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BIM+PT (2018) Introduction

from Reginald Brooks on Vimeo.

Part of TPISC III: Clarity

and

TPISC IV: Details

Copyright 2018, Reginald Brooks,

Brooks Design. All rights reserved.

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A brief synopsis...

Along the way, several new and very exciting findings have popped up! It's just like the **BIM** to just keep on giving!

The simple — and some of the not so simple — geometries of the

**BIM**and BIMtree have been laid out in a series of interactive canvases. Each canvas opens up on its own webpage. Once open, you just click/touch and drag the geometry around, apart, this way and that way, even rebuilding the initial geometry side by side until you can really see how the parts are so simply related. It's deconstruction!The entire geometry on canvas series was produced initially without any supporting text to keep it as clean, simple and intuitive as possible. The accompanying text will certainly help fill in the gaps, but it should be noted that the geometry on canvas series sequentially builds from one canvas to another.

The algebraic binomials and their square proofs are widely known. Here the very same proofs overlap with that of the

**BIM**, BIMtree and the ToPPTs! This, too, can be readily seen.The number 24 — and its factors of 1-24, 2-12, 3-8, and 4-6 have shown, once again, to be integral to both the

**BIM**in general (remember, the EVEN Inner Grid cell numbers are all evenly divisible by 4), and, the ToPPTs. HOW SO? Well, this is new, really new, so just the surface has been touched, but so far:- Marking (yellow) all
**BIM**cells evenly ÷ by 24 generates a striking diamond-grid, criss-crossing pattern with 4 additional yellow marked cells in the center of each diamond; (*TPISC IV: Details*) - Every PPT is found to exist on —and ONLY on — those Rows whose 1st Column grid cell are ÷ by 24 (yellow, with 1st Column grid cell marked as '
'), though not every such yellow marked Row contains a PPT. nPPT are only present on such Rows if accompanied by a PPT; (*Z**TPISC IV: Details*) - The “step-sister” of any given PT Row is found
-steps down the Axis from that Row (with 1st Column grid cell marked as '*r*') and it, too, always and only exists on a yellow marked Row (“*Z*_{f}” is part of the FTP originally derived from the Dickson Method for algebraically calculating all PTs); (*r**TPISC IV: Details*) - The PPT Row (and nPPT Row) always contains the 4
**A**(A=area) value of that PT and the “step-sister” PPT Row always contains the 8**A**value, both landing exclusively on yellow marked grid cells, giving a striking visualization of ALL PPTs and their-based “step-sisters;” (TPISC III: Clarity &*r**TPISC IV: Details*) - The significance of the “step-sister” is that it becomes the mathematical link to the “NEXT” PPT within the ToPPT — like the Russian-Doll model; (TPISC III: Clarity)
- The significance of the expanding and increasingly inter-connected PTs, as the
**BIM**itself expands, is one in which the perfect-symmetry geometry of regular shapes and solids — equal triangles, squares, circles,… of the**BIM**allows — at certain articulation nodes (i.e., Rows) — the introduction of the slightly less-perfect-symmetry geometry (i.e., bilateral symmetry) of the full rectangle and oval that the non-isosceles right triangle PTs represent, into the unfolding structural framework, working from the ground up, if you will. The roots of fractals-based self organization are first to grow here! (TPISC III: Clarity &*TPISC IV: Details*) more… - ALL self organization of any sort — be it force field or particulate matter — must have an organizing mathematical layer below driving it!

- Marking (yellow) all

Besides intimately tying the ToPPTs to a natural fractal pattern within the BIM:

- Thereafter, ALL PTs — primitive “parents” (PPT) and non-primitive “children” (nPPT) —are found on select Rows of the
**BIM**by simply following the squared numbers on the the Prime Diagonal down to intersecting Rows; ( TPISC I: Basics & TPISC II: Advanced) more… - The mathematical basis relating the PPTs was found, as was the consistent Fractal-Template Profile (FTP) that every PT follow; (TPISC II: Advanced & TPISC III: Clarity) more…
- The FTP allowed all the PPTs to be sorted out and organized into a definitive Tree of Primitive Pythagorean Triples (ToPPT) that co-extends infinitely throughout the infinitely expanding
**BIM**—>**BIMtree**or BIM-ToPPT; (TPISC III: Clarity) more… and more…

One could simply say that: subtract one from the squared values of any natural, whole integer number (WIN) and if it is evenly ÷ by 24, it is a candidate for being a PPT or 'step-sister' if that same squared value - 25 is ALSO ÷ by 24.

If the Row contains a

**Factor Pair Set**(two squared values that =and*a*^{2}) and the 4*b*^{2}**A**value ONLY, it is a PPT Row.If it contains BOTH the 4

**A**and 8**A**values, it is a 'step-sister' Row, i.e. (-1)/24 and (*c*^{2}-25)/24, if evenly divisible, are PPT and/or PPT 'step-sister' Rows, e.i. Row 17 is the PPT Row of the 8-15-17 PPT, but is also the 'step-sister' Row of the 5-12-13 PPT.*c*^{2}(NOTE: The PPTs intersect with grid cells/24 at: the subtraction of 1,5,7,11,13,17,19,23,25,… from the squared Axis number (

) and that gives a cell spacing of 4-2-4-2-4-2-4-2-… respectively, or a blank step ∆ between of 3,1,3,1,3,1,3,1,... )*c*^{2}- Thereafter, ALL PTs — primitive “parents” (PPT) and non-primitive “children” (nPPT) —are found on select Rows of the