Uniform honeycombs in hyperbolic space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.
Hyperbolic uniform honeycomb families
Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures, and the second includes affine subgroups.Compact uniform honeycomb families
The nine compact Coxeter groups are listed here with their Coxeter diagrams,in order of the relative volumes of their fundamental simplex domains.
These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is cited with the family below. Only two families are related as a mirror-removal halving: ↔ .
| Indexed | Fundamental simplex volume | Witt symbol | Coxeter notation | Commutator subgroup | Coxeter diagram | Honeycombs |
| H1 | 0.0358850633 | = + | 15 forms, 2 regular | |||
| H2 | 0.0390502856 | + | 9 forms, 1 regular | |||
| H3 | 0.0717701267 | + | 11 forms | |||
| H4 | 0.0857701820 | + | 9 forms | |||
| H5 | 0.0933255395 | + | 9 forms, 1 regular | |||
| H6 | 0.2052887885 | + | 9 forms | |||
| H7 | 0.2222287320 | ] | 6 forms | |||
| H8 | 0.3586534401 | + | 9 forms | |||
| H9 | 0.5021308905 | ] | ]+ | 6 forms |
There are just two radical subgroups with nonsimplectic domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is , represented by Coxeter diagrams an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔, which can be extended by restoring one mirror as. The other is , index 120 with a dodecahedral fundamental domain.
Paracompact hyperbolic uniform honeycombs
There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity.| Type | Coxeter groups |
| Linear graphs | |
| Tridental graphs | |
| Cyclic graphs | |
| Loop-n-tail graphs |
Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, with the constraint that at least one node must be ringed across infinite order branches.
| Dimension | Rank | Graphs |
| H3 | 5 | 3,5,3 familyThere are 9 forms, generated by ring permutations of the Coxeter group: orOne related non-wythoffian form is constructed from the vertex figure with 4 vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a tetrahedrally diminished dodecahedron. The bitruncated and runcinated forms contain the faces of two regular skew polyhedrons: and. 5,3,4 familyThere are 15 forms, generated by ring permutations of the Coxeter group: or.This family is related to the group by a half symmetry , or ↔, when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive ↔. 5,3,5 familyThere are 9 forms, generated by ring permutations of the Coxeter group: orThe bitruncated and runcinated forms contain the faces of two regular skew polyhedrons: and. 5,31,1 familyThere are 11 forms, generated by ring permutations of the Coxeter group: or. If the branch ring states match, an extended symmetry can double into the family, ↔.(4,3,3,3) familyThere are 9 forms, generated by ring permutations of the Coxeter group:The bitruncated and runcinated forms contain the faces of two regular skew polyhedrons: and. (5,3,3,3) familyThere are 9 forms, generated by ring permutations of the Coxeter group:The bitruncated and runcinated forms contain the faces of two regular skew polyhedrons: and. (4,3,4,3) familyThere are 6 forms, generated by ring permutations of the Coxeter group:. There are 4 extended symmetries possible based on the symmetry of the rings:,,, and.This symmetry family is also related to a radical subgroup, index 6, ↔, constructed by , and represents a trigonal trapezohedron fundamental domain. The truncated forms contain the faces of two regular skew polyhedrons: and. (4,3,5,3) familyThere are 9 forms, generated by ring permutations of the Coxeter group:The truncated forms contain the faces of two regular skew polyhedrons: and. (5,3,5,3) familyThere are 6 forms, generated by ring permutations of the Coxeter group:. There are 4 extended symmetries possible based on the symmetry of the rings:,,, and.The truncated forms contain the faces of two regular skew polyhedrons: and. Summary enumeration of compact uniform honeycombsThis is the complete enumeration of the 76 Wythoffian uniform honeycombs. The alternations are listed for completeness, but most are non-uniform. |