Click to Select:
Drag to Deconstruct
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:
Besides intimately tying the ToPPTs to a natural fractal pattern within the BIM:
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 = a2 and b2) and the 4A value ONLY, it is a PPT Row.
If it contains BOTH the 4A and 8A values, it is a 'step-sister' Row, i.e. (c2-1)/24 and (c2-25)/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.
(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 (c2) 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,... )