can always go from a trapezoid with slanted roof to being a rectangle, by merely a midpoint that etches out a right triangle tip which when folded down becomes a rectangle.
....,
Comment::
Here we see that a geometry diagram is ample proof of Fundamental Theorem of Calculus. And a geometry diagram is preferred for its simplicity and brevity, as the Pythagorean theorem started to do with mathematics in Ancient times. Other pieces and parts of the proof are scattered among the 10^60 facts and data of the Array of math. Here we show that a Statement is proven by a kernel of math knowledge-- you take a rectangle and procure a right triangle from the midpoint of the width and form a trapezoid for derivative and then reform back to a rectangle for integral. The derivative and integral pivot back and forth on a hinge at the midpoint of the rectangle width. The two operatives of integration and differentiation are reversible operators.
.....,
.....,
Fundamental Theorem of Algebra
...,
Old Math statement of FTA:: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number A is a complex number with an imaginary part equal to zero.
....,
Comment:: New Math statement theorem of FTA need not be so long, because in New Math we recognize that sqrt-1 is simply A/0 where A is a real-number (New Real Number).
Theorem-Statement of FTA:: given any polynomial with New Real Numbers, that there always exists at least one New Real Number, call it A, as a solution.
Proof of FTA: x^2 +1 = 0, goes to 1/x^2 = -1, with solution x=0 as 1/0 = sqrt-1, thus any polynomial is 1/polynomial = +-k has at least, one solution for A.
Comment:: Notice the beauty of the Statement of Theorem then Proof are almost identical in length of words. This is what happens when you have a true proof of a statement in mathematics.
....,
S_m
S_m+1
....,
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.
.....,
....,
....,
Fundamental Theorem of Arithmetic
Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.
Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.
....,
Comment:: Here we have the fine example of where the proof statement is smaller in length than the theorem-statement. But it only goes to show, that if you have a valid proof, you cut away the fat, and that is ironic since FTArithmetic is UPFAT, unique prime factorization theorem.
....,
Fundamental Theorem of Calculus
...,
Statement of Fundamental Theorem of Calculus: The integral of calculus is the area under the function graph, and the derivative of calculus is the dy/dx of a point (x_1, y_1) to the next successor point (x_2, y_2). The integral is area of the rectangle involved with (x_1, y_1) and (x_2, y_2), and the derivative is the dy/dx of y_2 - y_1 / x_2 - x_1. Prove that the integral and derivative are inverses of one another, meaning that given one, you can get the other, they are reversible.
....,
.....,
Proof of FTC::
From this:
/|
/ |
/----|
/ |
/ |
_____
To this:
______
| |
| |
| |
----------
You can always go from a trapezoid with slanted roof to being a rectangle, by merely a midpoint that etches out a right triangle tip which when folded down becomes a rectangle.
....,
Comment::
Here we see that a geometry diagram is ample proof of Fundamental Theorem of Calculus. And a geometry diagram is preferred for its simplicity and brevity, as the Pythagorean theorem started to do with mathematics in Ancient times. Other pieces and parts of the proof are scattered among the 10^60 facts and data of the Array of math. Here we show that a Statement is proven by a kernel of math knowledge-- you take a rectangle and procure a right triangle from the midpoint of the width and form a trapezoid for derivative and then reform back to a rectangle for integral. The derivative and integral pivot back and forth on a hinge at the midpoint of the rectangle width. The two operatives of integration and differentiation are reversible operators.
.....,
.....,
Fundamental Theorem of Algebra
...,
Old Math statement of FTA:: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number A is a complex number with an imaginary part equal to zero.
....,
Comment:: New Math statement theorem of FTA need not be so long, because in New Math we recognize that sqrt-1 is simply A/0 where A is a real-number (New Real Number).
Theorem-Statement of FTA:: given any polynomial with New Real Numbers, that there always exists at least one New Real Number, call it A, as a solution.
Proof of FTA: x^2 +1 = 0, goes to 1/x^2 = -1, with solution x=0 as 1/0 = sqrt-1, thus any polynomial is 1/polynomial = +-k has at least, one solution for A.
Comment:: Notice the beauty of the Statement of Theorem then Proof are almost identical in length of words. This is what happens when you have a true proof of a statement in mathematics.
....,
S_m
S_m+1
....,
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.
.....,
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.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium
On Monday, June 5, 2017 at 1:42:51 AM UTC-5, Archimedes Plutonium wrote:
Correction to Only 5 Regular Polyhedra proof
(snip)
Something on the order of this:: the vertex angle summation has to be always less than 360 degrees and always have more than two regular polygons participating.
....,The plane can be tiled (as can vertices) with hexagons (vertex angles are 60), but not pentagons (vertex angles are 108, I believe).
Yes I need another sentence in the proof explaining why the vertex angles have to be less than 360 degrees in summation. So that hexagons and even equilateral triangles cannot tile 360 and above.Something on the order of this:: the vertex angle summation has to be always less than 360 degrees and always have more than two regular polygons participating.
....,
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. The vertex angle summation has to be always less than 360 degrees and always have more than two regular polygons participating.
Otherwise, the surfaces would lie flat or disassemble or not form a solid in third dimension.
AP
Newsgroups: sci.math
Date: Mon, 5 Jun 2017 01:46:17 -0700 (PDT)
Subject: Correction to Only 5 Regular Polyhedra proof
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Mon, 05 Jun 2017 08:46:18 +0000
Correction to Only 5 Regular Polyhedra proof
...,
...,
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.
Proof Statement:: 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. We ask what is preventing odd deficient numbers from becoming perfect. We take any odd deficient and sum its factors and place the sum in the m grouping 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.
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.
AP
2, ARRAY of Math, governed by Conservation Principle
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.
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.
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Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have
a^2 +b^2 = c^2
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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
Democritus, Abdera Thrace, 460 - 370 BC Atomic theory
Theaetetus Athens 417- 369 BC
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.
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Theorem Statement:: Angle Sum of Triangle interior angles is 180 degrees.
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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.
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Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.
Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.
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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.
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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.
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Comments: The Fundamental theorems were entered in earlier in the Array in historical timeline. But here we repeat them as bunched together for it may spur new ideas.
Fundamental Theorem of Arithmetic
Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.
Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.
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Comment:: Here we have the fine example of where the proof statement is smaller in length than the theorem-statement. But it only goes to show, that if you have a valid proof, you cut away the fat, and that is ironic since FTArithmetic is UPFAT, unique prime factorization theorem.
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Fundamental Theorem of Calculus
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Statement of Fundamental Theorem of Calculus: The integral of calculus is the area under the function graph, and the derivative of calculus is the dy/dx of a point (x_1, y_1) to the next successor point (x_2, y_2). The integral is area of the rectangle involved with (x_1, y_1) and (x_2, y_2), and the derivative is the dy/dx of y_2 - y_1 / x_2 - x_1. Prove that the integral and derivative are inverses of one another, meaning that given one, you can get the other, they are reversible.
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Proof of FTC::
From this:
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To this:
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You can always go from a trapezoid with slanted roof to being a rectangle, by merely a midpoint that etches out a right triangle tip which when folded down becomes a rectangle.
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Comment::
Here we see that a geometry diagram is ample proof of Fundamental Theorem of Calculus. And a geometry diagram is preferred for its simplicity and brevity, as the Pythagorean theorem started to do with mathematics in Ancient times. Other pieces and parts of the proof are scattered among the 10^60 facts and data of the Array of math. Here we show that a Statement is proven by a kernel of math knowledge-- you take a rectangle and procure a right triangle from the midpoint of the width and form a trapezoid for derivative and then reform back to a rectangle for integral. The derivative and integral pivot back and forth on a hinge at the midpoint of the rectangle width. The two operatives of integration and differentiation are reversible operators.
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Fundamental Theorem of Algebra
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Old Math statement of FTA:: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number A is a complex number with an imaginary part equal to zero.
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Comment:: New Math statement theorem of FTA need not be so long, because in New Math we recognize that sqrt-1 is simply A/0 where A is a real-number (New Real Number).
Theorem-Statement of FTA:: given any polynomial with New Real Numbers, that there always exists at least one New Real Number, call it A, as a solution.
Proof of FTA: x^2 +1 = 0, goes to 1/x^2 = -1, with solution x=0 as 1/0 = sqrt-1, thus any polynomial is 1/polynomial = +-k has at least, one solution for A.
Comment:: Notice the beauty of the Statement of Theorem then Proof are almost identical in length of words. This is what happens when you have a true proof of a statement in mathematics.
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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.
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