Uniform honeycombs in hyperbolic space

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Four compact regular hyperbolic honeycombs
H3 534 CC center.png
{5,3,4}
H3 535 CC center.png
{5,3,5}
H3 435 CC center.png
{4,3,5}
H3 353 CC center.png
{3,5,3}
Poincaré ball model projections

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[edit]

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[edit]

The nine compact Coxeter groups are listed here with their Coxeter diagrams,[1] in order of the relative volumes of their fundamental simplex domains.[2]

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+].

Indexed Fundamental
simplex
volume[3]
Witt
symbol
Coxeter
notation
Commutator
subgroup
Coxeter
diagram
Honeycombs
H1 0.0358850633 {\bar{BH}}_3 [5,3,4] [(5,3)+,4,1+]
= [5,31,1]+
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 15 forms, 2 regular
H2 0.0390502856 {\bar{J}}_3 [3,5,3] [3,5,3]+ CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 9 forms, 1 regular
H3 0.0717701267 {\bar{DH}}_3 [5,31,1] [5,31,1]+ CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png 11 forms (7 overlap with [5,3,4] family, 4 are unique)
H4 0.0857701820 {\widehat{AB}}_3 [(4,3,3,3)] [(4,3,3,3)]+ CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png 9 forms
H5 0.0933255395 {\bar{K}}_3 [5,3,5] [5,3,5]+ CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 9 forms, 1 regular
H6 0.2052887885 {\widehat{AH}}_3 [(5,3,3,3)] [(5,3,3,3)]+ CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png 9 forms
H7 0.2222287320 {\widehat{BB}}_3 [(4,3)[2]] [(4,3+,4,3+)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png 6 forms
H8 0.3586534401 {\widehat{BH}}_3 [(3,4,3,5)] [(3,4,3,5)]+ CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png 9 forms
H9 0.5021308905 {\widehat{HH}}_3 [(5,3)[2]] [(5,3)[2]]+ CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png 6 forms

There are just two radial 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 [(4,3,4,3*)], represented by Coxeter diagrams CDel branch c1-2.pngCDel 4a4b.pngCDel branch.pngCDel labels.png an index 6 subgroup with a trigonal trapezohedron fundamental domainCDel node c1.pngCDel splitplit1u.pngCDel branch3u c2.pngCDel 3a3buc-cross.pngCDel branch3u c1.pngCDel splitplit2u.pngCDel node c2.png, which can be extended by restoring one mirror as CDel branchu c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel split2-44.pngCDel node.png. The other is [4,(3,5)*], index 120 with a dodecahedral fundamental domain.

Paracompact hyperbolic uniform honeycombs[edit]

Further information: paracompact 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
Type Coxeter groups
Linear graphs CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png | CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png | CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png | CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png | CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png | CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png | CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Tridental graphs CDel node.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel nodes.png | CDel node.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png | CDel node.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.png
Cyclic graphs CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.png | CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png | CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.png | CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png | CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png | CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.pngCDel label4.png | CDel node.pngCDel split1-44.pngCDel nodes.pngCDel split2.pngCDel node.png | CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png | CDel branch.pngCDel splitcross.pngCDel branch.png
Loop-n-tail graphs CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png | CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png | CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png | CDel node.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png

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.

Dimension Rank Graphs
H3 5
CDel node.pngCDel split1.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, CDel node.pngCDel split1-43.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, CDel node.pngCDel split1-44.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, CDel node.pngCDel split1-53.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, CDel node.pngCDel split1-63.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png
CDel branchu.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2-53.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-54.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-55.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-63.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-64.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-65.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-66.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-53.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1-43.pngCDel branchu.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-43.pngCDel branchu.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-44.pngCDel branchu.png, CDel branchu.pngCDel split2-54.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-55.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-63.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-64.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-65.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-66.pngCDel node.pngCDel split1.pngCDel branchu.png

[3,5,3] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png

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.[4]

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

# Honeycomb name
Coxeter diagram
and Schläfli
symbols
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
CDel node n2.pngCDel 5.pngCDel node n3.pngCDel 3.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 5.pngCDel node n3.png
1 icosahedral
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t0{3,5,3}
      (12)
Icosahedron.png
(3.3.3.3.3)
Order-3 icosahedral honeycomb verf.png H3 353 CC center.png
2 rectified icosahedral
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t1{3,5,3}
(2)
Dodecahedron.png
(5.5.5)
    (3)
Icosidodecahedron.png
(3.5.3.5)
Rectified icosahedral honeycomb verf.png H3 353 CC center 0100.png
3 truncated icosahedral
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t0,1{3,5,3}
(1)
Dodecahedron.png
(5.5.5)
    (3)
Truncated icosahedron.png
(5.6.6)
Truncated icosahedral honeycomb verf.png H3 353-0011 center ultrawide.png
4 cantellated icosahedral
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2{3,5,3}
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Triangular prism.png
(4.4.3)
  (2)
Small rhombicosidodecahedron.png
(3.5.4.5)
Cantellated icosahedral honeycomb verf.png H3 353-1010 center ultrawide.png
5 runcinated icosahedral
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,5,3}
(1)
Icosahedron.png
(3.3.3.3.3)
(5)
Triangular prism.png
(4.4.3)
(5)
Triangular prism.png
(4.4.3)
(1)
Icosahedron.png
(3.3.3.3.3)
Runcinated icosahedral honeycomb verf.png H3 353-1001 center ultrawide.png
6 bitruncated icosahedral
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{3,5,3}
(2)
Truncated dodecahedron.png
(3.10.10)
    (2)
Truncated dodecahedron.png
(3.10.10)
Bitruncated icosahedral honeycomb verf.png H3 353-0110 center ultrawide.png
7 cantitruncated icosahedral
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2{3,5,3}
(1)
Truncated dodecahedron.png
(3.10.10)
(1)
Triangular prism.png
(4.4.3)
  (2)
Great rhombicosidodecahedron.png
(4.6.10)
Cantitruncated icosahedral honeycomb verf.png H3 353-1110 center ultrawide.png
8 runcitruncated icosahedral
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,5,3}
(1)
Small rhombicosidodecahedron.png
(3.5.4.5)
(1)
Triangular prism.png
(4.4.3)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Truncated icosahedron.png
(5.6.6)
Runcitruncated icosahedral honeycomb verf.png H3 353-1101 center ultrawide.png
9 omnitruncated icosahedral
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,5,3}
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Great rhombicosidodecahedron.png
(4.6.10)
Omnitruncated icosahedral honeycomb verf.png H3 353-1111 center ultrawide.png
# Honeycomb name
Coxeter diagram
and Schläfli
symbols
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
CDel node n2.pngCDel 5.pngCDel node n3.pngCDel 3.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 5.pngCDel node n3.png
Alt
[77] partially diminished icosahedral
pd{3,5,3}[5]
(12)
Pentagonal antiprism.png
(3.3.3.5)
(4)
Dodecahedron.png
(5.5.5)
Partial truncation order-3 icosahedral honeycomb verf.png H3 353-pd center ultrawide.png
Nonuniform omnisnub icosahedral
CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{3,5,3}
(1)
Snub dodecahedron cw.png
(3.3.3.3.5)
(1)
Octahedron.png
(3.3.3.3
(1)
Octahedron.png
(3.3.3.3)
(1)
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Snub icosahedral honeycomb verf.png

[5,3,4] family[edit]

There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png.

This family is related to the group [5,31,1] by a half symmetry [5,3,4,1+], or CDel node c1.pngCDel 5.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png, when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive CDel node c1.pngCDel 5.pngCDel node c2.pngCDel split1.pngCDel nodes 10lu.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png.

# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 4.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 4.pngCDel node n4.png
2
CDel node n1.pngCDel 5.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 5.pngCDel node n2.pngCDel 3.pngCDel node n3.png
10 order-4 dodecahedral
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
- - - (8)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
(5.5.5)
Order-4 dodecahedral honeycomb verf.png H3 534 CC center.png
11 rectified order-4 dodecahedral
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes.png
(2)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png
(3.3.3.3)
- - (4)
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
(3.5.3.5)
Rectified order-4 dodecahedral honeycomb verf.png H3 534 CC center 0100.png
12 rectified order-5 cubic
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 11.png
(5)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png
(3.4.3.4)
- - (2)
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Icosahedron.png
(3.3.3.3.3)
Rectified order-5 cubic honeycomb verf.png H3 435 CC center 0100.png
13 order-5 cubic
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
(20)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png
(4.4.4)
- - - Order-5 cubic honeycomb verf.png H3 435 CC center.png
14 truncated order-4 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes.png
(1)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png
(3.3.3.3)
- - (4)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated dodecahedron.png
(3.10.10)
Truncated order-4 dodecahedral honeycomb verf.png H3 435-0011 center ultrawide.png
15 bitruncated order-5 cubic
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
(2)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated octahedron.png
(4.6.6)
- - (2)
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncated icosahedron.png
(5.6.6)
Bitruncated order-5 cubic honeycomb verf.png H3 534-0110 center ultrawide.png
16 truncated order-5 cubic
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
(5)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png
(3.8.8)
- - (1)
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Icosahedron.png
(3.3.3.3.3)
Truncated order-5 cubic honeycomb verf.png H3 534-0011 center ultrawide.png
17 cantellated order-4 dodecahedral
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 11.png
(1)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png
(3.4.3.4)
(2)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
(4.4.4)
- (2)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Small rhombicosidodecahedron.png
(3.4.5.4)
Cantellated order-4 dodecahedral honeycomb verf.png H3 534-1010 center ultrawide.png
18 cantellated order-5 cubic
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
(2)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Small rhombicuboctahedron.png
(3.4.4.4)
- (2)
CDel node.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Pentagonal prism.png
(4.4.5)
(1)
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
(3.5.3.5)
Cantellated order-5 cubic honeycomb verf.png H3 534-0101 center ultrawide.png
19 runcinated order-5 cubic
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
(1)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png
(4.4.4)
(3)
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png
Tetragonal prism.png
(4.4.4)
(3)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png
Pentagonal prism.png
(4.4.5)
(1)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
(5.5.5)
Runcinated order-5 cubic honeycomb verf.png H3 534-1001 center ultrawide.png
20 cantitruncated order-4 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
(1)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated octahedron.png
(4.6.6)
(1)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
(4.4.4)
- (2)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Great rhombicosidodecahedron.png
(4.6.10)
Cantitruncated order-4 dodecahedral honeycomb verf.png H3 534-1110 center ultrawide.png
21 cantitruncated order-5 cubic
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
(2)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Great rhombicuboctahedron.png
(4.6.8)
- (1)
CDel node.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Pentagonal prism.png
(4.4.5)
(1)
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncated icosahedron.png
(5.6.6)
Cantitruncated order-5 cubic honeycomb verf.png H3 534-0111 center ultrawide.png
22 runcitruncated order-4 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
(1)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png
Tetragonal prism.png
(4.4.4)
(2)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Decagonal prism.png
(4.4.10)
(1)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated dodecahedron.png
(3.10.10)
Runcitruncated order-4 dodecahedral honeycomb verf.png H3 534-1101 center ultrawide.png
23 runcitruncated order-5 cubic
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
(1)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png
(3.8.8)
(2)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Octagonal prism.png
(4.4.8)
(1)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png
Pentagonal prism.png
(4.4.5)
(1)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Small rhombicosidodecahedron.png
(3.4.5.4)
Runcitruncated order-5 cubic honeycomb verf.png H3 534-1011 center ultrawide.png
24 omnitruncated order-5 cubic
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
(1)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Great rhombicuboctahedron.png
(4.6.8)
(1)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Octagonal prism.png
(4.4.8)
(1)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Decagonal prism.png
(4.4.10)
(1)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Great rhombicosidodecahedron.png
(4.6.10)
Omnitruncated order-4 dodecahedral honeycomb verf.png H3 534-1111 center ultrawide.png
# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 4.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 4.pngCDel node n4.png
2
CDel node n1.pngCDel 5.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 5.pngCDel node n2.pngCDel 3.pngCDel node n3.png
Alt
[34] alternated order-5 cubic
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
(20)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Tetrahedron.png
(3.3.3)
    (12)
Icosahedron.png
(3.3.3.3.3)
Alternated order-5 cubic honeycomb verf.png Alternated order 5 cubic honeycomb.png
[35] cantic order-5 cubic
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
(1)
Icosidodecahedron.png
(3.5.3.5)
- (2)
Truncated icosahedron.png
(5.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
Truncated alternated order-5 cubic honeycomb verf.png H3 5311-0110 center ultrawide.png
[36] runcic order-5 cubic
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
(1)
Dodecahedron.png
(5.5.5)
- (3)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Tetrahedron.png
(3.3.3)
Runcinated alternated order-5 cubic honeycomb verf.png H3 5311-1010 center ultrawide.png
[37] runcicantic order-5 cubic
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
(1)
Truncated dodecahedron.png
(3.10.10)
- (2)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Truncated tetrahedron.png
(3.6.6)
Runcitruncated alternated order-5 cubic honeycomb verf.png H3 5311-1110 center ultrawide.png
Nonuniform snub rectified order-4 dodecahedral
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
(1)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Uniform polyhedron-43-h01.svg
(3.3.3.3.3)
(1)
CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node.png
Tetrahedron.png
(3.3.3)
- (2)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Alternated cantitruncated order-4 dodecahedral honeycomb verf.png
Irr. tridiminished icosahedron
Nonuniform runcic snub rectified order-4 dodecahedral
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
Rhombicuboctahedron uniform edge coloring.png
(3.4.4.4)
CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node 1.png
Cube rotorotational symmetry.png
(4.4.4.4)
- CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub dodecahedron cw.png
(3.3.3.3.5)
Tetrahedron.png
+(3.3.3)
Nonuniform omnisnub order-5 cubic
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
(1)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
Snub hexahedron.png
(3.3.3.3.4)
(1)
CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node h.png
Square antiprism.png
(3.3.3.4)
(1)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 2.pngCDel node h.png
Pentagonal antiprism.png
(3.3.3.5)
(1)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Snub order-4 dodecahedral honeycomb verf.png

[5,3,5] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: [5,3,5] or CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

The bitruncated and runcinated forms (29 and 30) contain the faces of two regular skew polyhedrons: {4,6|5} and {6,4|5}.

# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
1
CDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
2
CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png
3
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
25 (Regular) Order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t0{5,3,5}
      (20)
Dodecahedron.png
(5.5.5)
Order-5 dodecahedral honeycomb verf.png H3 535 CC center.png
26 rectified order-5 dodecahedral
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t1{5,3,5}
(2)
Icosahedron.png
(3.3.3.3.3)
    (5)
Icosidodecahedron.png
(3.5.3.5)
Rectified order-5 dodecahedral honeycomb verf.png H3 535 CC center 0100.png
27 truncated order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t0,1{5,3,5}
(1)
Icosahedron.png
(3.3.3.3.3)
    (5)
Truncated dodecahedron.png
(3.10.10)
Truncated order-5 dodecahedral honeycomb verf.png H3 535-0011 center ultrawide.png
28 cantellated order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t0,2{5,3,5}
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Pentagonal prism.png
(4.4.5)
  (2)
Small rhombicosidodecahedron.png
(3.5.4.5)
Cantellated order-5 dodecahedral honeycomb verf.png H3 535-1010 center ultrawide.png
29 Runcinated order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,3{5,3,5}
(1)
Dodecahedron.png
(5.5.5)
(3)
Pentagonal prism.png
(4.4.5)
(3)
Pentagonal prism.png
(4.4.5)
(1)
Dodecahedron.png
(5.5.5)
Runcinated order-5 dodecahedral honeycomb verf.png H3 535-1001 center ultrawide.png
30 bitruncated order-5 dodecahedral
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t1,2{5,3,5}
(2)
Truncated icosahedron.png
(5.6.6)
    (2)
Truncated icosahedron.png
(5.6.6)
Bitruncated order-5 dodecahedral honeycomb verf.png H3 535-0110 center ultrawide.png
31 cantitruncated order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t0,1,2{5,3,5}
(1)
Truncated icosahedron.png
(5.6.6)
(1)
Pentagonal prism.png
(4.4.5)
  (2)
Great rhombicosidodecahedron.png
(4.6.10)
Cantitruncated order-5 dodecahedral honeycomb verf.png H3 535-1110 center ultrawide.png
32 runcitruncated order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,1,3{5,3,5}
(1)
Small rhombicosidodecahedron.png
(3.5.4.5)
(1)
Pentagonal prism.png
(4.4.5)
(2)
Decagonal prism.png
(4.4.10)
(1)
Truncated dodecahedron.png
(3.10.10)
Runcitruncated order-5 dodecahedral honeycomb verf.png H3 535-1101 center ultrawide.png
33 omnitruncated order-5 dodecahedral
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t0,1,2,3{5,3,5}
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Decagonal prism.png
(4.4.10)
(1)
Decagonal prism.png
(4.4.10)
(1)
Great rhombicosidodecahedron.png
(4.6.10)
Omnitruncated order-5 dodecahedral honeycomb verf.png H3 535-1111 center ultrawide.png
# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
1
CDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
2
CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png
3
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Alt
Nonuniform omnisnub order-5 dodecahedral
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
ht0,1,2,3{5,3,5}
(1)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
Snub dodecahedron cw.png
(3.3.3.3.5)
(1)
CDel node h.pngCDel 2.pngCDel node h.pngCDel 5.pngCDel node h.png
Pentagonal antiprism.png
(3.3.3.5)
(1)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 2.pngCDel node h.png
Pentagonal antiprism.png
(3.3.3.5)
(1)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Snub order-5 dodecahedral honeycomb verf.png

[5,31,1] family[edit]

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 CDel nodes.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png. If the branch ring states match, an extended symmetry can double into the [5,3,4] family, CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 5.pngCDel node c3.pngCDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 5.pngCDel node c3.png.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 5a.pngCDel nodea.png
1
CDel nodes.pngCDel 2.pngCDel node.png
0'
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 5a.pngCDel nodea.png
3
CDel nodes.pngCDel split2.pngCDel node.png
34 alternated order-5 cubic
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
- - (12)
Icosahedron.png
(3.3.3.3.3)
(20)
Tetrahedron.png
(3.3.3)
Alternated order-5 cubic honeycomb verf.png Alternated order 5 cubic honeycomb.png
35 cantic order-5 cubic
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
(1)
Icosidodecahedron.png
(3.5.3.5)
- (2)
Truncated icosahedron.png
(5.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
Truncated alternated order-5 cubic honeycomb verf.png H3 5311-0110 center ultrawide.png
36 runcic order-5 cubic
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
(1)
Dodecahedron.png
(5.5.5)
- (3)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Tetrahedron.png
(3.3.3)
Runcinated alternated order-5 cubic honeycomb verf.png H3 5311-1010 center ultrawide.png
37 runcicantic order-5 cubic
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
(1)
Truncated dodecahedron.png
(3.10.10)
- (2)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Truncated tetrahedron.png
(3.6.6)
Runcitruncated alternated order-5 cubic honeycomb verf.png H3 5311-1110 center ultrawide.png
# Honeycomb name
Coxeter diagram
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 5.pngCDel node c3.pngCDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 5.pngCDel node c3.png
Cells by location
(and count around each vertex)
vertex figure Picture
0
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 5a.pngCDel nodea.png
1
CDel nodes.pngCDel 2.pngCDel node.png
3
CDel nodes.pngCDel split2.pngCDel node.png
Alt
[10] Order-4 dodecahedral
CDel nodes.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
(4)
Dodecahedron.png
(5.5.5)
- - Order-4 dodecahedral honeycomb verf.png H3 534 CC center.png
[11] rectified order-4 dodecahedral
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
(2)
Icosidodecahedron.png
(3.5.3.5)
- (2)
Uniform polyhedron-33-t1.png
(3.3.3.3)
Rectified alternated order-5 cubic honeycomb verf.png H3 534 CC center 0100.png
[12] rectified order-5 cubic
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
(1)
Icosahedron.png
(3.3.3.3.3)
- (5)
Uniform polyhedron-33-t02.png
(3.4.3.4)
Cantellated alternated order-5 cubic honeycomb verf.png H3 435 CC center 0100.png
[15] bitruncated order-5 cubic
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
(1)
Truncated icosahedron.png
(5.6.6)
- (2)
Uniform polyhedron-33-t012.png
(4.6.6)
Cantitruncated alternated order-5 cubic honeycomb verf.png H3 534-0110 center ultrawide.png
[14] truncated order-4 dodecahedral
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
(2)
Truncated dodecahedron.png
(3.10.10)
- (1)
Uniform polyhedron-33-t1.png
(3.3.3.3)
Bicantellated alternated order-5 cubic honeycomb verf.png H3 435-0011 center ultrawide.png
[17] cantellated order-4 dodecahedral
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
(1)
Small rhombicosidodecahedron.png
(3.4.5.4)
(2)
Uniform polyhedron 222-t012.png
(4.4.4)
(1)
Uniform polyhedron-33-t02.png
(3.4.3.4)
Runcicantellated alternated order-5 cubic honeycomb verf.png H3 534-1010 center ultrawide.png
[20] cantitruncated order-4 dodecahedral
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Uniform polyhedron 222-t012.png
(4.4.4)
(1)
Uniform polyhedron-33-t012.png
(4.6.6)
Omnitruncated alternated order-5 cubic honeycomb verf.png H3 534-1110 center ultrawide.png
Nonuniform snub rectified order-4 dodecahedral
CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
(2)
Snub dodecahedron cw.png
(3.3.3.3.5)
(1)
Uniform polyhedron-33-t0.png
(3.3.3)
(2)
Uniform polyhedron-33-s012.png
(3.3.3.3.3)
(4)
Uniform polyhedron-33-t2.png
+(3.3.3)
Alternated cantitruncated order-4 dodecahedral honeycomb verf.png
Irr. tridiminished icosahedron

[(4,3,3,3)] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png

The bitruncated and runcinated forms (41 and 42) contain the faces of two regular skew polyhedrons: {8,6|3} and {6,8|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
CDel nodea.pngCDel 3a.pngCDel branch.png
1
CDel nodeb.pngCDel 3b.pngCDel branch.png
2
CDel label4.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
3
CDel label4.pngCDel branch.pngCDel 3a.pngCDel nodea.png
Alt
38 tetrahedral-cubic
CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.png
{(3,3,3,4)}
(4)
Tetrahedron.png
(3.3.3)
- (4)
Hexahedron.png
(4.4.4)
(6)
Cuboctahedron.png
(3.4.3.4)
Uniform t0 4333 honeycomb verf.png H3 4333-1000 center ultrawide.png
39 tetrahedral-octahedral
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.png
{(3,3,4,3)}
(12)
Uniform polyhedron-33-t1.png
(3.3.3.3)
(8)
Tetrahedron.png
(3.3.3)
- (8)
Octahedron.png
(3.3.3.3)
Uniform t2 4333 honeycomb verf.png H3 4333-0100 center ultrawide.png
40 cyclotruncated tetrahedral-cubic
CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png
ct{(3,3,3,4)}
(3)
Truncated tetrahedron.png
(3.6.6)
(1)
Tetrahedron.png
(3.3.3)
(1)
Hexahedron.png
(4.4.4)
(3)
Truncated octahedron.png
(4.6.6)
Uniform t12 4333 honeycomb verf.png H3 4333-0110 center ultrawide.png
41 cyclotruncated cube-tetrahedron
CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch.png
ct{(4,3,3,3)}
(1)
Tetrahedron.png
(3.3.3)
(1)
Tetrahedron.png
(3.3.3)
(3)
Truncated hexahedron.png
(3.8.8)
(3)
Truncated hexahedron.png
(3.8.8)
Uniform t01 4333 honeycomb verf.png H3 4333-1100 center ultrawide.png
42 cyclotruncated tetrahedral-octahedral
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch 11.png
ct{(3,3,4,3)}
(4)
Truncated tetrahedron.png
(3.6.6)
(4)
Truncated tetrahedron.png
(3.6.6)
(1)
Octahedron.png
(3.3.3.3)
(1)
Octahedron.png
(3.3.3.3)
Uniform t23 4333 honeycomb verf.png H3 4333-0011 center ultrawide.png
43 rectified tetrahedral-cubic
CDel label4.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.png
r{(3,3,3,4)}
(1)
Uniform polyhedron-33-t1.png
(3.3.3.3)
(2)
Uniform polyhedron-33-t02.png
(3.4.3.4)
(1)
Cuboctahedron.png
(3.4.3.4)
(2)
Small rhombicuboctahedron.png
(3.4.4.4)
Uniform t02 4333 honeycomb verf.png H3 4333-1010 center ultrawide.png
44 truncated tetrahedral-cubic
CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.png
t{(3,3,3,4)}
(1)
Truncated tetrahedron.png
(3.6.6)
(1)
Uniform polyhedron-33-t02.png
(3.4.3.4)
(1)
Truncated hexahedron.png
(3.8.8)
(2)
Great rhombicuboctahedron.png
(4.6.8)
Uniform t012 4333 honeycomb verf.png H3 4333-1110 center ultrawide.png
45 truncated tetrahedral-octahedral
CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.png
t{(3,3,4,3)}
(2)
Uniform polyhedron-33-t012.png
(4.6.6)
(1)
Truncated tetrahedron.png
(3.6.6)
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
Truncated octahedron.png
(4.6.6)
Uniform t123 4333 honeycomb verf.png H3 4333-0111 center ultrawide.png
46 omnitruncated tetrahedral-cubic
CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.png
tr{(3,3,3,4)}
(1)
Uniform polyhedron-33-t012.png
(4.6.6)
(1)
Uniform polyhedron-33-t012.png
(4.6.6)
(1)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Great rhombicuboctahedron.png
(4.6.8)
Uniform t0123 4333 honeycomb verf.png H3 4333-1111 center ultrawide.png
Nonuniform omnisnub tetrahedral-cubic
CDel label4.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.png
sr{(3,3,3,4)}
(1)
Uniform polyhedron-33-s012.png
(3.3.3.3.3)
(1)
Uniform polyhedron-33-s012.png
(3.3.3.3.3)
(1)
Snub hexahedron.png
(3.3.3.3.4)
(1)
Snub hexahedron.png
(3.3.3.3.4)
(4)
Tetrahedron.png
+(3.3.3)
Snub 4333 honeycomb verf.png

[(5,3,3,3)] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png

The bitruncated and runcinated forms (50 and 51) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
CDel nodea.pngCDel 3a.pngCDel branch.png
1
CDel nodeb.pngCDel 3b.pngCDel branch.png
2
CDel label5.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
3
CDel label5.pngCDel branch.pngCDel 3a.pngCDel nodea.png
47 tetrahedral-dodecahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.png
(4)
Tetrahedron.png
(3.3.3)
- (4)
Dodecahedron.png
(5.5.5)
(6)
Icosidodecahedron.png
(3.5.3.5)
Uniform t0 5333 honeycomb verf.png H3 5333-1000 center ultrawide.png
48 tetrahedral-icosahedral
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.png
(30)
Uniform polyhedron-33-t1.png
(3.3.3.3)
(20)
Tetrahedron.png
(3.3.3)
- (12)
Icosahedron.png
(3.3.3.3.3)
Uniform t2 5333 honeycomb verf.png H3 5333-0010 center ultrawide.png
49 cyclotruncated tetrahedral-dodecahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png
(3)
Truncated tetrahedron.png
(3.6.6)
(1)
Tetrahedron.png
(3.3.3)
(1)
Dodecahedron.png
(5.5.5)
(3)
Truncated icosahedron.png
(5.6.6)
Uniform t12 5333 honeycomb verf.png H3 5333-0110 center ultrawide.png
52 rectified tetrahedral-dodecahedral
CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.png
(1)
Uniform polyhedron-33-t1.png
(3.3.3.3)
(2)
Uniform polyhedron-33-t02.png
(3.4.3.4)
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Small rhombicosidodecahedron.png
(3.4.5.4)
Uniform t02 5333 honeycomb verf.png H3 5333-1010 center ultrawide.png
53 truncated tetrahedral-dodecahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.png
(1)
Truncated tetrahedron.png
(3.6.6)
(1)
Uniform polyhedron-33-t02.png
(3.4.3.4)
(1)
Truncated dodecahedron.png
(3.10.10)
(2)
Great rhombicosidodecahedron.png
(4.6.10)
Uniform t012 5333 honeycomb verf.png H3 5333-1110 center ultrawide.png
54 truncated tetrahedral-icosahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.png
(2)
Uniform polyhedron-33-t012.png
(4.6.6)
(1)
Truncated tetrahedron.png
(3.6.6)
(1)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Truncated icosahedron.png
(5.6.6)
Uniform t123 5333 honeycomb verf.png H3 5333-0111 center ultrawide.png
# Honeycomb name
Coxeter diagram
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.png
Cells by location
(and count around each vertex)
vertex figure Picture
0,1
CDel nodea.pngCDel 3a.pngCDel branch.png
2,3
CDel label5.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
Alt
50 cyclotruncated dodecahedral-tetrahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.png
(2)
Tetrahedron.png
(3.3.3)
(6)
Truncated dodecahedron.png
(3.10.10)
Uniform t01 5333 honeycomb verf.png H3 5333-1100 center ultrawide.png
51 cyclotruncated tetrahedral-icosahedral
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 11.png
(10)
Truncated tetrahedron.png
(3.6.6)
(2)
Icosahedron.png
(3.3.3.3.3)
Uniform t23 5333 honeycomb verf.png H3 5333-0011 center ultrawide.png
55 omnitruncated tetrahedral-dodecahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.png
(2)
Uniform polyhedron-33-t012.png
(4.6.6)
(2)
Great rhombicosidodecahedron.png
(4.6.10)
Uniform t0123 5333 honeycomb verf.png H3 5333-1111 center ultrawide.png
Nonuniform omnisnub tetrahedral-dodecahedral
CDel label5.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.png
(2)
Uniform polyhedron-33-s012.png
(3.3.3.3.3)
(2)
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Snub 5333 honeycomb verf.png

[(4,3,4,3)] family[edit]

There are 6 forms, generated by ring permutations of the Coxeter group: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png. There are 4 extended symmetries possible based on the symmetry of the rings: CDel label4.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.pngCDel label4.png, CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label4.png, CDel label4.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel label4.png, and CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label4.png.

This symmetry family is also related to a radial subgroup, index 6, CDel branch c1-2.pngCDel 4a4b.pngCDel branch.pngCDel labels.pngCDel node c1.pngCDel splitplit1u.pngCDel branch3u c2.pngCDel 3a3buc-cross.pngCDel branch3u c1.pngCDel splitplit2u.pngCDel node c2.png, 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}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Pictures
0
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
1
CDel nodeb.pngCDel 3b.pngCDel branch.pngCDel label4.png
2
CDel label4.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
3
CDel label4.pngCDel branch.pngCDel 3a.pngCDel nodea.png
56 cubic-octahedral
CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label4.png
(6)
Octahedron.png
(3.3.3.3)
- (8)
Hexahedron.png
(4.4.4)
(12)
Cuboctahedron.png
(3.4.3.4)
Uniform t0 4343 honeycomb verf.png H3 4343-1000 center ultrawide.png
60 truncated cubic-octahedral
CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(1)
Truncated octahedron.png
(4.6.6)
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
Truncated hexahedron.png
(3.8.8)
(2)
Great rhombicuboctahedron.png
(4.6.8)
Uniform t012 4343 honeycomb verf.png H3 4343-1110 center ultrawide.png
# Honeycomb name
Coxeter diagram
CDel label4.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.pngCDel label4.png
Cells by location
(and count around each vertex)
vertex figure Picture
0,3
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
1,2
CDel nodeb.pngCDel 3b.pngCDel branch.pngCDel label4.png
Alt
57 cyclotruncated octahedral-cubic
CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(6)
Truncated octahedron.png
(4.6.6)
(2)
Hexahedron.png
(4.4.4)
Uniform t12 4343 honeycomb verf.png H3 4343-0110 center ultrawide.png
Nonuniform cyclosnub octahedral-cubic
CDel label4.pngCDel branch h0r.pngCDel 3ab.pngCDel branch h0l.pngCDel label4.png
(4)
Uniform polyhedron-43-h01.png
(3.3.3.3.3)
(2)
Tetrahedron.png
(3.3.3)
(4)
Octahedron.png
+(3.3.3.3)
Alternated truncated 3434 honeycomb verf.png
# Honeycomb name
Coxeter diagram
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label4.png
Cells by location
(and count around each vertex)
vertex figure Picture
0,1
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
2,3
CDel label4.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
58 cyclotruncated cubic-octahedral
CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label4.png
(2)
Octahedron.png
(3.3.3.3)
(6)
Truncated hexahedron.png
(3.8.8)
Uniform t01 4343 honeycomb verf.png H3 4343-0110 center ultrawide.png
# Honeycomb name
Coxeter diagram
CDel label4.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel label4.png
Cells by location
(and count around each vertex)
vertex figure Picture
0,2
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
1,3
CDel nodeb.pngCDel 3b.pngCDel branch.pngCDel label4.png
59 rectified cubic-octahedral
CDel label4.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(2)
Cuboctahedron.png
(3.4.3.4)
(4)
Small rhombicuboctahedron.png
(3.4.4.4)
Uniform t02 4343 honeycomb verf.png H3 4343-1010 center ultrawide.png
# Honeycomb name
Coxeter diagram
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label4.png
Cells by location
(and count around each vertex)
vertex figure Picture
0,1,2,3
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
Alt
61 omnitruncated cubic-octahedral
CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png
(4)
Great rhombicuboctahedron.png
(4.6.8)
Uniform t0123 4343 honeycomb verf.png H3 4343-1111 center ultrawide.png
Nonuniform omnisnub cubic-octahedral
CDel label4.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.pngCDel label4.png
(4)
Snub hexahedron.png
(3.3.3.3.4)
(4)
Tetrahedron.png
+(3.3.3)
Snub 4343 honeycomb verf.png

[(4,3,5,3)] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png

The truncated forms (65 and 66) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
1
CDel nodeb.pngCDel 3b.pngCDel branch.pngCDel label4.png
2
CDel label5.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
3
CDel label5.pngCDel branch.pngCDel 3a.pngCDel nodea.png
62 octahedral-dodecahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label4.png
(6)
Octahedron.png
(3.3.3.3)
- (8)
Dodecahedron.png
(5.5.5)
(1)
Icosidodecahedron.png
(3.5.3.5)
Uniform t0 5343 honeycomb verf.png H3 4353-0010 center ultrawide.png
63 cubic-icosahedral
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(30)
Cuboctahedron.png
(3.4.3.4)
(20)
Hexahedron.png
(4.4.4)
- (12)
Icosahedron.png
(3.3.3.3.3)
Uniform t2 5343 honeycomb verf.png H3 4353-1000 center ultrawide.png
64 cyclotruncated octahedral-dodecahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(3)
Truncated octahedron.png
(4.6.6)
(1)
Hexahedron.png
(4.4.4)
(1)
Dodecahedron.png
(5.5.5)
(3)
Truncated icosahedron.png
(5.6.6)
Uniform t12 5343 honeycomb verf.png H3 4353-0110 center ultrawide.png
67 rectified octahedral-dodecahedral
CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(1)
Cuboctahedron.png
(3.4.3.4)
(2)
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Small rhombicosidodecahedron.png
(3.4.5.4)
Uniform t02 5343 honeycomb verf.png H3 4353-0101 center ultrawide.png
68 truncated octahedral-dodecahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
(1)
Truncated octahedron.png
(4.6.6)
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
Truncated dodecahedron.png
(3.10.10)
(2)
Great rhombicosidodecahedron.png
(4.6.10)
Uniform t012 5343 honeycomb verf.png H3 4353-1110 center ultrawide.png
69 truncated cubic-dodecahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png
(2)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Truncated hexahedron.png
(3.8.8)
(1)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Truncated icosahedron.png
(5.6.6)
Uniform t123 5343 honeycomb verf.png H3 4353-0111 center ultrawide.png
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0,1
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label4.png
2,3
CDel label5.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
Alt
65 cyclotruncated dodecahedral-octahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label4.png
(2)
Octahedron.png
(3.3.3.3)
(8)
Truncated dodecahedron.png
(3.10.10)
Uniform t01 5343 honeycomb verf.png H3 4353-1100 center ultrawide.png
66 cyclotruncated cubic-icosahedral
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png
(10)
Truncated hexahedron.png
(3.8.8)
(2)
Icosahedron.png
(3.3.3.3.3)
Uniform t23 5343 honeycomb verf.png H3 4353-0011 center ultrawide.png
70 omnitruncated octahedral-dodecahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png
(2)
Great rhombicuboctahedron.png
(4.6.8)
(2)
Great rhombicosidodecahedron.png
(4.6.10)
Uniform t0123 5343 honeycomb verf.png H3 4353-1111 center ultrawide.png
Nonuniform omnisnub octahedral-dodecahedral
CDel label5.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.pngCDel label4.png
(2)
Snub hexahedron.png
(3.3.3.3.4)
(2)
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Snub 5343 honeycomb verf.png

[(5,3,5,3)] family[edit]

There are 6 forms, generated by ring permutations of the Coxeter group: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png. There are 4 extended symmetries possible based on the symmetry of the rings: CDel label5.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.pngCDel label5.png, CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label5.png, CDel label5.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel label5.png, and CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label5.png.

The truncated forms (72 and 73) contain the faces of two regular skew polyhedrons: {6,6|5} and {10,10|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel label5.png
1
CDel nodeb.pngCDel 3b.pngCDel branch.pngCDel label5.png
2
CDel label5.pngCDel branch.pngCDel 3b.pngCDel nodeb.png
3
CDel label5.pngCDel branch.pngCDel 3a.pngCDel nodea.png
Alt
71 dodecahedral-icosahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label5.png
(12)
Icosahedron.png
(3.3.3.3.3)
- (20)
Dodecahedron.png
(5.5.5)
(30)
Icosidodecahedron.png
(3.5.3.5)
Uniform t0 5353 honeycomb verf.png H3 5353-1000 center ultrawide.png
72 cyclotruncated icosahedral-dodecahedral
CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png
(3)
Truncated icosahedron.png
(5.6.6)
(1)
Dodecahedron.png
(5.5.5)
(1)
Dodecahedron.png
(5.5.5)
(3)
Truncated icosahedron.png
(5.6.6)
Uniform t12 5353 honeycomb verf.png H3 5353-0110 center ultrawide.png
73 cyclotruncated dodecahedral-icosahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label5.png
(1)
Icosahedron.png
(3.3.3.3.3)
(1)
Icosahedron.png
(3.3.3.3.3)
(3)
Truncated dodecahedron.png
(3.10.10)
(3)
Truncated dodecahedron.png
(3.10.10)
Uniform t01 5353 honeycomb verf.png H3 5353-1100 center ultrawide.png
74 rectified dodecahedral-icosahedral
CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Small rhombicosidodecahedron.png
(3.4.5.4)
Uniform t02 5353 honeycomb verf.png H3 5353-1010 center ultrawide.png
75 truncated dodecahedral-icosahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png
(1)
Truncated icosahedron.png
(5.6.6)
(1)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Truncated dodecahedron.png
(3.10.10)
(2)
Great rhombicosidodecahedron.png
(4.6.10)
Uniform t012 5353 honeycomb verf.png H3 5353-1101 center ultrawide.png
76 omnitruncated dodecahedral-icosahedral
CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Great rhombicosidodecahedron.png
(4.6.10)
Uniform t0123 5353 honeycomb verf.png H3 5353-1111 center ultrawide.png
Nonuniform omnisnub dodecahedral-icosahedral
CDel label5.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.pngCDel label5.png
(1)
Snub dodecahedron cw.png
(3.3.3.3.5)
(1)
Snub dodecahedron cw.png
(3.3.3.3.5)
(1)
Snub dodecahedron cw.png
(3.3.3.3.5)
(1)
Snub dodecahedron cw.png
(3.3.3.3.5)
(4)
Tetrahedron.png
+(3.3.3)
Snub 5353 honeycomb verf.png

Summary enumeration of compact uniform honeycombs[edit]

This is the complete enumeration of the 76 Wythoffian uniform honeycombs. The alternations are listed for completeness, but most are non-uniform.

Index Coxeter group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs
H1 {\bar{BH}}_3
[4,3,5]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[4,3,5]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 5.pngCDel node c4.png
15 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png | CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png | CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png | CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png | CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
[1+,4,(3,5)+] (2) CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png (= CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
[4,3,5]+ (1) CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
H2 {\bar{J}}_3
[3,5,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
[3,5,3]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 5.pngCDel node c3.pngCDel 3.pngCDel node c4.png
6 CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png | CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png | CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
[2+[3,5,3]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c1.png
5 CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png | CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png [2+[3,5,3]]+ (1) CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
H3 {\bar{DH}}_3
[5,31,1]
CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
[5,31,1]
CDel node c3.pngCDel 5.pngCDel node c4.pngCDel split1.pngCDel nodeab c1-2.png
4 CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png | CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
[1[5,31,1]]=[5,3,4]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png
(7) CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes.png | CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 11.png | CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 11.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png [1[5,31,1]]+
=[5,3,4]+
(1) CDel node h.pngCDel 5.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
H4 {\widehat{AB}}_3
[(4,3,3,3)]
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png
[(4,3,3,3)] 6 CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.png | CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.png | CDel label4.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.png | CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png | CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.png | CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.png
[2+[(4,3,3,3)]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.png
3 CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch.png | CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch 11.png | CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.png [2+[(4,3,3,3)]]+ (1) CDel label4.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.png
H5 {\bar{K}}_3
[5,3,5]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,5]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 5.pngCDel node c4.png
6 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
[2+[5,3,5]]
CDel branch c1.pngCDel 5a5b.pngCDel nodeab c2.png
3 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png [2+[5,3,5]]+ (1) CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
H6 {\widehat{AH}}_3
[(5,3,3,3)]
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png
[(5,3,3,3)] 6 CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.png | CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.png | CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.png | CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png | CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.png | CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.png
[2+[(5,3,3,3)]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.png
3 CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.png | CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 11.png | CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.png [2+[(5,3,3,3)]]+ (1) CDel label5.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.png
H7 {\widehat{BB}}_3
[(3,4)[2]]
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[(3,4)[2]] 2 CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label4.png | CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
[2+[(3,4)[2]]]
CDel label4.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel label4.png
1 CDel label4.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png
[2+[(3,4)[2]]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label4.png
1 CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[2+[(3,4)[2]]]
CDel label4.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.pngCDel label4.png
1 CDel label4.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png [2+[(3+,4)[2]]] (1) CDel label4.pngCDel branch h0r.pngCDel 3ab.pngCDel branch h0l.pngCDel label4.png
[(2,2)+[(3,4)[2]]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label4.png
1 CDel label4.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png [(2,2)+[(3,4)[2]]]+ (1) CDel label4.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.pngCDel label4.png
H8 {\widehat{BH}}_3
[(5,3,4,3)]
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[(5,3,4,3)] 6 CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label4.png | CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png | CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png | CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png | CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label4.png | CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png
[2+[(5,3,4,3)]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label5.png
3 CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label4.png | CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png | CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label4.png [2+[(5,3,4,3)]]+ (1) CDel label5.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.pngCDel label4.png
H9 {\widehat{HH}}_3
[(3,5)[2]]
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[(3,5)[2]] 2 CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.pngCDel label5.png | CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png
[2+[(3,5)[2]]]
CDel label5.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.pngCDel label5.png
1 CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png
[2+[(3,5)[2]]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label5.png
1 CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[2+[(3,5)[2]]]
CDel label5.pngCDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.pngCDel label5.png
1 CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label5.png
[(2,2)+[(3,5)[2]]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label5.png
1 CDel label5.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel label5.png [(2,2)+[(3,5)[2]]]+ (1) CDel label5.pngCDel branch hh.pngCDel 3ab.pngCDel branch hh.pngCDel label5.png

See also[edit]

Notes[edit]

  1. ^ Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [1]
  2. ^ Felikson, 2002
  3. ^ Felikson, 2002
  4. ^ Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [2]
  5. ^ http://www.bendwavy.org/klitzing/incmats/pt353.htm

References[edit]