The Pythagorean  Inverse Square Law Connection (TPISC)
A MathspeedST Supplement
Reginald Brooks
brooksdesignps.net
Copyright © 2014, Reginald Brooks
First published 2014
Published by Brooks Design:535 NW 107^{th} Ave, Portland, OR 97229, USA
Email: inverse2ed@gmail.com
Phone: 5036461548
URL: http://www.brooksdesignps.net
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Permission is hereby granted for single copies to be made for personal, noncommercial use for students and teachers of schools, colleges and universities provided that: either the entire paper, including figures and tables, is kept intact; or, any extracts of the text, or figures or tables (in part or whole), be properly and visibly cited as to authorship and source.
Original illustrations/ Photographs by Reginald Brooks
ISBN [ISBN number]
There is a simple whole number (integer) matrix grid table upon — and within — that every possible whole number Pythagorean Triangle — a.k.a. Pythagorean Triple — can be placed, and proved. The Brooks Base Square  Inverse Square Law (BBSISL) matrix is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their nonPrimitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c^{2}=a^{2}+b^{2} — is built into the BBSISL matrix.
Not only does the Inverse Square Law describe, define and quantify our most important energies (and their expressions as force) — gravity, light, sound, electromagnetism… — the BBSISL matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate interconnection way beyond a simple, casual association has been revealed. The Dickson Method confirms, validates and provides insight into the generation of ALL Pythagorean Triples.
The Pythagorean Theorem and the Inverse Square Law share the same code. In this respect, the c^{2} of c^{2}=a^{2}+b^{2} equals the r^{2} of 1/r^{2} (ISL). Embedded within the BBSISL matrix — that is essentially based on the squares of whole numbers — are the Pythagorean Triples — whole number rightangle triangles of uneven legs whose area squares equal in sum that of their longer hypotenuse side. Numbers of inevitability.
The Pythagorean  Inverse Square Law Connection
(TPISC)
Introduction
The Pythagorean  Inverse Square Law Connection or “How the Pythagorean Theorem is intimately embedded within the The Architecture Of SpaceTime (TAOST) as described by the BBSISL matrix” is first and foremost the topic of this paper. It is a MathspeedST Supplement.
As one of the most influential mathematical descriptions of all time, there is no shortage of writings, descriptions and/or proofs of this most profound theorem.
The reason that this paper is being presented is precisely because the very nature of that geometric relationship of the Pythagorean Triangle — whereby the area of the square of the long side (hypotenuse) is equal to the sum of the squares of the two shorter sides (legs) — is so intimately related to the Inverse Square Law (ISL), as depicted in the BBSISL matrix (Brooks Base Square  Inverse Square Law matrix), that a cause and effect relationship between the two is unavoidable.
That the BBSISL matrix also provides perhaps the simplest, most intuitively obvious proof to the Theorem directly on the grid only supports the argument. Every possible Pythagorean Triple (PT), and its proof, is visually and mathematical present. The Dickson Method confirms this.
The Pythagorean  Inverse Square Law Connection will often be referenced as TPISC.
A NEW short INTRO has been added!
The brief history of the development of the BBSISL matrix is summarily listed below. For the full treatment, please go to
http://www.brooksdesignps.net/Reginald_Brooks/Code/Html/arthry5.htm where the individual papers are presented. You can find individual links to the key papers in Appendix A of this paper.
1985: L.U.F.E. (The Layman’s Unified Field Exposé) was published. The first seeds of the BBSISL planted.
1987: GoMAS (The Geometry of Music, Art and Structure, first edition) was published. The ISL informs it all!
200106: PIN (Pattern In Number) various papers on geometric number patterns, primes and the axial structure of DNA — ISL
200308: various papers on Quantum and Gravitational physics, including The Conservation of SpaceTime
2009: GoMAS, Part I — ISL
2012: GoMAC, Part II & III — ISL
200912: Brooks (Base) Square and The Inverse Square Law published — 200 BASIC BBSISL “Rules” on TAOST (The Architecture Of SpaceTime) and TCAOP (The Complete Absence Of Primes)
201113: Brooks (Base) Square Interactive Matrix — hands on ISL
2011: TAOST (The Architecture Of SpaceTime) — ISL
2012: Numbers of Inevitability — ISL
2013: AFPOP (A Fresh Piece Of Pi(e) and the √2, too…FractalFractalFractal) — more Archimedean Geometry to backup the ISL
201213: LightspeedST (Why is the speed of light constant?), a series of 8 papers, hundreds of images and 50 short videos ultimately recast into an interactive ebook published in the iBookstore. LightspeedST forms the complementary work to MathspeedST.
201314: MathspeedST (How can information be ubiquitous, effectively traveling FASTER than the speed of light?). This freely available interactive ebook brings together under one publication the Brooks (Base) SquareInverse Square Law, or BBSISL matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of testyourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISLderived mathematical basis of the matrix. Available for free in the iBookstore
The BBSISL matrix has shown remarkable interconnections to all forms of geometry and number theory including a section on the distribution of the nonPrimitive Pythagorean Triples. The Architecture Of SpaceTime as defined by the BBS matrix and the ISL is certainly one of boldest and farreaching claims. The TCAOP (The Complete Absence Of Primes) brought out the Periodic Table of Primes (PTOP) is a hidden gem. And now, we have every possible PT — PPT and nPPT — presenting itself and its proof on the BBSISL matrix.
The history, derivation, use and the hundreds of proofs of the Pythagorean Theorem would fill a library all onto itself. Classic proofs from Pythagoras, Euclid, da Vinci, Newton, Bhaskara, Einstein, Garfield, and the list goes on, are well documented — many with beautiful and clever animations. These proofs generally rely on rearrangement, similarity, or division of the triangle to reveal the equality of the areas formed. Algebraic and other forms of abstract mathematics can also do the trick.
Although there are a large number of references cited, here are three that will bring you up to speed as to terminology, proofs and their history:
1 .http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Pythag/pythag.html
2. http://www.cuttheknot.org/pythagoras/index.shtml
3. http://en.wikipedia.org/wiki/Pythagorean_theorem
The Pythagorean Triangle (PT) is rightangle triangle composed of whole number (integer) sides. The two shorter sides or legs, a and b, are of differing lengths, as is the longer or hypotenuse side. Note that other rightangle triangles with fractional sides follow the Pythagorean Theorem — but are not Pythagorean Triangles.
The PTs can be divided into two groups: Primitives and nonPrimitives. The Primitive PT (PPT) is one that is irreducible. The nonPrimitive (nPPT) is simply a multiple of a Primitive, e.i. a 6810 nPPT is simply a double of the 345 PPT.
The pervasive distribution of all the possible PTs (PPT and nPPT) is amazing — going to infinity. Quoting Professor Knott in the first reference above: ”Can any number be a side in some Pythagorean Triangle? is Yes!”
Brooks Base Square (BBS) is a matrix grid of the Inverse Square Law (ISL), referred to as simply the BBSISL matrix. Every number — integer — value on the matrix grid is made from the Prime Diagonal (PD). The PD is made from the squares of the Axial numbers (0), 1,2,3,… It divides the matrix into two symmetrical halves — the lower of which gets our attention solely for convenience of display.
The Inner Grid simply refers to that part of the matrix between the Axis and PD. It contains no even numbers not divisible by four. Take away the first Parallel Diagonal to the PD (3,5,7,…) and the Inner Grid — now referred to as the Strict Inner Grid — also does not contain any prime numbers. All other numbers are present.
Between the PD, Axis and Inner Grid, the BBSISL matrix can reveal remarkable, amazing, even mindblowing interconnected patterns — many that have peen presented in the BASICS of MathspeedST.
Included in those first 200 “Rules,” Rules 8186 reveal that over 50% (or more) of the entire matrix grid — a grid that is infinitely expandable — consists of the many multiples of the Pythagorean Triple (PT) triangle 345. The multiples, or siblings, of the Primitive are simply 2x, 3x, 4x, … that of the parent. The 121620 is the smallest PT to fit on the grid diagonally. All such multiples — and there are generations of them — line up their SIDE values (i.e. Legs and hypotenuse) along a given Diagonal, parallel or perpendicular, to the PD. These are their “SIDE” values as opposed to the the “AREA” values formed from the SIDES values.
New in the MathspeedST Supplement are the PT “AREA” values of every possible Primitive (PPT) or nonPrimitive (nPPT) triangle laid out on the matrix with c^{2} at the PD and a^{2} and b^{2} on the ROW leading to it!
TPISC icon
Before delving into the specifics, the major takeaway on TPISC can be found right in the icon/logo used to identify the MathspeedST Supplement.
Within this simplified 5x5 BBSISL matrix grid we have the Prime Diagonal (PD) down the center. The square Areas are marked 1,4,9,16 and 25.
The first PPT has Sides 345 as abc and their Sides squared (Areas) as a^{2}b^{2}c^{2} or 91625. The latter show up in two places on the grid.
First, in the actual PD Square Areas, we can see 3^{2}=9—4^{2}=16—5^{2}=25.
Notice that the Areas are stacked on top of each other like layers. Each Area starts at 0. Also notice that the 9 little squares of Area 3^{2}=9 (a^{2}), when added to the 16 little squares of Area 4^{2}=16 (b^{2}), the resulting sum is 25 little squares of Area 5^{2}=25 (c^{2}), or the Pythagorean Theorem of a^{2}=b^{2}=c^{2}. C^{2}=a^{2}=b^{2} is visually proved right on the BBSISL matrix.
Secondly, the same AREA information shows up directly on Axis Row 5. Remember, 5 is c=hypotenuse and c^{2}=25 is found along Axis Row 5 at the PD. The a^{2}=3^{2}=9 and b^{2}=4^{2}=16 AREAS are found on the same Axis Row 5.
This scenario is true for ALL PTs and they — and their proofs — are ALL found on the BBSISL matrix.
Finally, you might ask, just what determines the actual placement — what determines the actual Primitive Pythagorean Triangle (PPT) and subsequently what rows do these primordial triangles occupy? What comes first, the location or the identity, or are they established as one? The childrensibling nonPrimitive Pythagorean Triangle (nPPT) multiples are simply that — 2x, 3x, 4x, …of their parent Primitives. But what determines PPT values? This big, big question is addressed (no pun!) toward the end of the paper, but here is a hint:
If all quantities are enumerated by some counting scheme — whether it is counting on your fingers (BASE_10), your computer (BASE_2), or something else or between (BASE_3,_4,_5_6_8) — if a pattern emerges consistently between all such BASES and the quantities that they represent, surely this represents some deeper connection — an interconnection — between the two, the describer and the described. The PPT seems to have this connection with the ISL as laid out in the BBSISL matrix.
Also, another “locator” pattern is this: If you take the difference (∆) between the c=hypotenuse values of every PPT, it is ALWAYS a ∆ of a multiple of 4, i.e. 1x=4, 2x=8, 3x=12, 4x=16,…where x=4. This is NOT true for the sibling nPPT multiples!
For some, reading this paper in reverse order — back to front — would provide the greatest benefit as it reveals the larger view first and thus gives a context for that that comes before.
Reading in the traditional order will be addressed by the outline shown below. Feel free to jump ahead, back and forth — especially with the large number of visuals — to generate this larger view as you go.
Outline for this section:
A. Classic Proofs of the Pythagorean Theorem on the BBSISL matrix
B. New Proofs of the Pythagorean Theorem on the BBSISL matrix
C. Pythagorean Triples on the BBSISL matrix
1. Primitives (PTT) — “parents”
2. NonPrimitives (nPPT) — “sibling” multiples
3. Distribution
a. SIDES
b. AREAS
C. SIDES In Place of AREAS
In the Interconnection section that follows:
A. BASE_2_3_4_5_6_8_10 PT
B. BASE_3
C. BASE_4
D. BASE_5
E. CirclePentagon Displays
A. Classic Proofs on the BBSISL matrix
The Pythagorean Theorem describes any rightangle triangle with side “c” (hypotenuse) > side “b” > side “a” such that the Area formed from side c= the sum of the Areas of side a and side b — as c^{2}=a^{2}+b^{2}.
While there are somewhere near 1000 proofs, only those rightangle triangles composed solely of whole numbers (integers) will be considered here. Integer Pythagorean triangles are called Pythagorean Triples/Triads (PT).
The PTs can be further divided into Primitive (PPT) and nonPrimitive (nPPT) categories. The PPTs, or parent PT, are irreducible, while the nPPTs, or childsibling PT, are simply 2x, 3x, 4x.. Multiples of the unique parent PT.
Since all presentations and proofs of the PTs will be upon the BBSISL matrix, the following format will be utilized:
This paper presents both translation and inventiondiscovery. The simple beauty of the Classic Proofs of the Pythagorean Theorem are translated onto the BBSISL matrix where it has been discovered that these very same PTs have a nearly ubiquitous presence. Indeed, their very simple proofs are simplified even further!
~Pythagorean Proof Sequence~
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~Pythagorean Proofs Classic~
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B. New Proofs on the BBSISL matrix
~New Proofs −1~
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~New Proofs −2~
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~New Proofs −3~
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~New Proofs −4~
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~Classic & New Proofs Examples 110~
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C. Triples Proofs and Distribution on the BBSISL matrix
~Triples Proofs~
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Basically, Primitives are the unique Pythagorean Triples that are irreducible — they cannot be factored any smaller. They are the parents.
The Primitive Pythagorean Triple (PPT) occupies a unique geometric form. Like the Primes, it represents that which is a fundamental entity that defies any other form of distillation. It is the pure form of the PT.
As will be seen, there are several numeric patterns on the BBSISL matrix that suggest ALL potential PTs can be predicted and found.
Numerous references have shown enormous efforts in the listings of algebraic solutions to revealing ALL possible PTs. TPSIC is a unique demonstration of how All possible PTs can be found, proved and predicted within the BBSISL matrix.
Basically, nonPrimitives are the multiple copies of the parent Primitive Pythagorean Triples — they are reducible — they can be infinitely reproduced as 2x, 3x, 4x, … their Primitive source. They are the childrensiblings and they are nearly ubiquitous.
So, if the Primitive PTs are so fundamentally pure, what does that say of the masses — the redundant, reiterative, selfsimilar copies — the fractallike nonPrimitive Pythagorean Triples (nPPT)?
Both the PPTs and nPPTs have an infinite number of possible forms, the nPPT far, far outnumber the PPT simply because they are multiples of the parent form.
The distribution of the PTs is perhaps the most interesting aspect of their existence. Once we understand how they are all over the BBSISL matrix, the real question becomes how do these SIDES and AREAS of the Squares formed become distributed over the grid. Since ALL possible PTs and their proofs exist on the matrix — though not on every Row, Column or Diagonal — what are the patterns of SIDES and AREAS telling us about:
The BBSISL Matrix:
Formation of the Pythagorean Triples as revealed in the BBSISL matrix:
As one can see, there is a hugh overlap between the Formation of the PTs and the BBSISL matrix. In addition, when a standard Multiplication Table is overlaid upon the BBSISL matrix, in the IDENTICAL cells on the grid, an identical PT pattern is found but with a difference. That difference is that on the BBSISL matrix the PTs found on the Axial Rows are ALL made up of the AREAS while on the standard Multiplication Table these same PT slots are made up of SIDEs. More on this will follow further down in this section. Right now, focus will be on TPISC — The Pythagorean Inverse Square Connection.
Table I. Differences (∆) in the hypotenuse values (“c”) in PRIMITIVE PTs.
Click HERE for a PNG version.
Table II. All Pythagorean Triples from hypotenuse “c” = 5 to 1225. NonPrimitives have their Primitive source shown in parenthesis.Click here to go to original source. Click here for a PDF version, or HERE for a GIF version or HERE for a PNG version.
SIDES are simply the integer values for the sides (two legs, a and b, + the hypotenuse, c). The PTs are designated by their sides, abc.
SIDES appear within the BBSISL as multiple generations, all either parallel or perpendicular to the Prime Diagonal, PD.
The nPPT SIDES have been showcased extensively in Rules 8186 of the original Brooks (Base) Square and The Inverse Square Law (ISL) work that was to become MathspeedST. Now, one can see that the Primitive SIDES are ALSO displayed Diagonally.
AREAS are simply the integer values for the squares of the sides (two legs, a and b, + the hypotenuse, c). The PTs are designated by their sides, abc, but a great deal of new information is revealed in studying their AREAS.
AREAS appear within the BBSISL as complete AREA sets — a pairset of the two leg AREAS + the hypotenuse AREA — the former along the Axial Row and culminating in the latter at the PD.
Neither the PPT or the nPPT AREAS have been showcased in the original Brooks (Base) Square and The Inverse Square Law (ISL) work that was to become MathspeedST. This is new, creativediscovery work revealed here for the first time.
The real value in working with the AREAS of the PTs, both individually as PPT or nPPT, and together, is that one gets an absolutely unique, visual look at how both of these PTs are embedded within the BBSISL matrix.
As was discovered only very recently, a new and very interesting pattern has emerged when the BBSISL matrix is overlaid with a matching standard Multiplication Table. In the identical cells of both grids, the same PTs emerge, only now in the Multiplication Table, it is the SIDEs — well, sort of, as you will see — that are now in the place of the AREAS on the BBSISL matrix! This is an important section with some very new information!
On the BBSISL matrix:
AREAS are simply the integer values for the squares of the sides (two legs, a and b, + the hypotenuse, c). The PTs are designated by their sides, abc, but a great deal of new information is revealed in studying their AREAS.
AREAS appear within the BBSISL as complete AREA sets — a pairset of the two leg AREAS + the hypotenuse AREA — the former along the Axial Row and culminating in the latter at the PD.
On the standard Multiplication Table:
A new and very interesting pattern has emerged when the BBSISL matrix is overlaid with a matching standard Multiplication Table. In the identical cells of both grids, the SAME PTs emerge, only now in the Multiplication Table, it is the SIDEs — well, sort of, as you will see — that are now in the place of the AREAS on the BBSISL matrix!
SIDES are simply the integer values for the sides (two legs, “a” and “b,” + the hypotenuse, “c”). The PTs are designated by their sides, abc.
SIDES appear within the BBSISL along Diagonals as multiple generations, all either parallel or perpendicular to the Prime Diagonal, PD.
SIDES in the Multiplication Table appear in the Axial Rows of the table.
Their location is identical to that of the AREAS for the same PT on the BBSISL matrix, only the cell values designate SIDE values — NOT AREA values. The location is the same. Please take your time as this becomes sorted out.
The distribution is the same, the PTs are true PTs, yet all the numbers for the two “legs” are different — the PD numbers for the hypotenuse are the same numbers, but with different meanings.
For one, the table is a Multiplication Table. Each cell is a product of its two Axial numbers. BBSISL is NOT a multiplication table.
For two, the actual SIDE number values are different: the 345 PPT located on Row 5 shows 152025, or 5x the SIDE values.
For three, every PT — PPT and nPPT — is a product of the original PT SIDEs values x it Row number.
Naturally, this means that All these PTs are now multiples and thus nonPrimitives.
Every conventional PT found on the BBSISL matrix is now — in the exact SAME location — transformed from AREAS to SIDES, and, raised in value by the Row number x the original SIDES value.
Surely, this points to yet another deep connection between the Primitives and their multiple offspring, the nonPrimitives in a way that is much more than the latter simply being whole number duplicates of the former. Or does it? Actually, what it shows is the deeper connection between the BBSISL and a standard Multiplication Table — more than any new connection to the PTs.
The Multiplication Table can be transformed into the BBSISL matrix by:
1. Adding the two Axis numbers that make up that cell, e.i. 15 = 3x5, so 3+5=8;
2. Multiplying that sum by the BBSISL submatrix 1 cell value, e.i. 2, so 2x8=16 (16 is the cell value on the BBSISL).
3. The PD values on BOTH are identical in numerical value and this is the key.
The entire BBSISL matrix grid is formed from the PD values. Ultimately, the same is true for the Multiplication Table. The Multiplication Table is less restrictive upfront because it does contain ALL whole integer numbers within its “Inner Grid” borders — including even numbers not divisible by 4 — and, like the BBSISL, outside of the single (x1 for the MT and the 1^{st} Diagonal for the BBSISL) it also contains NO PRIMES. Set this part aside.
It may be helpful to look at the connection between the Multiplication Table, BBSISL and the PTs by plotting all three on a reduced BBSISL matrix and utilize this to reference the Table that follows.
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(These 5 links all go to the same webpage)
Image Reference 9
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(These 5 links all go to the same webpage)
In Tables IIIV below, the interconnections between the standard Multiplication Table presentation of the PTs and that on the BBSISL matrix have been charted. Initially, it seemed — and rather confusingly so — that the product of the SIDES of the MT x the AREAS of the BBSISL was merely pointing at that deeper connection of the MT to the BBSISL. And yet, it looks from the view of Table III that the initial 345 Primitive PT is actually forming all other PTs, including all other Primitives.
Upon further examination, a clearer view was found. The translations of the column headers in Table IV below show the process.
The final simplification of the BBSISL Matrix — Multiplication Table (MT) Interconnections is shown in Table V below.
The emphasis now is back onto the BBSISL matrix. The related parameters are expressed simply in terms of abc (or “x” when combined).
Clearly, the 345 Primitive Pythagorean Triangle is the fundamental PT for ALL PTs — Primitive and nonPrimitives alike. Every PT is relatable back to the 345 PPT by a factor of 60 (abc), 144 (a^{2}b^{2}) or 12 (ab), the √ of the latter.
It turns out that those confusing MT values of what was thought to be the SIDES of PTs located at the exact same cell position, and previously referred to in the diagrams and charts above as a_{m} and b_{m}, are really the product of the SIDES ac and bc, respectively.
The easiest and most informative way to see the intimate interconnections between the abc’s of ANY PT is to plot them out following the template shown below in Table VIa.
Notice that, like a simple multiplication table grid, the simple combinations of abc are plotted on each axis. The key products have been given some light shading emphasis within the grid.
Below, in Tables VIX, four PPTs and one nPPT have been laid out on the “Foundation Table.” All PTs can be done exactly the same way following the values given in Table V above.
In the 345 PPT above in Table VI, the key cell values are:
In the 51213 PPT above in Table VII, the key cell values are:
In the 81517 PPT above in Table VIII, the key cell values are:
In the 72425 PPT above in Table IX, the key cell values are:
In the 152025 nPPT above in Table X the key cell values are:
The cell values for the first 15 PTs — PPT and nPPT — have been compiled again in Table XI below, now with a few “Additions.” These additions are literally taking the sums,∑, of a + b + c. When these sums are divided into the products of a x b x c — abc — yet another bit of interrelated information is acquired.
The 345 PPT is uniquely distinguished in several ways:
Yes, some of these findings are the result from simply rearranging the parameters, as one result follows from the other. And yet, this is uniquely true for the 345 PT.
Yes, dividing a number by itself will always result in 1. This was shown in several columns to spotlight the uniqueness of the 345 PT. All other PTs — PPT and nPPT — are multiples of the 345 PT. They can be factored down to the 345. In the x/60 column, we can see that all subsequent PTs, when divided by 60, are 8, 13, 27, … times that of the 345 value. All whole integer numbers. Notice, also, that here, the nPPT have an exponential factoring connected to their values, the PPT do not.
Under the column a^{2}b^{2}/144, all subsequent PTs, after the 345, when divided by 144, are 16=4^{2}, 25=5^{2}, 81=9^{2},… times that of the 345 value. Here, there is no difference between PPT and nPPT as to exponential factoring.
For example, the 51213 PT is 13x the 345 PT for abc — 780/60=13 for the 51213 and 60/60=1 for the 345. Its (a^{2}b^{2}^{)}/144 value is 25=5^{2 } times that of the 345 PT value of 1. The 51213 PPT, nor any of the other PTs, does NOT enjoy the unique properties of the 345 (listed above.)
An interconnection between the products and sums of the 345 PT and all other PTs has been found. A single, unique 345 PT informs all other PTs.
What about predictions? Extensive analysis may reveal a pattern leading to predictions of any and all PTs. Right now, the structural pattern of the BBSISL matrix itself is the best visual lead we have.
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Example: 345 has:
251=24,
254=21,
259=16*,
2516=9* Row sequence values. Notice that the last *two values crisscross in value with the PD sequence values. This is a PT. The Row values of 16 + 9 = 25 is b^{2} + a^{2}^{ }=^{ }c^{2}.
Example: 6810 has:
1001=99,
1004=96,
1009=91,
10016=84,
10025=75,
10036=64*,
10049=51,
10064=36*,
10081=19 Row sequence values. Notice that the *36–64 and *64–36 values crisscross. The Row values of 64 + 36 = 100 is b^{2} + a^{2}^{ }=^{ }c^{2}.
Example: 51213 has:
1691=168,
1694=165,
1699=160,
16916=153,
16925=144*,
16936=133,
16949=120,
16964=105,
16981=88,
169100=69,
169121=48,
169144=25* Row sequence values. Notice that the *25–144 and *144–25 values crisscross. The Row values of 144 + 25 = 169 is b^{2} + a^{2}^{ }=^{ }c^{2}.
And so it goes. The PD driven BBSISL matrix simply defines the pattern and that pattern simply defines the Pythagorean Triples! If a SQUARE number appears within the matrix, it is always part of Pythagorean Triple.
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Since all PTs are defined this way: what defines the Primitive versus nonPrimitives?
Lastly, for those wishing to delve deeper, taking a closer look at the role of the PD in defining not only the entire BBSISL matrix grid but the whole spectrum of Pythagorean Triples, here is a more detailed look. Herein lies the Rosetta Stone!
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The link below spotlights the various documented ways to generate Pythagorean Triples.
Formulas for generating Pythagorean triples
Although the Pythagorean Trees and Fibonacci Boxes are of special interest, it was Dickson’s Method for generating ALL PTs that drew notice — including Josef Rukavicka’s (2013) simple proof. Notice that his s, t values for each PT are in fact equivalent to the number of STEPS from the PD to the a^{2},b^{2} AREA Squares on the given Row defining that said PT! Again, the AREAS and SIDES crisscross complement each other as shown in Table XIII.
Dickson’s Method states: “To find integer solutions to , find positive integers r, s, and t such that is a square.
As “ is any even integer and that s and t are factors of .”
TPISC AREA Proof, Row Distribution and Interrelationships with the Dickson Method for generating Pythagorean Triples image shown below reveals an intriguing connection.
The r, s, t parameters that his method utilizes refer directly to the SIDES values of abc, yet when plotted out graphically with the usual BBSISL matrix AREAs approach, one quickly sees that these r, s, t values point to the number of STEPS along the pathways — pathways along the Row, the Column, and the PD — to reach those same abc values. And, these point not to the SIDES but rather to the AREA values, reversing the “a” with the “b” and the “b” with the “a” parameter. Once again, the uncanny manner in which the BBSISL matrix visually reveals the relationships between numbers only begs further research!
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(Pythagorean_BBSISL_Dickson.png)
Simplification!
Knowing that the Dickson Method works, knowing that the BBSISL matrix shows ALL PTs, and now knowing that the Dickson Method is pointing to key PT arrangements on the BBSISL matrix (as just shown), a true simplification is in order. As the images below shows:
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((Pythagorean_BBSISL_Dickson2.png)
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(Pythagorean_BBSISL_Dickson3.png)
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(Pythagorean_BBSISL_Dickson4.png)
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Table XIV. The BBSISL matrix  Dickson Method Master Chart summarizes the information presented.
If you start with a blank page, using the Dickson Method, you can calculate All PTs by r= even numbers, such that r^{2}=2st.
If you start with the BBSISL matrix, you can find All PTs by picking out the PD pairsets on the Rows (or Columns).
The BBSISL geometric presentation of All PTs appears in three forms:
Overlapping the two methods — algebraic and geometric — one sees that they end with the same PT. The Dickson Method’s s,t values pointing to the STEPS or FactorPairs between the Legs and Hypotenuse of a given PT.
Whereas without the Dickson Method, one found the PTs by scanning all the Rows (and Columns), now with the Dickson Method, one has the added benefit of the s,t FactorPairs informing which Rows (and Columns) will contain them. The pairsets of the BBSISL matrix format are just the Dickson FactorPairs/STEPS that are contained — and numerically allowed — in any active PT Row (or Column) leading up to its PD as c^{2} value.
Notice in Table 14, how the Dickson Method of r^{2}/2=st, when further divided by two, as in r^{2}/4= a PD number. The natural sequence progression of the r=an even number falls into sets of 1491625364964… relating directly to the very structure of the BBSISL matrix.
Notice that every such set, with the exception of the first, always has a minimum of two FactorPairs: A base of s=1 and s=2. The first set, the r=2 that forms the 345 PPT contains both the 1 and 2 factors in its one FactorPair. This overlap precludes it having any more FactorPairs.
For example, pick a random even number, say 6 (r=6), and apply r^{2}/2=st to get (6)^{2}/2=18. FactorPair this and you get:
Now you know that there are 3 PTs in this set. Scan the grid visually, or, place straight lines *up the Inner Grid between 1, 2 and 3, and separately between 18, 9 and 6 STEPS in from the PD. The two straight lines will tend to converge at the top. Your 3 PT Rows will cross these lines. The difference between FactorPairs is, of course, the difference in STEPS between the two AREA Squares of a^{2}b^{2} of that PT.
Remember that while the s,t sets are Multiplication FactorPairs of Dickson’s r^{2}=2st, they translate to Addition STEPS, additional STEPS between PD and the legAREA Squares. Those AREA Squares are Added together to form the PD AREA Square of the hypotenuse. Dickson’s Method ultimately rejoins the r,s,t values by addition, as well, to reach the same spot.
This all becomes incredibly simple once one gets familiar with the process!
*Start with the 118 first and work back up the grid. Lines will ultimately converge.
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(Pythagorean_BBSISL_Dickson6.png)
In this brief section, a quick look at TPSIC and the BBSISL matrix in different BASEs_ will reveal an even deeper connection.
As the previous section on the Distribution of the PTs has shown, there is a very deep connection — an interconnection, if you will — between the PTs and the very structure of the BBSISL matrix. There are no coincidences.
And while the earlier section on the visualization and proof of any PT directly on the BBSISL matrix was shown to be simple, direct and even intuitively obvious — that is, easily discoverable — once one brought the PT idea to the matrix table, digging deeper into the mechanics of distribution certainly added some mystery back.
Sure, one can state and easily see that (*most) every PT is presented on the BBSISL matrix in one of two ways.
First as SIDES, the abc values of ALL nPPTs are shown as diagonally linked arrays of triple sets. Because the matrix Strict Inner Grid contains NO prime numbers, nor any even numbers not divisible by four, only the multiple siblings of the Primitives are revealed. Parallel and perpendicular to the PD, these SIDES are nearly ubiquitously patterned over much of the entire matrix grid.
Second, as AREAS, the a^{2}b^{2}c^{2} —as b^{2} + a^{2} = c^{2} — absolutely ALL PTs are revealed on select Rows (and Columns) of the grid, with b^{2} and a^{2} on the Row culminating with c^{2} on the PD. This AREA presence is both simple, direct and intimately tied to the underlying mechanics of the BBSISL matrix .
What about other interconnections? Can they shed some new light on how to think about the distribution — and, ultimately the interconnection — between the PTs and the BBSISL matrix?
The ISL, the foundation of the BBSISL matrix, is fundamentally based on distribution — the distribution of quantity. That quantity may be a composite parameter like force, energy or other influence, or it may be the measuring mechanism itself like the area or volume geometry.
The BBSISL enumerates the ISLbase quantities as whole number integers within an everexpanding matrix grid.
That matrix is enumerated in our common BASE_10 system: 0123456789.
What about enumerating in other BASEs_ like BASE_2_3_4_5_6_8? Does the BBSISL matrix in other BASEs_ still hold up?
The answer is YES. The quantity relationship interconnections that form the basis of the matrix remain intact regardless of the counting system.
Does the examination of the PT AREA distribution in the BBSISL matrix in other BASE_ counting systems reveal any clues as to the distribution of the PTs — PPT and nPPT?
The answer is YES. Examination of the PTs across BASE_2_3_4_5_6_8_10 reveals select pointers to the PTs in general and to specific pointers exclusive to the PPTs. BASE_5 tended to be more mixed, as in BASE_10.
As presented above, the PPT AREAS value c^{2} only ends with 1/5/9 in BASE_10. It only ends in 1 in BASE_2_3_4_6_8 as examined in all PTs with c^{2} up to 10,000. Looking at the c=hypotenuse SIDE values, Primitives always end with 1 in BASE_2_4; end in either 1/2 in BASE_3, and end in either 1/5 in BASE_6_8.
It would appear that in order for a Primitive Pythagorean Triple to form, both of two requirements must be met:
1. Based on c = hypotenuse (SIDE)
BASE_2=1
BASE_3=1/2
BASE_4=1
BASE_6=1/5
BASE_8=1/5
2. Based on c^{2} = hypotenuse^{2} (AREA)
BASE_2=1
BASE_3=1
BASE_4=1
BASE_6=1
BASE_8=1
The nonPrimitive Pythagorean Triples overlap these specific pointers but do NOT satisfy both conditions.
*Since the PT SIDES contain even numbers not divisible by 4 and/or prime numbers, and, the BBSISL matrix grid contains neither, only multiple siblings appear.
~BASE_3~
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~BASE_4~
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~BASE_5~
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~CirclePentagon Displays~
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The interesting — and, perhaps, not so interesting — thing about the Circle Displays is how completely untuned they are to any specific meaning.
Take any long enough sequence of random whole number integers in BASE_10 (0123456789), which is what we are using, and it will naturally form the double pentagram/pentagon (decagram/decagon) pattern. If that starting random sequence is reduced to just 5 of the 10 numbers — as in 13579 (ODDS) or 02468 (EVENS) as in the 1^{st} number “ones” of the various Diagonals parallel to the PD in the BBSISL matrix — only the single pentagram/pentagon forms. The same occurs visually, if the 6 numbers of the PD/Row/Column are used, in that the two ends of this symmetrically repeating pattern are separated by only one number. Visually, this is masked in the composite view.
Nevertheless, the extraordinary beauty of the decagram is witness to the beauty of Nature first hand. First the triangle, then the circle, the square…and then the quantum jump…the pentagon/pentagram. As Nature replicates, so does her geometry. The doublepentagon provides the perfect form containing the Golden Mean/Section/Ratio, as well as all of the geometric jewels of the simpler forms below it — π, √2, etc.. No wonder it is the blueprint for DNA.
A lot of ground has been covered here. This might be expected when a new lens, such that the BBSISL matrix certainly is, comes to bear on an ancient, universal theme — the Pythagorean Triples (PT).
More than adequately covered in the literature, writings, songs and art of all cultures, it has been an attempt here to present an entirely new manner of looking at the Pythagorean Triples, their proofs, distribution and ultimately their interconnections. As it turns out, the interconnections with the BBSISL matrix are what has become the main theme. As the Inverse Square Law (ISL) upon which the BBSISL matrix is formed is, in many respects, one of if not the main “laws” or relationship informing principles in all of science — most especially physics and mathematics. That the PT should be so intimately born, described and proved within that matrix is not without great introspection. If the ISL is fundamentally about how influence is distributed — spread out and diluted — over SpaceTime (ST), then the Pythagorean Theorem is a natural, builtin measure of that.
Like many pivotal relationships, that of The Pythagorean  Inverse Square Connection (TPISC), one must not let the initial simplicity blind us to the deeper attributes offered. To keep things straight, one must keep in mind the distinctions between how the PTs are presented on the BBSISL matrix.
Typically, the abc SIDES are presented Diagonally as sets of triples within the Inner Grid of the matrix.
The a^{2}b^{2}c^{2} AREAS are presented horizontally as Rows (or vertically as Columns when looking at the upper half of the symmetrical matrix) but with a difference. The b^{2} and a^{2} values are on a Row (or Column) within the Inner Grid, but the c^{2} resides on the Prime Diagonal (PD) at the end of that Row, and that is key.
While the SIDES values represent the multiple siblings of both Primitives (PPT) and nonPrimitives (nPPT) and are nearly ubiquitously patterned over the entire expanding matrix grid, the AREAS values, while also representing all PTs, both PPT and nPPT, are more uniquely presented. They are presented only once upon a given Row. Even their multiple sibling nPPTs are presented, as AREAS, only once on the entire grid, at their designated Row (or Column).
It is this singular representation — or call it presentation — upon a Row (or Column) that both defines the PT and sheds light on the wheres and whys of their distribution. As the BBSISL matrix is defined by the odd number summation series 13579… that when added sequentially become 1491625… (PD numbers) which is none other than the Squares of the whole number integers 12345…, every PT is ultimately defined by this same propagator. As every Row (or Column) is itself defined by the PD sequence 1491625… — or its propagator 13579… — the PTs are defined by the same. Since not every Row (or Column) has a PT, the quest becomes to determine what builtin relationships or interconnections within the BBSISL matrix itself determine when a Row is hot, active, sparkling with the PT that is onboard!
For one thing, only select Rows (or Columns) contain a Squared number. When they do they always come in pairs or pairsets. In every instance, one of these pairsets is a PT. Once a Row is hot, duplicates occur like clockwork patterns further out on the grid.
Another clue, all Squares on the PD, and therefore all Squares on any Row (or Column) end in one of the 6 possible values 1/4/9/6/5/0, as do the nPPTs, but the PPT ONLY end in 1/5/9. Any SQUARE ending in 4, 6 or 0 is a nonPrimitive Pythagorean Triple.
Detailed in the latter part of the Distribution section is a detailed look at the Dickson Method for generating ALL PTs. The abc and a^{2}b^{2}c^{2} values of PT are deconstructed down to FactorPair sets — sets of two factors (s,t) that when multiplied together equal r^{2}/2, where r=an even number. The additions of the r, s, an t values for each FactorPair set, in combinations, will give the PT values.
Further deconstruction and overlapping with the BBSISL matrix has shown that the Dickson Method acts like a Rosetta Stone in that the very same FactorPair sets are equivalent to the STEPS separating the AREA squares of the PT on the grid.
Both the Dickson algebraic method and the BBSISL geometric matrix method reveal exactly the same PT. The FactorPairs = STEPS pair sets.
The BBSISL matrix method has the added advantage in that it presents ALL PTs in an AREA Proofs visual manner as well as in listing the AREAS of each PT on the Row (or Columns). Additionally, All PT SIDES (above a minimum size) are presented Diagonally over the entire matrix.
The interconnections between these two methods are so tight that little daylight has any chance of separating the two.
Yet more clues are had when comparing the distribution of PTs in BASE_ systems other than our common BASE_10. That the Primitives once again are filtered out across the BASEs_2_3_4_6_8 further validates the true intimacy of TPISC.
Perhaps the most the most profound finding is that when the abc components are broken down individually and reexamined within a multiplication grid — showing new combinations of a, b and c — that ALL PTs — PPT and nPPT — can be shown to be related to and derived from the one single 345 Primitive.
Summary: The BBSISL matrix reveals ALL PTs.
~~~~~
It is ultimately the nature of how dimensions — line, area, volume —numerically interact with scaling that defines the geometry of the Universe. The addition of time dimensions, as well as additional space dimensions, via rotation, movement, scaling, included in the mix adds to the mysterious richness and complexity that issues forth. It belies us to examine the central, fundamental role that line, area, and volume play in all realizations of spacetime (ST).
The journey has been long, yet revealing. The formidable numerical relationships embedded within the BBSISL matrix through TPISC, The Pythagorean — Inverse Square Connection, only reaffirms the fundamental nature of the Inverse Square Law in defining the very Nature within which we exist!
All references are valuable. Unordered list.
art theory 101 / database index titles and topics: 























A NEW short TPISC: INTRO has been added to ease into this paper! 