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.

+ Four compact regular hyperbolic honeycombsOrder-4 dodecahedral honeycombOrder-5 dodecahedral honeycombOrder-5 cubic honeycombIcosahedral honeycombPoincaré ball model projections

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. Two known examples are 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+].

IndexedFundamentalsimplexvolumeWittsymbolCoxeternotationCommutatorsubgroupCoxeterdiagramHoneycombsH10.0358850633 [5,3,4] [(5,3)+,4,1+]= [5,31,1]+ 15 forms, 2 regularH20.0390502856 [3,5,3] [3,5,3]+ 9 forms, 1 regularH30.0717701267 [5,31,1] [5,31,1]+ 11 forms (7 overlap with [5,3,4] family, 4 are unique)H40.0857701820 [(4,3,3,3)] [(4,3,3,3)]+ 9 formsH50.0933255395 [5,3,5] [5,3,5]+ 9 forms, 1 regularH60.2052887885 [(5,3,3,3)] [(5,3,3,3)]+ 9 formsH70.2222287320 [(4,3)] [(4,3+,4,3+)] 6 formsH80.3586534401 [(3,4,3,5)] [(3,4,3,5)]+ 9 formsH90.5021308905 [(5,3)] [(5,3)]+ 6 forms

There are just two radical subgroups with non-simplicial 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.

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. DimensionRankGraphsH35 , , , , , , , , , , , , , , , , , , , , , , , , , , ,

[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) Another is constructed with 2 antipodal vertices removed.

The bitruncated and runcinated forms (5 and 6) contain the faces of two regular skew polyhedra: {4,10|3} and {10,4|3}.

#Honeycomb nameCoxeter diagramand SchläflisymbolsCell counts/vertexand positions in honeycombVertex figurePicture01231icosahedral (ikhon)t0{3,5,3} (12)(3.3.3.3.3)2rectified icosahedral (rih)t1{3,5,3}(2)(5.5.5) (3)(3.5.3.5)3truncated icosahedral (tih)t0,1{3,5,3}(1)(5.5.5) (3)(5.6.6)4cantellated icosahedral (srih)t0,2{3,5,3}(1)(3.5.3.5)(2)(4.4.3) (2)(3.5.4.5)5runcinated icosahedral (spiddih)t0,3{3,5,3}(1)(3.3.3.3.3)(5)(4.4.3)(5)(4.4.3)(1)(3.3.3.3.3)6bitruncated icosahedral (dih)t1,2{3,5,3}(2)(3.10.10) (2)(3.10.10)7cantitruncated icosahedral (grih)t0,1,2{3,5,3}(1)(3.10.10)(1)(4.4.3) (2)(4.6.10)8runcitruncated icosahedral (prih)t0,1,3{3,5,3}(1)(3.5.4.5)(1)(4.4.3)(2)(4.4.6)(1)(5.6.6)9omnitruncated icosahedral (gipiddih)t0,1,2,3{3,5,3}(1)(4.6.10)(1)(4.4.6)(1)(4.4.6)(1)(4.6.10)

#Honeycomb nameCoxeter diagramand SchläflisymbolsCell counts/vertexand positions in honeycombVertex figurePicture0123Altpartially diminished icosahedral (pidih)pd{3,5,3}(12)(3.3.3.5)(4)(5.5.5)semi-partially diminished icosahedral (spidih)spd{3,5,3}(6)(3.3.3.5)(6)(3.3.3.3.3)(2)(5.5.5)Nonuniformomnisnub icosahedral (snih)ht0,1,2,3{3,5,3}(1)(3.3.3.3.5)(1)(3.3.3.3(1)(3.3.3.3)(1)(3.3.3.3.5)(4)+(3.3.3)

[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 ↔ .

#Name of honeycombCoxeter diagramCells by location and count per vertexVertex figurePicture012310order-4 dodecahedral (doehon) ↔ - - -(8)(5.5.5)11rectified order-4 dodecahedral (riddoh) ↔ (2)(3.3.3.3) - -(4)(3.5.3.5)12rectified order-5 cubic (ripech) ↔ (5)(3.4.3.4) - -(2)(3.3.3.3.3) 13order-5 cubic (pechon)(20)(4.4.4) - - -14truncated order-4 dodecahedral (tiddoh) ↔ (1)(3.3.3.3) - -(4)(3.10.10)15bitruncated order-5 cubic (ciddoh) ↔ (2)(4.6.6) - -(2)(5.6.6)16truncated order-5 cubic (tipech)(5)(3.8.8) - -(1)(3.3.3.3.3)17cantellated order-4 dodecahedral (sriddoh) ↔ (1)(3.4.3.4)(2)(4.4.4) -(2)(3.4.5.4)18cantellated order-5 cubic (sripech)(2)(3.4.4.4) -(2)(4.4.5)(1)(3.5.3.5)19runcinated order-5 cubic (sidpicdoh)(1)(4.4.4)(3)(4.4.4)(3)(4.4.5)(1)(5.5.5)20cantitruncated order-4 dodecahedral (griddoh) ↔ (1)(4.6.6)(1)(4.4.4) -(2)(4.6.10)21cantitruncated order-5 cubic (gripech)(2)(4.6.8) -(1)(4.4.5)(1)(5.6.6)22runcitruncated order-4 dodecahedral (pripech)(1)(3.4.4.4)(1)(4.4.4)(2)(4.4.10)(1)(3.10.10)23runcitruncated order-5 cubic (priddoh)(1)(3.8.8)(2)(4.4.8)(1)(4.4.5)(1)(3.4.5.4)24omnitruncated order-5 cubic (gidpicdoh)(1)(4.6.8)(1)(4.4.8)(1)(4.4.10)(1)(4.6.10)

#Name of honeycombCoxeter diagramCells by location and count per vertexVertex figurePicture0123Altalternated order-5 cubic (apech) ↔ (20)(3.3.3) (12)(3.3.3.3.3)cantic order-5 cubic (tapech) ↔ (1)(3.5.3.5) -(2)(5.6.6)(2)(3.6.6)runcic order-5 cubic (birapech) ↔ (1)(5.5.5) - (3)(3.4.5.4) (1)(3.3.3)runcicantic order-5 cubic (bitapech) ↔ (1)(3.10.10) -(2)(4.6.10)(1)(3.6.6)Nonuniformsnub rectified order-4 dodecahedral(1)(3.3.3.3.3)(1)(3.3.3) -(2)(3.3.3.3.5)(4)+(3.3.3)Irr. tridiminished icosahedronNonuniformruncic snub rectified order-4 dodecahedral(3.4.4.4)(4.4.4.4) -(3.3.3.3.5)+(3.3.3)Nonuniformomnisnub order-5 cubic(1)(3.3.3.3.4)(1)(3.3.3.4)(1)(3.3.3.5)(1)(3.3.3.3.5)(4)+(3.3.3)

[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 polyhedra: {4,6|5} and {6,4|5}.

#Name of honeycombCoxeter diagramCells by location and count per vertexVertex figurePicture012325(Regular) Order-5 dodecahedral (pedhon)t0{5,3,5} (20)(5.5.5)26rectified order-5 dodecahedral (ripped)t1{5,3,5}(2)(3.3.3.3.3) (5)(3.5.3.5)27truncated order-5 dodecahedral (tipped)t0,1{5,3,5}(1)(3.3.3.3.3) (5)(3.10.10)28cantellated order-5 dodecahedral (sripped)t0,2{5,3,5}(1)(3.5.3.5)(2)(4.4.5) (2)(3.5.4.5)29Runcinated order-5 dodecahedral (spidded)t0,3{5,3,5}(1)(5.5.5)(3)(4.4.5)(3)(4.4.5)(1)(5.5.5)30bitruncated order-5 dodecahedral (diddoh)t1,2{5,3,5}(2)(5.6.6) (2)(5.6.6)31cantitruncated order-5 dodecahedral (gripped)t0,1,2{5,3,5}(1)(5.6.6)(1)(4.4.5) (2)(4.6.10)32runcitruncated order-5 dodecahedral (pripped)t0,1,3{5,3,5}(1)(3.5.4.5)(1)(4.4.5)(2)(4.4.10)(1)(3.10.10)33omnitruncated order-5 dodecahedral (gipidded)t0,1,2,3{5,3,5}(1)(4.6.10)(1)(4.4.10)(1)(4.4.10)(1)(4.6.10)

#Name of honeycombCoxeter diagramCells by location and count per vertexVertex figurePicture0123AltNonuniformomnisnub order-5 dodecahedralht0,1,2,3{5,3,5}(1)(3.3.3.3.5)(1)(3.3.3.5)(1)(3.3.3.5)(1)(3.3.3.3.5)(4)+(3.3.3)

[5,31,1] 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, ↔ .

#Honeycomb nameCoxeter diagramCells by location(and count around each vertex)Vertex figurePicture010'334alternated order-5 cubic (apech) ↔ - -(12)(3.3.3.3.3)(20)(3.3.3)35cantic order-5 cubic (tapech) ↔ (1)(3.5.3.5) -(2)(5.6.6)(2)(3.6.6)36runcic order-5 cubic (birapech) ↔ (1)(5.5.5) - (3)(3.4.5.4) (1)(3.3.3)37runcicantic order-5 cubic (bitapech) ↔ (1)(3.10.10) -(2)(4.6.10)(1)(3.6.6)

#Honeycomb nameCoxeter diagram ↔ Cells by location(and count around each vertex)Vertex figurePicture013Alt Order-4 dodecahedral (doehon) ↔ (4)(5.5.5) - - rectified order-4 dodecahedral (riddoh) ↔ (2)(3.5.3.5) - (2)(3.3.3.3) rectified order-5 cubic (ripech) ↔ (1)(3.3.3.3.3) - (5)(3.4.3.4) bitruncated order-5 cubic (ciddoh) ↔ (1)(5.6.6) - (2)(4.6.6) truncated order-4 dodecahedral (tiddoh) ↔ (2)(3.10.10) - (1)(3.3.3.3) cantellated order-4 dodecahedral (sriddoh) ↔ (1)(3.4.5.4) (2)(4.4.4) (1)(3.4.3.4) cantitruncated order-4 dodecahedral (griddoh) ↔ (1)(4.6.10) (1)(4.4.4) (1)(4.6.6)Nonuniformsnub rectified order-4 dodecahedral ↔ (2)(3.3.3.3.5)(1)(3.3.3)(2)(3.3.3.3.3)(4)+(3.3.3)Irr. tridiminished icosahedron

[(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 polyhedra: {8,6|3} and {6,8|3}.

#Honeycomb nameCoxeter diagramCells by location(and count around each vertex)Vertex figurePicture0123Alt38tetrahedral-cubic (gadtatdic){(3,3,3,4)} (4)(3.3.3) - (4)(4.4.4) (6)(3.4.3.4)39tetrahedral-octahedral (gacocaddit){(3,3,4,3)} (12)(3.3.3.3) (8)(3.3.3) - (8)(3.3.3.3)40cyclotruncated tetrahedral-cubic (cytitch)ct{(3,3,3,4)} (3)(3.6.6) (1)(3.3.3) (1)(4.4.4) (3)(4.6.6)41cyclotruncated cube-tetrahedron (cyticth)ct{(4,3,3,3)} (1)(3.3.3) (1)(3.3.3) (3)(3.8.8) (3)(3.8.8)42cyclotruncated octahedral-tetrahedral (cytoth)ct{(3,3,4,3)} (4)(3.6.6) (4)(3.6.6) (1)(3.3.3.3) (1)(3.3.3.3)43rectified tetrahedral-cubic (ritch)r{(3,3,3,4)} (1)(3.3.3.3) (2)(3.4.3.4) (1)(3.4.3.4) (2)(3.4.4.4)44truncated tetrahedral-cubic (titch)t{(3,3,3,4)} (1)(3.6.6) (1)(3.4.3.4) (1)(3.8.8) (2)(4.6.8)45truncated tetrahedral-octahedral (titdoh)t{(3,3,4,3)} (2)(4.6.6) (1)(3.6.6) (1)(3.4.4.4) (1)(4.6.6)46omnitruncated tetrahedral-cubic (otitch)tr{(3,3,3,4)} (1)(4.6.6) (1)(4.6.6) (1)(4.6.8) (1)(4.6.8)Nonuniformomnisnub tetrahedral-cubicsr{(3,3,3,4)}(1)(3.3.3.3.3)(1)(3.3.3.3.3)(1)(3.3.3.3.4)(1)(3.3.3.3.4)(4)+(3.3.3)

[(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 polyhedra: {10,6|3} and {6,10|3}.

Uniform honeycombs in hyperbolic space

Bron en licentie