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George Duke wrote on Sun, Jun 28, 2009 08:09 PM UTC:
That cubic Ferz triangulates. Take 8 cubes 2x2x2 labelled ccw below 1-2-3-4 and above '1' 5-6-7-8 ccw. Ferz goes 2->7->5->2, or else 2742, or 2572, or 2542, or 2452 or 2->4->7->2. There are no more starting with cube-2. Since any of the other seven cubes can begin similar patterns, there are (6x8) 48 possible Ferz triangulating sequences in 2x2x2. It's the ''normal'' thing, literally and figuratively as Gilman would say, since there are several pathways. Noticing 336 (and 392) familiarly as 8x7x6 and 8x7x7, here there are 8 concocted starting cubes, and since triangulating involves two more different cubes, 7, then 6 with the last step the same numbered cube as the first, so 8x7x6x1 possibilities. 336 thus represents # cube-sequences without regard to the Ferz move-rule in quasi-minimal 2x2x2. Real Ferz actually has the above 48 of this maximum 336 possible. Now 336 is the same number in Falcon USP5690334 as number of square-based pathways 2-D  allowed Jetan's Chieftain with no ''doubling back,'' with 392 also for less restricted Chieftain interpretation. There in the text over a decade now, (8x7x7) and (8x7x6) the elements are the 8 chess directions of each step in Chieftain's three-step pathways. Falcon uses precisely 48 of Chieftain's 336 (or 392, depending on strict or loose interpretation of Chieftain). It is well-known Falcon has those exact 48 pathways to her 16 squares. Dis-allowing 135 degrees is another way to arrive at only 336 rather than 392. In computing pathways 336 for Chieftain three-steps, it matters whether changes of direction are 135- or 90-degree and exact orthogonal-diagonal order and positioning, not allowing revisiting spaces already passed over;  but those are the numbers reached arithmetically on sufficently-large lay-out, 10x10 being ideal. Cubes 2x2x2 and squares 10x10 and both numbers 48 and 336 out from their very different geometries.  After all, Gilman's ND root-3 shows important comparability between space-filling squares and hexagons in that number; so such other more-so tailored parallels are not surprising from cubic and square geometries taken together. 48 and 336 paired and emerging in different contexts, one cubic, the other one ordinary flat decimal board in compelling pre-cognitive patterning.

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