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.
Four compact regular hyperbolic honeycombs
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 (finite subgroups), and the second includes affine subgroups.
Compact uniform honeycomb families
The nine compact Coxeter groups are listed here with their Coxeter diagrams,Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2   in order of the relative volumes of their fundamental simplex domains.Felikson, 2002
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 {3,5,3} family below.
Only two families are related as a mirror-removal halving: [5,31,1] ↔ [5,3,4,1+].
There are just two radical subgroups with non-symplectic 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 [(4,3,4,3*)], 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 [4,(3,5)*], 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.
Hyperbolic paracompact group summary
Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains (double tetrahedra) 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.
[3,5,3] family
There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or
One related non-wythoffian form is constructed from the {3,5,3} vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a tetrahedrally diminished dodecahedron.Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005)
The bitruncated and runcinated forms (5 and 6) contain the faces of two regular skew polyhedrons: {4,10|3} and {10,4|3}.
[5,3,4] family
There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or .
This family is related to the group [5,31,1] by a half symmetry [5,3,4,1+], 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] family
There are 9 forms, generated by ring permutations of the Coxeter group: [5,3,5] or
The bitruncated and runcinated forms (29 and 30) contain the faces of two regular skew polyhedrons: {4,6|5} and {6,4|5}.
[5,3<sup>1,1</sup>] family
There are 11 forms (and only 4 not shared with [5,3,4] family), generated by ring permutations of the Coxeter group: [5,31,1] or .
If the branch ring states match, an extended symmetry can double into the [5,3,4] family,  ↔ .
[(4,3,3,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group:
The bitruncated and runcinated forms (41 and 42) contain the faces of two regular skew polyhedrons: {8,6|3} and {6,8|3}.
[(5,3,3,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group:
The bitruncated and runcinated forms (50 and 51) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.
[(4,3,4,3)] family
There 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 [(4,3,4,3*)], and represents a trigonal trapezohedron fundamental domain.
The truncated forms (57 and 58) contain the faces of two regular skew polyhedrons: {6,6|4} and {8,8|3}.
[(4,3,5,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group:
The truncated forms (65 and 66) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.
[(5,3,5,3)] family
There 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 (72 and 73) contain the faces of two regular skew polyhedrons: {6,6|5} and {10,10|3}.
Summary enumeration of compact uniform honeycombs
This is the complete enumeration of the 76 Wythoffian uniform honeycombs.
The alternations are listed for completeness, but most are non-uniform.
See also
Uniform tilings in hyperbolic plane
List of regular polytopes#Tessellations of hyperbolic 3-space
Notes
References
James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, ,  (Chapter 10, Regular Honeycombs in Hyperbolic Space)
Coxeter, Regular Polytopes, 3rd.
ed., Dover Publications, 1973. .
(Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
Jeffrey R. Weeks The Shape of Space, 2nd edition  (Chapters 16–17: Geometries on Three-manifolds I,II)
Coxeter Decompositions of Hyperbolic Tetrahedra, arXiv/PDF, A. Felikson, December 2002
C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Can.
J. Math. 19, 1179–1186, 1967.
PDF
Norman Johnson, Geometries and Transformations (2018), Chapters 11,12,13
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups 1999, Volume 4, Issue 4, pp 329–353
N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups H3: p130.
