TPISC I — Basics

Be prepared to find two very simple geometries ubiquitously joined together!

TPISC I: Basics
TPISC I: The Pythagorean - Inverse Square Connection: Basics reveals the presence, proofs and and overall idea of the distribution of the Pythagorean Triples (PTs) upon the BBS-ISL Matrix. Information is presented by a number of graphic means on the initial discovery and the profoundly simple geometric proofs of all PTs straight on the matrix. 

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Abstract
There is a simple whole number (integer) matrix grid table that every possible whole number Pythagorean Triangle — a.k.a. Pythagorean Triple — can  be found and proved. The Brooks Base Square - Inverse Square Law (BBS-ISL Matrix) is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their non-Primitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem —  c²=a²+b² —  is built into the BBS-ISL 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 BBS-ISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate inter-connection way beyond a simple, casual association has been revealed. The Dickson Method confirms, validates and provides insight into the generation of ALL Pythagorean Triples.

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Introduction
The reason that this work is being presented is precisely because the very nature of that geometric relationship of the Pythagorean Triangle (PT) — 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 BBS-ISL Matrix, that a cause and effect relationship between the two is unavoidable. 

That the BBS-ISL Matrix provides the simplest, most intuitively obvious proof to the Theorem directly on the grid only supports the argument. Every possible PT, and its proof, is visually and mathematical present. The Dickson Method confirms this.

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Conclusion
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 inter-connections. As it turns out, the inter-connections with the BBS-ISL Matrix are what has become the main theme. As the Inverse Square Law (ISL), upon which the BBS-ISL 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, built-in measure of that.