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| Pythagorean Triples: Be prepared to find two very simple geometries ubiquitously joined together!There is a simple whole number (integer) matrix grid that every possible whole number Pythagorean Triangle — a.k.a. Pythagorean Triple — can be found and proved — the BBS-ISL Matrix. The BBS-ISL Matrix is composed of Pythagorean Triples!
By: Reginald Brooks, Brooks Design 
TPISC_ TPISC I: The Pythagorean - Inverse Square Connection: Basics reveals the presence, proofs and and overall distribution of the Pythagorean Triples (PTs) upon the BBS-ISL Matrix. This work is being presented 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) 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. ~~~ 1. What are Natural Whole numbers? Integer numbers 0,1, 2, 3, … 2. What is the ISL (Inverse Square Law)? Universal law of Nature that describes how influence (energy, force, light, sound,…) is diluted with distance from the source — as in density ∝ 1/r2. 3. What is the BBS-ISL Matrix? A matrix grid of Natural Whole numbers that describes the ISL. 4. Is the BBS-ISL Matrix a multiplication table? No! Only the Primary/Prime Diagonal (PD), which divides the matrix into two equal, symmetrical triangles is composed of the multiplication product (the “squares”) of the two Axis numbers. 5. Why does the BBS-ISL Matrix look so complex? New things often “look” complicated, even when they are extremely simple! 6. How can you let me see that? Easy, can you add 2 to any number? 7. Yes! Then you can easily make the 1st diagonal—1st diagonal running parallel next to the PD. Start with 3 + 2 = 5. Now to that sum of 5, add 2 again, and so on. 3 + 2 = 5 5 + 2 = 7... 8. OK, that was easy! But where did the 3 come from? PD 4 - PD 1 = 3. 9. What’s next? Can you add 2 + 2? 10. Yes, of course! On the 2nd parallel diagonal from the PD, start with 8 and add 4 to each subsequent sum. 8 + 4 = 12 12 +4 = 16... 11. Where did the 8 come from? PD 9 - PD 1 = 8 12. So let me guess, the next diagonal—the 3rd parallel to the PD will increase by either 6 or 8? The 3rd diagonal, starting with 15, increases by 6. 15 + 6 = 21 21 + 6 = 27… And, the starting number came from the difference (∆) between the PDs— top of the column, and the end of the row, where they intersect the PD. PD 16 - PD 1 = 15. TIP: The number (#) we are adding each time to the specific starting number is simply double (2x) the Axis number of that diagonal! 13. Are the rest of the parallel diagonals all formed the same way? Yes! Absolutely. Congratulations! 14. Hold on a minute! What about the other half? Remember, it is a symmetrical mirror of the lower half. ... 15. Does the PD play any other roles in the matrix? Yes, absolutely. For one, every grid cell value is simply the ∆ between the PD values at the row and column intercepts. 16. So are there any other things this BBS-ISL Matrix can do? I getting kinda bored, ya know! Yes! Hundreds. A very interesting geometry that really sticks out—the Pythagorean Triples (PTs). 17. What’s that? Everyone, sooner or later, learns about the Pythagorean Theorem. The Pythagorean Triples (PTs) are 90° right-triangles composed solely of Natural Whole numbers—just like in the BBS-ISL Matrix. 18. So what does that have to do with the BBS-ISL Matrix? Every PT resides uniquely on a row... 19. Does every row have a PT? 20. Explain? 21. Can the same PT be on different rows? 22. Can more than one PT be on the same row? 23. Are the PTs connected in any way? Yes, absolutely! Every PT is ultimately connected to every other PT and that Number Sequence Patterning is presented in TPISC I: Basics (free) and TPISC II: Advanced. 24. OK, without my reading a math book—even one only about Natural Whole numbers—why should I care about this? The short answer: Everything! 25. Well, you’ve got me there! But, so what? Glad you asked!... 26. OK. Squares or rectangles, so what’s the big deal? 27. Where do these nodes occur? 28. That’s it? We (Nature) can make things out of rectangles in addition to circles, squares and isosceles triangles? 29. Aren’t there other forms—including fractal-like forms, the golden rectangle, etc. that are so obviously in play in the ST forms we see with our eyes (and telescopes) everyday? Indeed, and the... ~~~ Conclusion The inter-connections with the BBS-ISL Matrix are what has become the main theme. 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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is a multi-touch, interactive ebook made with iBooks Author® and published in the Apple iBooks Store®. Currently, it requires an iOS device or a Mac running OS X 10.9 and above, and the free iBooks® app to download and read. It is available for free. Author's TPISC I: Basics webpage: http://www.brooksdesign- Author'sTPISC II: Advanced webpage: Contains Selected Screenshots. End
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