an intimate look at a girl’s best friend
Each atom in a diamond crystal is bonded to four neighbors. Games of go played on such a lattice ought to be similar in some ways to games played on a standard board – unlike on a board tiled with triangles, for instance.
what is it????
The polychoron viewer Jenn was originally conceived to generate uniform go boards in spherical 3-space. The “biggest” such figure (not counting duoprisms) is the omnitruncated 120/600-cell with 14400 vertices.
It is an interesting “lattice”. It keeps company with no less than {3,3,4,3} and 5_21 (ie E8), as both are quarter-cubics. Þat is, you take a body-centred cubic as two cubics, and remove alternating vertices from each.
The ‘t-diamond’ tiling consists of two consecutive nodes on the An ring: its dual is t-catseye. Yes, the voronii cells of this thing in N dimensions, taken as a simplex-symmetry, is what you get looking at a cubic cut perpendicular to a long diagonal of a cubic in N dimensions.
The most efficient packing of spheres in 9d is also a quarter-cubic, but if you shake it gently enough it will fall into two semicubics: the packing rattles!
t-catseye is actually the voronii cells of the edges of t-diamond, not the vertices.
In 3d, the voronii cell of the t-diamond is in the Conway-Hart notation 3ktT, You take a tetrahedron T, truncate it to get tT “truncated tetrahedron”, and then raise pyramids on the triangles, being 1/4 of the total height (trikis-tT = 3ktT).
trikis just means ‘thrice’.