Unsolved problem in mathematics :
Find the complete set of hyperbolic uniform honeycombs.
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.
Contents
1 Hyperbolic uniform honeycomb families
1.1 Compact uniform honeycomb families 1.2 Paracompact hyperbolic uniform honeycombs 1.3 [3,5,3] family 1.4 [5,3,4] family 1.5 [5,3,5] family 1.6 [5,31,1 ] family 1.7 [(4,3,3,3)] family 1.8 [(5,3,3,3)] family 1.9 [(4,3,4,3)] family 1.10 [(4,3,5,3)] family 1.11 [(5,3,5,3)] family 1.12 Other non-Wythoffians 1.13 Summary enumeration of compact uniform honeycombs
2 See also 3 Notes 4 References
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 ,[ 1] fundamental simplex domains .[ 2] 1,1 ] ↔ [5,3,4,1+ ].
Indexed
Fundamental[ 2]
Witt
Coxeter
Commutator
Coxeter
Honeycombs
H1
0.0358850633
B H ¯ 3 [5,3,4]
[(5,3)+ ,4,1+ ]1,1 ]+
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png 15 forms, 2 regular
H2
0.0390502856
J ¯ 3 [3,5,3]
[3,5,3]+
File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png 9 forms, 1 regular
H3
0.0717701267
D H ¯ 3 [5,31,1 ]
[5,31,1 ]+
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes.png 11 forms (7 overlap with [5,3,4] family, 4 are unique)
H4
0.0857701820
A B ^ 3 [(4,3,3,3)]
[(4,3,3,3)]+
File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png 9 forms
H5
0.0933255395
K ¯ 3 [5,3,5]
[5,3,5]+
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png 9 forms, 1 regular
H6
0.2052887885
A H ^ 3 [(5,3,3,3)]
[(5,3,3,3)]+
File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png 9 forms
H7
0.2222287320
B B ^ 3 [(4,3)[2] ]
[(4,3+ ,4,3+ )]
File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png 6 forms
H8
0.3586534401
B H ^ 3 [(3,4,3,5)]
[(3,4,3,5)]+
File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png 9 forms
H9
0.5021308905
H H ^ 3 [(5,3)[2] ]
[(5,3)[2] ]+
File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png 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 File:CDel branch c1-2.png File:CDel 4a4b.png File:CDel branch.png File:CDel labels.png trigonal trapezohedron fundamental domain ↔ File:CDel node c1.png File:CDel splitplit1u.png File:CDel branch3u c2.png File:CDel 3a3buc-cross.png File:CDel branch3u c1.png File:CDel splitplit2u.png File:CDel node c2.png File:CDel branchu c1-2.png File:CDel 3ab.png File:CDel branch c2-1.png File:CDel split2-44.png File:CDel node.png * ], 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.
Dimension
Rank
Graphs
H3
5
File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel 2a2b-cross.png File:CDel nodes.png File:CDel node.png File:CDel split1-43.png File:CDel nodes.png File:CDel 2a2b-cross.png File:CDel nodes.png File:CDel node.png File:CDel split1-44.png File:CDel nodes.png File:CDel 2a2b-cross.png File:CDel nodes.png File:CDel node.png File:CDel split1-53.png File:CDel nodes.png File:CDel 2a2b-cross.png File:CDel nodes.png File:CDel node.png File:CDel split1-63.png File:CDel nodes.png File:CDel 2a2b-cross.png File:CDel nodes.png File:CDel branchu.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-43.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-43.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-44.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-44.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-53.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-54.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-55.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-63.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-64.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-65.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2-66.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel ultra.png File:CDel node.png File:CDel branchu.png File:CDel split2.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-43.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-53.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-44.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-43.png File:CDel node.png File:CDel split1-43.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-44.png File:CDel node.png File:CDel split1-43.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-44.png File:CDel node.png File:CDel split1-44.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-54.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-55.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-63.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-64.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-65.png File:CDel node.png File:CDel split1.png File:CDel branchu.png File:CDel branchu.png File:CDel split2-66.png File:CDel node.png File:CDel split1.png File:CDel branchu.png
[3,5,3] family There are 9 forms, generated by ring permutations of the Coxeter group : [3,5,3] or File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png 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 .[ 3] [ 4] regular skew polyhedrons : {4,10|3} and {10,4|3}.
#
Honeycomb nameCoxeter diagram Schläfli
Cell counts/vertex
Vertex figure
Picture
0File:CDel node n2.png File:CDel 5.png File:CDel node n3.png File:CDel 3.png File:CDel node n4.png
1File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png File:CDel 3.png File:CDel node n4.png
2File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 2.png File:CDel node n4.png
3File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 5.png File:CDel node n3.png
1
icosahedral (ikhon)File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png 0 {3,5,3}
(12)File:Icosahedron.png (3.3.3.3.3)
File:Order-3 icosahedral honeycomb verf.svg File:H3 353 CC center.png
2
rectified icosahedral (rih)File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png 1 {3,5,3}
(2)File:Dodecahedron.png (5.5.5)
(3)File:Icosidodecahedron.png (3.5.3.5)
File:Rectified icosahedral honeycomb verf.png File:H3 353 CC center 0100.png
3
truncated icosahedral (tih)File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png 0,1 {3,5,3}
(1)File:Dodecahedron.png (5.5.5)
(3)File:Truncated icosahedron.png (5.6.6)
File:Truncated icosahedral honeycomb verf.png File:H3 353-0011 center ultrawide.png
4
cantellated icosahedral (srih)File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 0,2 {3,5,3}
(1)File:Icosidodecahedron.png (3.5.3.5)
(2)File:Triangular prism.png (4.4.3)
(2)File:Small rhombicosidodecahedron.png (3.5.4.5)
File:Cantellated icosahedral honeycomb verf.png File:H3 353-1010 center ultrawide.png
5
runcinated icosahedral (spiddih)File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {3,5,3}
(1)File:Icosahedron.png (3.3.3.3.3)
(5)File:Triangular prism.png (4.4.3)
(5)File:Triangular prism.png (4.4.3)
(1)File:Icosahedron.png (3.3.3.3.3)
File:Runcinated icosahedral honeycomb verf.png File:H3 353-1001 center ultrawide.png
6
bitruncated icosahedral (dih)File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 1,2 {3,5,3}
(2)File:Truncated dodecahedron.png (3.10.10)
(2)File:Truncated dodecahedron.png (3.10.10)
File:Bitruncated icosahedral honeycomb verf.png File:H3 353-0110 center ultrawide.png
7
cantitruncated icosahedral (grih)File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 0,1,2 {3,5,3}
(1)File:Truncated dodecahedron.png (3.10.10)
(1)File:Triangular prism.png (4.4.3)
(2)File:Great rhombicosidodecahedron.png (4.6.10)
File:Cantitruncated icosahedral honeycomb verf.png File:H3 353-1110 center ultrawide.png
8
runcitruncated icosahedral (prih)File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,1,3 {3,5,3}
(1)File:Small rhombicosidodecahedron.png (3.5.4.5)
(1)File:Triangular prism.png (4.4.3)
(2)File:Hexagonal prism.png (4.4.6)
(1)File:Truncated icosahedron.png (5.6.6)
File:Runcitruncated icosahedral honeycomb verf.png File:H3 353-1101 center ultrawide.png
9
omnitruncated icosahedral (gipiddih)File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {3,5,3}
(1)File:Great rhombicosidodecahedron.png (4.6.10)
(1)File:Hexagonal prism.png (4.4.6)
(1)File:Hexagonal prism.png (4.4.6)
(1)File:Great rhombicosidodecahedron.png (4.6.10)
File:Omnitruncated icosahedral honeycomb verf.png File:H3 353-1111 center ultrawide.png
[5,3,4] family There are 15 forms, generated by ring permutations of the Coxeter group : [5,3,4] or File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png 1,1 ] by a half symmetry [5,3,4,1+ ], or File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel split1.png File:CDel nodeab c3.png File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 4.png File:CDel node h0.png File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel split1.png File:CDel nodes 10lu.png File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node h1.png
[5,3,5] family There are 9 forms, generated by ring permutations of the Coxeter group : [5,3,5] or File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png regular skew polyhedrons : {4,6|5} and {6,4|5}.
#
Name of honeycombCoxeter diagram
Cells by location and count per vertex
Vertex figure
Picture
0File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png
1File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 5.png File:CDel node.png
2File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 2.png File:CDel node.png
3File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png
25
(Regular) Order-5 dodecahedral (pedhon)File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png 0 {5,3,5}
(20)File:Dodecahedron.png (5.5.5)
File:Order-5 dodecahedral honeycomb verf.png File:H3 535 CC center.png
26
rectified order-5 dodecahedral (ripped)File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png 1 {5,3,5}
(2)File:Icosahedron.png (3.3.3.3.3)
(5)File:Icosidodecahedron.png (3.5.3.5)
File:Rectified order-5 dodecahedral honeycomb verf.png File:H3 535 CC center 0100.png
27
truncated order-5 dodecahedral (tipped)File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png 0,1 {5,3,5}
(1)File:Icosahedron.png (3.3.3.3.3)
(5)File:Truncated dodecahedron.png (3.10.10)
File:Truncated order-5 dodecahedral honeycomb verf.png File:H3 535-0011 center ultrawide.png
28
cantellated order-5 dodecahedral (sripped)File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png 0,2 {5,3,5}
(1)File:Icosidodecahedron.png (3.5.3.5)
(2)File:Pentagonal prism.png (4.4.5)
(2)File:Small rhombicosidodecahedron.png (3.5.4.5)
File:Cantellated order-5 dodecahedral honeycomb verf.png File:H3 535-1010 center ultrawide.png
29
Runcinated order-5 dodecahedral (spidded)File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node 1.png 0,3 {5,3,5}
(1)File:Dodecahedron.png (5.5.5)
(3)File:Pentagonal prism.png (4.4.5)
(3)File:Pentagonal prism.png (4.4.5)
(1)File:Dodecahedron.png (5.5.5)
File:Runcinated order-5 dodecahedral honeycomb verf.png File:H3 535-1001 center ultrawide.png
30
bitruncated order-5 dodecahedral (diddoh)File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png 1,2 {5,3,5}
(2)File:Truncated icosahedron.png (5.6.6)
(2)File:Truncated icosahedron.png (5.6.6)
File:Bitruncated order-5 dodecahedral honeycomb verf.png File:H3 535-0110 center ultrawide.png
31
cantitruncated order-5 dodecahedral (gripped)File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png 0,1,2 {5,3,5}
(1)File:Truncated icosahedron.png (5.6.6)
(1)File:Pentagonal prism.png (4.4.5)
(2)File:Great rhombicosidodecahedron.png (4.6.10)
File:Cantitruncated order-5 dodecahedral honeycomb verf.png File:H3 535-1110 center ultrawide.png
32
runcitruncated order-5 dodecahedral (pripped)File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node 1.png 0,1,3 {5,3,5}
(1)File:Small rhombicosidodecahedron.png (3.5.4.5)
(1)File:Pentagonal prism.png (4.4.5)
(2)File:Decagonal prism.png (4.4.10)
(1)File:Truncated dodecahedron.png (3.10.10)
File:Runcitruncated order-5 dodecahedral honeycomb verf.png File:H3 535-1101 center ultrawide.png
33
omnitruncated order-5 dodecahedral (gipidded)File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png 0,1,2,3 {5,3,5}
(1)File:Great rhombicosidodecahedron.png (4.6.10)
(1)File:Decagonal prism.png (4.4.10)
(1)File:Decagonal prism.png (4.4.10)
(1)File:Great rhombicosidodecahedron.png (4.6.10)
File:Omnitruncated order-5 dodecahedral honeycomb verf.png File:H3 535-1111 center ultrawide.png
[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 File:CDel nodes.png File:CDel split2.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel nodeab c1.png File:CDel split2.png File:CDel node c2.png File:CDel 5.png File:CDel node c3.png File:CDel node h0.png File:CDel 4.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 5.png File:CDel node c3.png
[(4,3,3,3)] family There are 9 forms, generated by ring permutations of the Coxeter group : File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png regular skew polyhedrons : {8,6|3} and {6,8|3}.
#
Honeycomb nameCoxeter diagram
Cells by location
vertex figure
Picture
0File:CDel nodea.png File:CDel 3a.png File:CDel branch.png
1File:CDel nodeb.png File:CDel 3b.png File:CDel branch.png
2File:CDel label4.png File:CDel branch.png File:CDel 3b.png File:CDel nodeb.png
3File:CDel label4.png File:CDel branch.png File:CDel 3a.png File:CDel nodea.png
Alt
38
tetrahedral-cubic (gadtatdic)File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch.png (4)File:Tetrahedron.png (3.3.3)
-
(4)File:Hexahedron.png (4.4.4)
(6)File:Cuboctahedron.png (3.4.3.4)
File:Uniform t0 4333 honeycomb verf.png File:H3 4333-1000 center ultrawide.png
39
tetrahedral-octahedral (gacocaddit)File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 10l.png (12)File:Uniform polyhedron-33-t1.svg (3.3.3.3)
(8)File:Tetrahedron.png (3.3.3)
-
(8)File:Octahedron.png (3.3.3.3)
File:Uniform t2 4333 honeycomb verf.png File:H3 4333-0100 center ultrawide.png
40
cyclotruncated tetrahedral-cubic (cytitch)File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 10l.png (3)File:Truncated tetrahedron.png (3.6.6)
(1)File:Tetrahedron.png (3.3.3)
(1)File:Hexahedron.png (4.4.4)
(3)File:Truncated octahedron.png (4.6.6)
File:Uniform t12 4333 honeycomb verf.png File:H3 4333-0110 center ultrawide.png
41
cyclotruncated cube-tetrahedron (cyticth)File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch.png (1)File:Tetrahedron.png (3.3.3)
(1)File:Tetrahedron.png (3.3.3)
(3)File:Truncated hexahedron.png (3.8.8)
(3)File:Truncated hexahedron.png (3.8.8)
File:Uniform t01 4333 honeycomb verf.png File:H3 4333-1100 center ultrawide.png
42
cyclotruncated octahedral-tetrahedral (cytoth)File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 11.png (4)File:Truncated tetrahedron.png (3.6.6)
(4)File:Truncated tetrahedron.png (3.6.6)
(1)File:Octahedron.png (3.3.3.3)
(1)File:Octahedron.png (3.3.3.3)
File:Uniform t23 4333 honeycomb verf.png File:H3 4333-0011 center ultrawide.png
43
rectified tetrahedral-cubic (ritch)File:CDel label4.png File:CDel branch 01r.png File:CDel 3ab.png File:CDel branch 10l.png (1)File:Uniform polyhedron-33-t1.svg (3.3.3.3)
(2)File:Uniform polyhedron-33-t02.png (3.4.3.4)
(1)File:Cuboctahedron.png (3.4.3.4)
(2)File:Small rhombicuboctahedron.png (3.4.4.4)
File:Uniform t02 4333 honeycomb verf.png File:H3 4333-1010 center ultrawide.png
44
truncated tetrahedral-cubic (titch)File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 10l.png (1)File:Truncated tetrahedron.png (3.6.6)
(1)File:Uniform polyhedron-33-t02.png (3.4.3.4)
(1)File:Truncated hexahedron.png (3.8.8)
(2)File:Great rhombicuboctahedron.png (4.6.8)
File:Uniform t012 4333 honeycomb verf.png File:H3 4333-1110 center ultrawide.png
45
truncated tetrahedral-octahedral (titdoh)File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 11.png (2)File:Uniform polyhedron-33-t012.png (4.6.6)
(1)File:Truncated tetrahedron.png (3.6.6)
(1)File:Small rhombicuboctahedron.png (3.4.4.4)
(1)File:Truncated octahedron.png (4.6.6)
File:Uniform t123 4333 honeycomb verf.png File:H3 4333-0111 center ultrawide.png
46
omnitruncated tetrahedral-cubic (otitch)File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 11.png (1)File:Uniform polyhedron-33-t012.png (4.6.6)
(1)File:Uniform polyhedron-33-t012.png (4.6.6)
(1)File:Great rhombicuboctahedron.png (4.6.8)
(1)File:Great rhombicuboctahedron.png (4.6.8)
File:Uniform t0123 4333 honeycomb verf.png File:H3 4333-1111 center ultrawide.png
Nonuniform
omnisnub tetrahedral-cubic File:CDel label4.png File:CDel branch hh.png File:CDel 3ab.png File:CDel branch hh.png (1)File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
(1)File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
(1)File:Snub hexahedron.png (3.3.3.3.4)
(1)File:Snub hexahedron.png (3.3.3.3.4)
(4)File:Tetrahedron.png (3.3.3)
File:Snub 4333 honeycomb verf.png
[(5,3,3,3)] family There are 9 forms, generated by ring permutations of the Coxeter group : File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png 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 : File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png File:CDel label4.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c1-2.png File:CDel label4.png File:CDel label4.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png File:CDel label4.png File:CDel label4.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c2-1.png File:CDel label4.png File:CDel label4.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c1.png File:CDel label4.png File:CDel branch c1-2.png File:CDel 4a4b.png File:CDel branch.png File:CDel labels.png File:CDel node c1.png File:CDel splitplit1u.png File:CDel branch3u c2.png File:CDel 3a3buc-cross.png File:CDel branch3u c1.png File:CDel splitplit2u.png File:CDel node c2.png * )], 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 : File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png 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 : File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png File:CDel label5.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c1-2.png File:CDel label5.png File:CDel label5.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png File:CDel label5.png File:CDel label5.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c2-1.png File:CDel label5.png File:CDel label5.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c1.png File:CDel label5.png regular skew polyhedrons : {6,6|5} and {10,10|3}.
Other non-Wythoffians There are infinitely many known non-Wythoffian uniform compact hyperbolic honeycombs, and there may be more undiscovered ones. Two have been listed above as diminishings of the icosahedral honeycomb {3,5,3}.[ 6] 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. The case p = 2 degenerates to the Euclidean cubic honeycomb .[ 6] noncompact families. The tessellation File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png truncated cubes File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png order-8 triangular tilings File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png compact uniform honeycomb consisting only of the truncated cubes. (So they are analogous to the hemi-faces of spherical hemipolyhedra .)[ 6] [ 7] File:CDel nodes 11.png File:CDel split2-43.png File:CDel node.png File:CDel 8.png File:CDel node.png small rhombicuboctahedra File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png order-8 triangular tilings File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png order-8 square tilings File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 8.png File:CDel node.png 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.[ 8] File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node.png truncated octahedra File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png order-6 square tilings File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node.png cuboctahedra File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel nodes 11.png File:CDel split2-53.png File:CDel node.png File:CDel 4.png File:CDel node.png small rhombicosidodecahedra File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png octahedra File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png order-4 pentagonal tilings File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 4.png File:CDel node.png pentagonal prisms File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 4.png File:CDel node.png [ 9] octahedra and eight snub cubes at each vertex.[ 6] File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel p.png File:CDel node h.png snub dodecahedral vertex figure respectively), paracompact for p=6 (with a snub trihexagonal tiling as the vertex figure), and hypercompact for p>6. Again, the truncated and rectified versions of these honeycombs are still uniform.[ 6] parallelepiped domains. Two known forms generalise the cubic-octahedral honeycomb , having distorted small rhombicuboctahedral vertex figures. One form has small rhombicuboctahedra, cuboctahedra, and cubes; another has small rhombicosidodecahedra, icosidodecahedra , and cubes. (The version with tetrahedral-symmetry polyhedra is the cubic-octahedral honeycomb, using cuboctahedra, octahedra, and cubes).[ 9]
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.
Index
Coxeter group
Extended
Honeycombs
Chiral
Alternation honeycombs
H1
B H ¯ 3 File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png [4,3,5]File:CDel node c1.png File:CDel 4.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 5.png File:CDel node c4.png
15
File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png [1+ ,4,(3,5)+ ]
(2)
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node h1.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes 10lu.png File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png
[4,3,5]+
(1)
File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node h.png
H2
J ¯ 3 File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png [3,5,3]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 5.png File:CDel node c3.png File:CDel 3.png File:CDel node c4.png
6
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
[2+ [3,5,3]]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 5.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png
5
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png [2+ [3,5,3]]+
(1)
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png
H3
D H ¯ 3 1,1 ]File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes.png [5,31,1 ]File:CDel node c3.png File:CDel 5.png File:CDel node c4.png File:CDel split1.png File:CDel nodeab c1-2.png
4
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes 10lu.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes 10lu.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel split1.png File:CDel nodes 10lu.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel split1.png File:CDel nodes 10lu.png
[1[5,31,1 ]]=[5,3,4]File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel split1.png File:CDel nodeab c3.png File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 4.png File:CDel node h0.png
(7)
File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes 11.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel nodes 11.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png [1[5,31,1 ]]+ +
(1)
File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel split1.png File:CDel nodes hh.png
H4
A B ^ 3 File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png [(4,3,3,3)]
6
File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel branch 01r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 11.png
[2+ [(4,3,3,3)]]File:CDel label4.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png
3
File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 11.png [2+ [(4,3,3,3)]]+
(1)
File:CDel label4.png File:CDel branch hh.png File:CDel 3ab.png File:CDel branch hh.png
H5
K ¯ 3 File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png [5,3,5]File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 5.png File:CDel node c4.png
6
File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node 1.png
[2+ [5,3,5]]File:CDel branch c1.png File:CDel 5a5b.png File:CDel nodeab c2.png
3
File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png [2+ [5,3,5]]+
(1)
File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 5.png File:CDel node h.png
H6
A H ^ 3 File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png [(5,3,3,3)]
6
File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png File:CDel branch 01r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 11.png
[2+ [(5,3,3,3)]]File:CDel label5.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png
3
File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 11.png [2+ [(5,3,3,3)]]+
(1)
File:CDel label5.png File:CDel branch hh.png File:CDel 3ab.png File:CDel branch hh.png
H7
B B ^ 3 [2] ]File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png [(3,4)[2] ]
2
File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png
[2+ [(3,4)[2] ]]File:CDel label4.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c2-1.png File:CDel label4.png
1
File:CDel label4.png File:CDel branch 01r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png
[2+ [(3,4)[2] ]]File:CDel label4.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png File:CDel label4.png
1
File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png
[2+ [(3,4)[2] ]]File:CDel label4.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c1-2.png File:CDel label4.png
1
File:CDel label4.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png [2+ [(3+ ,4)[2] ]]
(1)
File:CDel label4.png File:CDel branch h0r.png File:CDel 3ab.png File:CDel branch h0l.png File:CDel label4.png
[(2,2)+ [(3,4)[2] ]]File:CDel label4.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c1.png File:CDel label4.png
1
File:CDel label4.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label4.png [(2,2)+ [(3,4)[2] ]]+
(1)
File:CDel label4.png File:CDel branch hh.png File:CDel 3ab.png File:CDel branch hh.png File:CDel label4.png
H8
B H ^ 3 File:CDel label4.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png [(5,3,4,3)]
6
File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel label5.png File:CDel branch 01r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label4.png File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label4.png
[2+ [(5,3,4,3)]]File:CDel label4.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png File:CDel label5.png
3
File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch.png File:CDel label4.png File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label4.png File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label4.png [2+ [(5,3,4,3)]]+
(1)
File:CDel label5.png File:CDel branch hh.png File:CDel 3ab.png File:CDel branch hh.png File:CDel label4.png
H9
H H ^ 3 [2] ]File:CDel label5.png File:CDel branch.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png [(3,5)[2] ]
2
File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png
[2+ [(3,5)[2] ]]File:CDel label5.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c2-1.png File:CDel label5.png
1
File:CDel label5.png File:CDel branch 01r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png
[2+ [(3,5)[2] ]]File:CDel label5.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c2.png File:CDel label5.png
1
File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch.png File:CDel label5.png
[2+ [(3,5)[2] ]]File:CDel label5.png File:CDel branch c1-2.png File:CDel 3ab.png File:CDel branch c1-2.png File:CDel label5.png
1
File:CDel label5.png File:CDel branch 10r.png File:CDel 3ab.png File:CDel branch 10l.png File:CDel label5.png
[(2,2)+ [(3,5)[2] ]]File:CDel label5.png File:CDel branch c1.png File:CDel 3ab.png File:CDel branch c1.png File:CDel label5.png
1
File:CDel label5.png File:CDel branch 11.png File:CDel 3ab.png File:CDel branch 11.png File:CDel label5.png [(2,2)+ [(3,5)[2] ]]+
(1)
File:CDel label5.png File:CDel branch hh.png File:CDel 3ab.png File:CDel branch hh.png File:CDel label5.png
See also Notes
↑ Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [1]
↑ 2.0 2.1 Felikson, 2002
↑ Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [2]
↑ 4.0 4.1 "Spd{3,5,3" .↑ "Pd{3,5,3" .↑ 6.0 6.1 6.2 6.3 6.4 "Hyperbolic Tesselations" .↑ "x4x3o8o" .↑ "lt-o8o4xb3x" .↑ 9.0 9.1 "Hyperbolic Tessellations – Triangular Prismatic Domains" .
References J. Humphreys (1990), Reflection Groups and Coxeter Groups , Cambridge studies in advanced mathematics, 29H.S.M. Coxeter (1954), "Regular Honeycombs in Hyperbolic Space" Proceedings of the International Congress of Mathematicians , vol. 3, North-Holland, pp. 155–169. Reprinted as Ch. 10 in Coxeter (1999), The Beauty of Geometry: Twelve Essays , Dover, ISBN 0-486-40919-8 H.S.M. Coxeter (1973), Regular Polytopes ISBN 0-486-61480-8 . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)J. Weeks The Shape of Space , 2nd ed. ISBN 0-8247-0709-5 , Chapters 16–17: Geometries on Three-manifolds I, IIA. Felikson (2002), "Coxeter Decompositions of Hyperbolic Tetrahedra" (preprint) arXiv :math/0212010
C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Can. J. Math. 19, 1179–1186, 1967. PDF [3] Archived 2015-04-02 at the Wayback Machine
N. W. Johnson (2018), Geometries and Transformations , Chapters 11–13N. W. Johnson, R. Kellerhals , J. G. Ratcliffe, S. T. Tschantz (1999), The size of a hyperbolic Coxeter simplex , Transformation Groups, Volume 4, Issue 4, pp 329–353 [4]
N. W. Johnson, R. Kellerhals , J.G. Ratcliffe, S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups H3 : p130. [5]
Klitzing, Richard. "Hyperbolic honeycombs H3 compact" .
