User:Tomruen/testx

Spherical uniform honeycombs (finite polytopes)
The number of convex nonprismatic polytopes is largely determined by 7 families. The nonsymmetric linear graph families Bn, Hn have (2n-1) uniform polytopes. The symmetric linear graph families: An and F4 have alittle more than half as many. The Dn has about 3/4 as many, but half of those are shared with the Bn family, so actually have (2n-2) unique polytopes above n=4.

Euclidean honeycombs
The Euclidean uniform honeycombs can be enumerated similarly. The $${\tilde{A}}_n$$ family is enumerated by the reverseable necklace sequence (A000029). The symmetric $${\tilde{C}}_n$$ family is a bit over half the possible permutations. The bifurcating $${\tilde{B}}_n$$ family is about 3/4 of the permutations, and 2/3 of those shared by $${\tilde{C}}_n$$.

Compact hyperbolic honeycombs
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