Single Comment

Tetrahedra have 4 faces, 6 edges. This pyramid stands on edge instead of face. Two edge-pair's midpoints determine one line, and 2 such lines determine a plane (the 4x4 here). That cross-section is a rectangle not usually square. So are all cross-sections of all other parallel planes intersecting. Cleverly and arbitrarily, Mark Thompson chooses 5 such planes and 2 more planes fashioned out of two edges, totalling 7, in order to get notional-3D 84 spaces(year 2002 84-square contest). They then divide conveniently into two(1x7), two (2x6), two (3x5), and one (4x4) making 84. How is that related also to 84 as tetrahedral number? 84 as tetrahedral number(think sphere-stacking oranges) sequences one(1), supported by 3 making four(4), supported by 6 making ten(10)[this is experimental too], then 10 making twenty(20): 1,4,10,20,35,56,84,120... For the five piece-type differentiation, the six edges each have two directions for 12 altogether. Contrast these 12 to the 26 directions necessary for complete interpretation in awkward standard cubic chesses (6 orthogonal, 12 diagonal and 8 triagonal or trigonal). See in the tripartite diagram the distinguishment between King and Rook.