TPISC IV:

*TPISC ( pisque—silent “T”): *The Pythagorean — Inverse Square Connection

Teach your children well.
The world of numbers is yes!
Yes heaven, yes hell.
 
So where do they start?
Why not 0, 1, 2, 3,…
Now go square them — smart!
 
Place them on a grid,
Axis and diagonal.
Clever you! Done did.
 
The numbers between —
Subtract a squared from a squared —
Are there to be seen!
 
A row that ripples
Of squared numbers, gives to you:
Triangle “Triples!”
 
~ ~ ~
 
The story of the BIM (BBS-ISL Matrix) and the connection with the Pythagorean Theorem — TPISC (The Pythagorean - Inverse Square Connection) — is fundamentally simple!

In fact, keep the 3-Steps to Nirvana in mind as we parse out some of the details.

3-Steps to Nirvana:
Make the BIM: draw a square with evenly spaced grid lines place 0-1-2-3-… along top and side (left) and their squared values along diagonal (PD=Prime Diagonal) fill in the rest of the grid cells with the difference (∆) between the horizontal and vertical PD values. Locate the PTs (Pythagorean Triples): from the PD, starting with 9, drop down vertically until a squared number on a Row is found that squared number will always point to its complementary squared number (found on both the PD and along that SAME Row) this PT Row — with the two a2 and b2 values — terminates with c2 at the interception with the PD. Connect the dots to make the ToPPT (Tree of Primitive Pythagorean Triples): every PT — both parental Primitive Pythagorean Triples (PPTs) and their children non-Primitive Pythagorean Triples (nPPTs) — will occupy a unique Row (with some occasional overlaps) every PT will follow the detailed r-value based Template, differing in only its specific a2, b2, c2, 4A, 8A, p, ƒ and ƒ2... variables connecting the dots (values) between the Universal Template and the Specific variables gives the ToPPT. ~~~~~ 
1 banana, 2 banana, 3 banana, 4 … and there’s so much more!
     
A simple curiosity on how even simpler Whole Integer Numbers (WINs), when squared, form a law that rules the Universe — indeed, SpaceTime (ST) itself — has been a source of awe and wonder for quite some time. This is the Inverse Square Law (ISL). BIM (BBS-ISL Matrix) is the matrix grid that mathematically describes the ubiquitous nature of the ISL. 

The “inverse” means it’s per area. When the WINs are squared, you get the area.

The forms and forces of Nature obey the ISL. The initial concentration of an impulse or influence of energy dissipates in a known manner as it expands out from its source. A wonderful and ubiquitous law in forming the ST that builds the Universe. And it doesn’t even need to travel to get there…it is known!

Now, on top of this beautiful law and all the magnificent forms and expressions it can offer, comes yet another sublimely beautiful law — or theorem — that also relates areas to significant form — the Pythagorean Theorem: the long, hypotenuse side squared = the sum of the two squared shorter sides of a 90°-right triangle.

When these triangles are made solely of WINs, we have Pythagorean Triples (PTs).

The PTs are a major subset of the ISL and as such are both infinite in number and ubiquitous in distribution along with the ISL. Their structural beauty lies in the fractal-nodes of slight asymmetry they give to the expanding expression of the ISL. They give new forms!

Not only can these PT fractals be located throughout the ISL Matrix, together they form a linked structure relating all the parts to the whole, in the form the Tree of Primitive Pythagorean Triples (ToPPT).

This clarity and simplification of form has become the focus of this work. The non-isosceles right triangles, rectangles and ovals of the ToPPT join with the isosceles right triangles, squares and circles of the ISL to generate our ST Universe — giving expression to the Universal mathematical constructs.

3 potato, 4 potato, 5 potato squared …
~~~~~ 
TPISC IV contains all of the above plus:
PRIMES vs NO-PRIMES,  PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture, PRIMES On the BIM, and  *DESQEC & CaCoST as well as “details” of the earlier TPISC I-III works.
*DSEQEC (Double-Slit Experiment—Quantum Entanglement Conjecture); CaCoST (Creation and Conservation of SpaceTime).

Details: here we examine the little devils more closely!