Convex Uniform Honeycomb - Hyperbolic Forms

Hyperbolic Forms

There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family.

From these 9 families, there are a total of 76 unique honeycombs generated:

• : - 9 forms
• : - 15 forms
• : - 9 forms
• : - 11 forms (7 overlap with family, 4 are unique)
• : - 9 forms
• : - 6 forms
• : - 9 forms
• : - 9 forms
• : - 6 forms

The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is in the {3,5,3} family.

There are also 23 noncompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity: