The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist.
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 .
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.
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.
[6] In 1997 Wendy Krieger discovered an infinite series of uniform hyperbolic honeycombs with pseudoicosahedral vertex figures, made from 8 cubes and 12 p-gonal prisms at a vertex for any integer p. In the case p = 4, all cells are cubes and the result is the order-5 cubic honeycomb.
However the latter intersect the sphere at infinity orthogonally, having exactly the same curvature as the hyperbolic space, and can be replaced by mirror images of the remainder of the tessellation, resulting in a compact uniform honeycomb consisting only of the truncated cubes.
The order-8 square tilings already intersect the sphere at infinity orthogonally, and if the order-8 triangular tilings are augmented with a set of triangular prisms, the surface passing through their centre points also intersects the sphere at infinity orthogonally.
After replacing with mirror images, the result is a compact honeycomb containing the small rhombicuboctahedra and the triangular prisms.
A compact uniform honeycomb is taken by discarding the order-6 square tilings they have in common, using only the truncated octahedra and cuboctahedra.
These two likewise have the same circumradius, and a compact uniform honeycomb is taken by using only the finite cells of both, discarding the order-4 pentagonal tilings they have in common.
Two known forms generalise the cubic-octahedral honeycomb, having distorted small rhombicuboctahedral vertex figures.