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uniform honeycombs in hyperbolic space

{{short description|Tiling of hyperbolic 3-space by uniform polyhedra}}

{{unsolved|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.

{{Clear}}

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|+ Four compact regular hyperbolic honeycombs

bgcolor="#d0e0ff" align=center

|200px
Order-4 dodecahedral honeycomb
{{math|{5,3,4} }}

|200px
Order-5 dodecahedral honeycomb
{{math|{5,3,5} }}

bgcolor="#d0e0ff" align=center

|200px
Order-5 cubic honeycomb
{{math|{4,3,5} }}

|200px
Icosahedral honeycomb
{{math|{3,5,3} }}

colspan=2|Poincaré ball model projections

Hyperbolic uniform honeycomb families

Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups.

= Compact uniform honeycomb families =

The nine compact Coxeter groups are listed here with their Coxeter diagrams,Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [https://books.google.com/books?id=ODfjmOeNLMUC&pg=PA141]

in order of the relative volumes of their fundamental simplex domains.Felikson, 2002

These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. Two known examples are cited with the {3,5,3} family below. Only two families are related as a mirror-removal halving: [5,31,1] ↔ [5,3,4,1+].

class=wikitable

!Indexed

!Fundamental
simplex
volume

!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]+|| {{CDD|node|5|node|3|node|4|node}} || 15 forms, 2 regular

H2

|0.0390502856

|{\bar{J}}_3 || [3,5,3]|| [3,5,3]+|| {{CDD|node|3|node|5|node|3|node}} || 9 forms, 1 regular

H3

|0.0717701267

|{\bar{DH}}_3 || [5,31,1]|| [5,31,1]+|| {{CDD|node|5|node|split1|nodes}} || 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)]+|| {{CDD|label4|branch|3ab|branch}} || 9 forms

H5

|0.0933255395

|{\bar{K}}_3 || [5,3,5]|| [5,3,5]+|| {{CDD|node|5|node|3|node|5|node}} || 9 forms, 1 regular

H6

|0.2052887885

|{\widehat{AH}}_3 || [(5,3,3,3)]|| [(5,3,3,3)]+|| {{CDD|label5|branch|3ab|branch}} || 9 forms

H7

|0.2222287320

|{\widehat{BB}}_3 || [(4,3)[2]]|| [(4,3+,4,3+)]|| {{CDD|label4|branch|3ab|branch|label4}} || 6 forms

H8

|0.3586534401

|{\widehat{BH}}_3 || [(3,4,3,5)]|| [(3,4,3,5)]+|| {{CDD|label5|branch|3ab|branch|label4}} || 9 forms

H9

|0.5021308905

|{\widehat{HH}}_3 || [(5,3)[2]]|| [(5,3)[2]]+|| {{CDD|label5|branch|3ab|branch|label5}} || 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 {{CDD|branch_c1-2|4a4b|branch|labels}} an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔ {{CDD|node_c1|splitplit1u|branch3u_c2|3a3buc-cross|branch3u_c1|splitplit2u|node_c2}}, which can be extended by restoring one mirror as {{CDD|branchu_c1-2|3ab|branch_c2-1|split2-44|node}}. The other is [4,(3,5)*], index 120 with a dodecahedral fundamental domain.

= Paracompact hyperbolic uniform honeycombs =

{{further|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.

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|+ Hyperbolic paracompact group summary

!Type

!Coxeter groups

align=center

!Linear graphs

|{{CDD|node|6|node|3|node|3|node}} | {{CDD|node|4|node|4|node|3|node}} | {{CDD|node|6|node|3|node|4|node}} | {{CDD|node|6|node|3|node|5|node}} | {{CDD|node|4|node|4|node|4|node}} | {{CDD|node|3|node|6|node|3|node}} | {{CDD|node|6|node|3|node|6|node}}

align=center

!Tridental graphs

| {{CDD|node|3|node|split1-44|nodes}} | {{CDD|node|6|node|split1|nodes}} | {{CDD|node|4|node|split1-44|nodes}}

align=center

!Cyclic graphs

| {{CDD|label6|branch|3ab|branch|2}} | {{CDD|label6|branch|3ab|branch|label4}} | {{CDD|label4|branch|4-4|branch}} | {{CDD|label6|branch|3ab|branch|label5}} | {{CDD|label6|branch|3ab|branch|label6}} | {{CDD|label4|branch|4-4|branch|label4}} | {{CDD|node|split1-44|nodes|split2|node}} | {{CDD|node|split1|branch|split2|node}} | {{CDD|branch|splitcross|branch}}

align=center

!Loop-n-tail graphs

|{{CDD|node|3|node|split1|branch}} | {{CDD|node|4|node|split1|branch}} | {{CDD|node|5|node|split1|branch}} | {{CDD|node|6|node|split1|branch}}

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.

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!Dimension

!Rank

!Graphs

H3

!5

|

: {{CDD|node|split1|nodes|2a2b-cross|nodes}}, {{CDD|node|split1-43|nodes|2a2b-cross|nodes}}, {{CDD|node|split1-44|nodes|2a2b-cross|nodes}}, {{CDD|node|split1-53|nodes|2a2b-cross|nodes}}, {{CDD|node|split1-63|nodes|2a2b-cross|nodes}}

: {{CDD|branchu|split2|node|3|node|ultra|node}}, {{CDD|branchu|split2|node|4|node|ultra|node}}, {{CDD|branchu|split2-43|node|3|node|ultra|node}}, {{CDD|branchu|split2-43|node|4|node|ultra|node}}, {{CDD|branchu|split2-44|node|3|node|ultra|node}}, {{CDD|branchu|split2-44|node|4|node|ultra|node}}

: {{CDD|branchu|split2-53|node|3|node|ultra|node}}, {{CDD|branchu|split2-54|node|3|node|ultra|node}}, {{CDD|branchu|split2-55|node|3|node|ultra|node}}, {{CDD|branchu|split2-63|node|3|node|ultra|node}}, {{CDD|branchu|split2-64|node|3|node|ultra|node}}, {{CDD|branchu|split2-65|node|3|node|ultra|node}}, {{CDD|branchu|split2-66|node|3|node|ultra|node}}

: {{CDD|branchu|split2|node|split1|branchu}}, {{CDD|branchu|split2-43|node|split1|branchu}}, {{CDD|branchu|split2-53|node|split1|branchu}}, {{CDD|branchu|split2-44|node|split1|branchu}}, {{CDD|branchu|split2-43|node|split1-43|branchu}}, {{CDD|branchu|split2-44|node|split1-43|branchu}}, {{CDD|branchu|split2-44|node|split1-44|branchu}}, {{CDD|branchu|split2-54|node|split1|branchu}}, {{CDD|branchu|split2-55|node|split1|branchu}}, {{CDD|branchu|split2-63|node|split1|branchu}}, {{CDD|branchu|split2-64|node|split1|branchu}}, {{CDD|branchu|split2-65|node|split1|branchu}}, {{CDD|branchu|split2-66|node|split1|branchu}}

= [3,5,3] family =

There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or {{CDD|node|3|node|5|node|3|node}}

One related non-wythoffian form is constructed from the {3,5,3} vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a tetrahedrally diminished dodecahedron.Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [http://symmetry.hu/oldsite/content/aus_journal_content_abs_2005_16_2.html] Another is constructed with 2 antipodal vertices removed.

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

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!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
and Schläfli
symbols

!colspan=4|Cell counts/vertex
and positions in honeycomb

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

!0
{{CDD|node_n2|5|node_n3|3|node_n4}}

!1
{{CDD|node_n1|2|2|node_n3|3|node_n4}}

!2
{{CDD|node_n1|3|node_n2|2|node_n4}}

!3
{{CDD|node_n1|3|node_n2|5|node_n3}}

align=center BGCOLOR="#f0e0e0"

!1

|icosahedral (ikhon)
{{CDD|node_1|3|node|5|node|3|node}}
t0{3,5,3}

| 

| 

| 

|(12)
40px
(3.3.3.3.3)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!2

|rectified icosahedral (rih)
{{CDD|node|3|node_1|5|node|3|node}}
t1{3,5,3}

|(2)
40px
(5.5.5)

| 

| 

|(3)
40px
(3.5.3.5)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!3

|truncated icosahedral (tih)
{{CDD|node_1|3|node_1|5|node|3|node}}
t0,1{3,5,3}

|(1)
40px
(5.5.5)

| 

| 

|(3)
40px
(5.6.6)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!4

|cantellated icosahedral (srih)
{{CDD|node_1|3|node|5|node_1|3|node}}
t0,2{3,5,3}

|(1)
40px
(3.5.3.5)

|(2)
40px
(4.4.3)

| 

|(2)
40px
(3.5.4.5)

|100px

|120px

align=center BGCOLOR="#e0f0e0"

!5

|runcinated icosahedral (spiddih)
{{CDD|node_1|3|node|5|node|3|node_1}}
t0,3{3,5,3}

|(1)
40px
(3.3.3.3.3)

|(5)
40px
(4.4.3)

|(5)
40px
(4.4.3)

|(1)
40px
(3.3.3.3.3)

|100px

|120px

align=center BGCOLOR="#e0f0e0"

!6

|bitruncated icosahedral (dih)
{{CDD|node|3|node_1|5|node_1|3|node}}
t1,2{3,5,3}

|(2)
40px
(3.10.10)

| 

| 

|(2)
40px
(3.10.10)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!7

|cantitruncated icosahedral (grih)
{{CDD|node_1|3|node_1|5|node_1|3|node}}
t0,1,2{3,5,3}

|(1)
40px
(3.10.10)

|(1)
40px
(4.4.3)

| 

|(2)
40px
(4.6.10)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!8

|runcitruncated icosahedral (prih)
{{CDD|node_1|3|node_1|5|node|3|node_1}}
t0,1,3{3,5,3}

|(1)
40px
(3.5.4.5)

|(1)
40px
(4.4.3)

|(2)
40px
(4.4.6)

|(1)
40px
(5.6.6)

|100px

|120px

align=center BGCOLOR="#e0f0e0"

!9

|omnitruncated icosahedral (gipiddih)
{{CDD|node_1|3|node_1|5|node_1|3|node_1}}
t0,1,2,3{3,5,3}

|(1)
40px
(4.6.10)

|(1)
40px
(4.4.6)

|(1)
40px
(4.4.6)

|(1)
40px
(4.6.10)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
and Schläfli
symbols

!colspan=5|Cell counts/vertex
and positions in honeycomb

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

!0
{{CDD|node_n2|5|node_n3|3|node_n4}}

!1
{{CDD|node_n1|2|2|node_n3|3|node_n4}}

!2
{{CDD|node_n1|3|node_n2|2|node_n4}}

!3
{{CDD|node_n1|3|node_n2|5|node_n3}}

!Alt

align=center BGCOLOR="#e0f0f0"

|[77]

|partially diminished icosahedral (pidih)
pd{3,5,3}{{Cite web | url=http://www.bendwavy.org/klitzing/incmats/pt353.htm | title=Pd{3,5,3}|author=Dr. Richard Klitzing|publisher=bendwavy.org}}

|

|

|

|(12)
40px
(3.3.3.5)

|(4)
40px
(5.5.5)

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|[78]

|semi-partially diminished icosahedral
spd{3,5,3}{{Cite web | url=http://www.bendwavy.org/klitzing/incmats/spt353.htm | title=Spd{3,5,3}|author=Dr. Richard Klitzing|publisher=bendwavy.org}}

|

|

|

|(6)
40px
(3.3.3.5)
(6)
40px
(3.3.3.3.3)

|(2)
40px
(5.5.5)

|

|

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub icosahedral (snih)
{{CDD|node_h|3|node_h|5|node_h|3|node_h}}
ht0,1,2,3{3,5,3}

|(1)
40px
(3.3.3.3.5)

|(1)
40px
(3.3.3.3

|(1)
40px
(3.3.3.3)

|(1)
40px
(3.3.3.3.5)

|(4)
40px
+(3.3.3)

|100px

|

= [5,3,4] family =

There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or {{CDD|node|5|node|3|node|4|node}}.

This family is related to the group [5,31,1] by a half symmetry [5,3,4,1+], or {{CDD|node_c1|5|node_c2|split1|nodeab_c3}} ↔ {{CDD|node_c1|5|node_c2|3|node_c3|4|node_h0}}, when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive {{CDD|node_c1|5|node_c2|split1|nodes_10lu}} ↔ {{CDD|node_c1|5|node_c2|3|node|4|node_h1}}.

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!rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram

!colspan=4|Cells by location and count per vertex

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node_n2|3|node_n3|4|node_n4}}

!1
{{CDD|node_n1|2|node_n3|4|node_n4}}

!2
{{CDD|node_n1|5|node_n2|2|node_n4}}

!3
{{CDD|node_n1|5|node_n2|3|node_n3}}

BGCOLOR="#f0e0e0" align=center

!10

|order-4 dodecahedral (doehon)
{{CDD|node_1|5|node|3|node|4|node}} ↔ {{CDD|node_1|5|node|split1|nodes}}

| -

| -

| -

|(8)
{{CDD|node_1|5|node|3|node}}
40px
(5.5.5)

|100px

|100px

BGCOLOR="#f0e0e0" align=center

!11

|rectified order-4 dodecahedral (riddoh)
{{CDD|node|5|node_1|3|node|4|node}} ↔ {{CDD|node|5|node_1|split1|nodes}}

|(2)
{{CDD|node_1|3|node|4|node}}
40px
(3.3.3.3)

| -

| -

|(4)
{{CDD|node|5|node_1|3|node}}
40px
(3.5.3.5)

|100px

|120px

BGCOLOR="#e0e0f0" align=center

!12

|rectified order-5 cubic (ripech)
{{CDD|node|5|node|3|node_1|4|node}} ↔ {{CDD|node|5|node|split1|nodes_11}}

|(5)
{{CDD|node|3|node_1|4|node}}
40px
(3.4.3.4)

| -

| -

|(2)
{{CDD|node|5|node|3|node_1}}
40px
(3.3.3.3.3)

|100px

| 120px

BGCOLOR="#e0e0f0" align=center

!13

|order-5 cubic (pechon)
{{CDD|node|5|node|3|node|4|node_1}}

|(20)
{{CDD|node|3|node|4|node_1}}
40px
(4.4.4)

| -

| -

| -

|100px

|120px

BGCOLOR="#f0e0e0" align=center

!14

|truncated order-4 dodecahedral (tiddoh)
{{CDD|node_1|5|node_1|3|node|4|node}} ↔ {{CDD|node_1|5|node_1|split1|nodes}}

|(1)
{{CDD|node_1|3|node|4|node}}
40px
(3.3.3.3)

| -

| -

|(4)
{{CDD|node_1|5|node_1|3|node}}
40px
(3.10.10)

|100px

|120px

BGCOLOR="#e0f0e0" align=center

!15

|bitruncated order-5 cubic (ciddoh)
{{CDD|node|5|node_1|3|node_1|4|node}} ↔ {{CDD|node|5|node_1|split1|nodes_11}}

|(2)
{{CDD|node_1|3|node_1|4|node}}
40px
(4.6.6)

| -

| -

|(2)
{{CDD|node|5|node_1|3|node_1}}
40px
(5.6.6)

|100px

|120px

BGCOLOR="#e0e0f0" align=center

!16

|truncated order-5 cubic (tipech)
{{CDD|node|5|node|3|node_1|4|node_1}}

|(5)
{{CDD|node|3|node_1|4|node_1}}
40px
(3.8.8)

| -

| -

|(1)
{{CDD|node|5|node|3|node_1}}
40px
(3.3.3.3.3)

|100px

|120px

BGCOLOR="#f0e0e0" align=center

!17

|cantellated order-4 dodecahedral (sriddoh)
{{CDD|node_1|5|node|3|node_1|4|node}} ↔ {{CDD|node_1|5|node|split1|nodes_11}}

|(1)
{{CDD|node|3|node_1|4|node}}
40px
(3.4.3.4)

|(2)
{{CDD|node_1|2|node_1|4|node}}
40px
(4.4.4)

| -

|(2)
{{CDD|node_1|5|node|3|node_1}}
40px
(3.4.5.4)

|100px

|120px

BGCOLOR="#e0e0f0" align=center

!18

|cantellated order-5 cubic (sripech)
{{CDD|node|5|node_1|3|node|4|node_1}}

|(2)
{{CDD|node_1|3|node|4|node_1}}
40px
(3.4.4.4)

| -

|(2)
{{CDD|node|5|node_1|2|node_1}}
40px
(4.4.5)

|(1)
{{CDD|node|5|node_1|3|node}}
40px
(3.5.3.5)

|100px

|120px

BGCOLOR="#e0f0e0" align=center

!19

|runcinated order-5 cubic (sidpicdoh)
{{CDD|node_1|5|node|3|node|4|node_1}}

|(1)
{{CDD|node|3|node|4|node_1}}
40px
(4.4.4)

|(3)
{{CDD|node_1|2|node|4|node_1}}
40px
(4.4.4)

|(3)
{{CDD|node_1|5|node|2|node_1}}
40px
(4.4.5)

|(1)
{{CDD|node_1|5|node|3|node}}
40px
(5.5.5)

|100px

|120px

BGCOLOR="#f0e0e0" align=center

!20

|cantitruncated order-4 dodecahedral (griddoh)
{{CDD|node_1|5|node_1|3|node_1|4|node}} ↔ {{CDD|node_1|5|node_1|split1|nodes_11}}

|(1)
{{CDD|node_1|3|node_1|4|node}}
40px
(4.6.6)

|(1)
{{CDD|node_1|2|node_1|4|node}}
40px
(4.4.4)

| -

|(2)
{{CDD|node_1|5|node_1|3|node_1}}
40px
(4.6.10)

|100px

|120px

BGCOLOR="#e0e0f0" align=center

!21

|cantitruncated order-5 cubic (gripech)
{{CDD|node|5|node_1|3|node_1|4|node_1}}

|(2)
{{CDD|node_1|3|node_1|4|node_1}}
40px
(4.6.8)

| -

|(1)
{{CDD|node|5|node_1|2|node_1}}
40px
(4.4.5)

|(1)
{{CDD|node|5|node_1|3|node_1}}
40px
(5.6.6)

|100px

|120px

BGCOLOR="#f0e0e0" align=center

!22

|runcitruncated order-4 dodecahedral (pripech)
{{CDD|node_1|5|node_1|3|node|4|node_1}}

|(1)
{{CDD|node_1|3|node|4|node_1}}
40px
(3.4.4.4)

|(1)
{{CDD|node_1|2|node|4|node_1}}
40px
(4.4.4)

|(2)
{{CDD|node_1|5|node_1|2|node_1}}
40px
(4.4.10)

|(1)
{{CDD|node_1|5|node_1|3|node}}
40px
(3.10.10)

|100px

|120px

BGCOLOR="#e0e0f0" align=center

!23

|runcitruncated order-5 cubic (priddoh)
{{CDD|node_1|5|node|3|node_1|4|node_1}}

|(1)
{{CDD|node|3|node_1|4|node_1}}
40px
(3.8.8)

|(2)
{{CDD|node_1|2|node_1|4|node_1}}
40px
(4.4.8)

|(1)
{{CDD|node_1|5|node|2|node_1}}
40px
(4.4.5)

|(1)
{{CDD|node_1|5|node|3|node_1}}
40px
(3.4.5.4)

|100px

|120px

BGCOLOR="#e0f0e0" align=center

!24

|omnitruncated order-5 cubic (gidpicdoh)
{{CDD|node_1|5|node_1|3|node_1|4|node_1}}

|(1)
{{CDD|node_1|3|node_1|4|node_1}}
40px
(4.6.8)

|(1)
{{CDD|node_1|2|node_1|4|node_1}}
40px
(4.4.8)

|(1)
{{CDD|node_1|5|node_1|2|node_1}}
40px
(4.4.10)

|(1)
{{CDD|node_1|5|node_1|3|node_1}}
40px
(4.6.10)

|100px

|120px

class=wikitable

!rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram

!colspan=5|Cells by location and count per vertex

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node_n2|3|node_n3|4|node_n4}}

!1
{{CDD|node_n1|2|node_n3|4|node_n4}}

!2
{{CDD|node_n1|5|node_n2|2|node_n4}}

!3
{{CDD|node_n1|5|node_n2|3|node_n3}}

!Alt

BGCOLOR="#d0f0f0" align=center

|[34]

|alternated order-5 cubic (apech)
{{CDD|node|5|node|3|node|4|node_h1}} ↔ {{CDD|node|5|node|split1|nodes_10lu}}

|(20)
{{CDD|node|3|node|4|node_h1}}
40px
(3.3.3)

| 

| 

|(12)
40px
(3.3.3.3.3)

|

|100px

|100px

BGCOLOR="#d0f0f0" align=center

|[35]

|cantic order-5 cubic (tapech)
{{CDD|node|5|node_1|3|node|4|node_h1}} ↔ {{CDD|node|5|node_1|split1|nodes_10lu}}

|(1)
40px
(3.5.3.5)

| -

|(2)
40px
(5.6.6)

|(2)
40px
(3.6.6)

|

|100px

|120px

|

BGCOLOR="#d0f0f0" align=center

|[36]

|runcic order-5 cubic (birapech)
{{CDD|node_1|5|node|3|node|4|node_h1}} ↔ {{CDD|node_1|5|node|split1|nodes_10lu}}

| (1)
40px
(5.5.5)

| -

| (3)
40px
(3.4.5.4)

| (1)
40px
(3.3.3)

|

|100px

|120px

|

BGCOLOR="#d0f0f0" align=center

|[37]

|runcicantic order-5 cubic (bitapech)
{{CDD|node_1|5|node_1|3|node|4|node_h1}} ↔ {{CDD|node_1|5|node_1|split1|nodes_10lu}}

|(1)
40px
(3.10.10)

| -

|(2)
40px
(4.6.10)

|(1)
40px
(3.6.6)

|

|100px

|120px

|

BGCOLOR="#d0f0f0" align=center

|Nonuniform

|snub rectified order-4 dodecahedral
{{CDD|node_h|5|node_h|3|node_h|4|node}}

|(1)
{{CDD|node_h|3|node_h|4|node}}
40px
(3.3.3.3.3)

|(1)
{{CDD|node_h|2x|node_h|4|node}}
40px
(3.3.3)

| -

|(2)
{{CDD|node_h|5|node_h|3|node_h}}
40px
(3.3.3.3.5)

|(4)
40px
+(3.3.3)

|100px
Irr. tridiminished icosahedron

|

BGCOLOR="#d0f0f0" align=center

|Nonuniform

|runcic snub rectified order-4 dodecahedral
{{CDD|node_h|5|node_h|3|node_h|4|node_1}}

|{{CDD|node_h|3|node_h|4|node_1}}
40px
(3.4.4.4)

|{{CDD|node_h|2x|node_h|4|node_1}}
40px
(4.4.4.4)

| -

|{{CDD|node_h|5|node_h|3|node_h}}
40px
(3.3.3.3.5)

|40px
+(3.3.3)

|

|

BGCOLOR="#d0f0f0" align=center

|Nonuniform

|omnisnub order-5 cubic
{{CDD|node_h|5|node_h|3|node_h|4|node_h}}

|(1)
{{CDD|node_h|3|node_h|4|node_h}}
40px
(3.3.3.3.4)

|(1)
{{CDD|node_h|2x|node_h|4|node_h}}
40px
(3.3.3.4)

|(1)
{{CDD|node_h|5|node_h|2x|node_h}}
40px
(3.3.3.5)

|(1)
{{CDD|node_h|5|node_h|3|node_h}}
40px
(3.3.3.3.5)

|(4)
40px
+(3.3.3)

|100px

|

= [5,3,5] family =

There are 9 forms, generated by ring permutations of the Coxeter group: [5,3,5] or {{CDD|node|5|node|3|node|5|node}}

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

class=wikitable

!rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram

!colspan=4|Cells by location and count per vertex

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node|3|node|5|node}}

!1
{{CDD|node|2|node|5|node}}

!2
{{CDD|node|5|node|2|node}}

!3
{{CDD|node|5|node|3|node}}

align=center BGCOLOR="#f0e0e0"

!25

|(Regular) Order-5 dodecahedral (pedhon)
{{CDD|node_1|5|node|3|node|5|node}}
t0{5,3,5}

| 

| 

| 

|(20)
40px
(5.5.5)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!26

|rectified order-5 dodecahedral (ripped)
{{CDD|node|5|node_1|3|node|5|node}}
t1{5,3,5}

|(2)
40px
(3.3.3.3.3)

| 

| 

|(5)
40px
(3.5.3.5)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!27

|truncated order-5 dodecahedral (tipped)
{{CDD|node_1|5|node_1|3|node|5|node}}
t0,1{5,3,5}

|(1)
40px
(3.3.3.3.3)

| 

| 

|(5)
40px
(3.10.10)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!28

|cantellated order-5 dodecahedral (sripped)
{{CDD|node_1|5|node|3|node_1|5|node}}
t0,2{5,3,5}

|(1)
40px
(3.5.3.5)

|(2)
40px
(4.4.5)

| 

|(2)
40px
(3.5.4.5)

|100px

|120px

align=center BGCOLOR="#e0f0e0"

!29

|Runcinated order-5 dodecahedral (spidded)
{{CDD|node_1|5|node|3|node|5|node_1}}
t0,3{5,3,5}

|(1)
40px
(5.5.5)

|(3)
40px
(4.4.5)

|(3)
40px
(4.4.5)

|(1)
40px
(5.5.5)

|100px

|120px

align=center BGCOLOR="#e0f0e0"

!30

|bitruncated order-5 dodecahedral (diddoh)
{{CDD|node|5|node_1|3|node_1|5|node}}
t1,2{5,3,5}

|(2)
40px
(5.6.6)

| 

| 

|(2)
40px
(5.6.6)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!31

|cantitruncated order-5 dodecahedral (gripped)
{{CDD|node_1|5|node_1|3|node_1|5|node}}
t0,1,2{5,3,5}

|(1)
40px
(5.6.6)

|(1)
40px
(4.4.5)

| 

|(2)
40px
(4.6.10)

|100px

|120px

align=center BGCOLOR="#f0e0e0"

!32

|runcitruncated order-5 dodecahedral (pripped)
{{CDD|node_1|5|node_1|3|node|5|node_1}}
t0,1,3{5,3,5}

|(1)
40px
(3.5.4.5)

|(1)
40px
(4.4.5)

|(2)
40px
(4.4.10)

|(1)
40px
(3.10.10)

|100px

|120px

align=center BGCOLOR="#e0f0e0"

!33

|omnitruncated order-5 dodecahedral (gipidded)
{{CDD|node_1|5|node_1|3|node_1|5|node_1}}
t0,1,2,3{5,3,5}

|(1)
40px
(4.6.10)

|(1)
40px
(4.4.10)

|(1)
40px
(4.4.10)

|(1)
40px
(4.6.10)

|100px

|120px

class=wikitable

!rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram

!colspan=5|Cells by location and count per vertex

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node|3|node|5|node}}

!1
{{CDD|node|2|node|5|node}}

!2
{{CDD|node|5|node|2|node}}

!3
{{CDD|node|5|node|3|node}}

!Alt

BGCOLOR="#d0f0f0" align=center

|Nonuniform

|omnisnub order-5 dodecahedral
{{CDD|node_h|5|node_h|3|node_h|5|node_h}}
ht0,1,2,3{5,3,5}

|(1)
{{CDD|node_h|3|node_h|5|node_h}}
40px
(3.3.3.3.5)

|(1)
{{CDD|node_h|2x|node_h|5|node_h}}
40px
(3.3.3.5)

|(1)
{{CDD|node_h|5|node_h|2x|node_h}}
40px
(3.3.3.5)

|(1)
{{CDD|node_h|5|node_h|3|node_h}}
40px
(3.3.3.3.5)

|(4)
40px
+(3.3.3)

|100px

|

= [5,3<sup>1,1</sup>] family =

There are 11 forms (and only 4 not shared with [5,3,4] family), generated by ring permutations of the Coxeter group: [5,31,1] or {{CDD|nodes|split2|node|5|node}}. If the branch ring states match, an extended symmetry can double into the [5,3,4] family, {{CDD|nodeab_c1|split2|node_c2|5|node_c3}} ↔ {{CDD|node_h0|4|node_c1|3|node_c2|5|node_c3}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=4|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|nodea|5a|nodea}}

!1
{{CDD|nodes|2|node}}

!0'
{{CDD|nodea|3a|nodea|5a|nodea}}

!3
{{CDD|nodes|split2|node}}

align=center

!34

|alternated order-5 cubic (apech)
{{CDD|nodes_10ru|split2|node|5|node}} ↔ {{CDD|node_h1|4|node|3|node|5|node}}

| -

| -

|(12)
40px
(3.3.3.3.3)

|(20)
40px
(3.3.3)

|100px

|100px

align=center

!35

|cantic order-5 cubic (tapech)
{{CDD|nodes_10ru|split2|node_1|5|node}} ↔ {{CDD|node_h1|4|node|3|node_1|5|node}}

|(1)
40px
(3.5.3.5)

| -

|(2)
40px
(5.6.6)

|(2)
40px
(3.6.6)

|100px

|120px

align=center

!36

|runcic order-5 cubic (birapech)
{{CDD|nodes_10ru|split2|node|5|node_1}} ↔ {{CDD|node_h1|4|node|3|node|5|node_1}}

| (1)
40px
(5.5.5)

| -

| (3)
40px
(3.4.5.4)

| (1)
40px
(3.3.3)

|100px

|120px

align=center

!37

|runcicantic order-5 cubic (bitapech)
{{CDD|nodes_10ru|split2|node_1|5|node_1}} ↔ {{CDD|node_h1|4|node|3|node_1|5|node_1}}

|(1)
40px
(3.10.10)

| -

|(2)
40px
(4.6.10)

|(1)
40px
(3.6.6)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
{{CDD|nodeab_c1|split2|node_c2|5|node_c3}} ↔ {{CDD|node_h0|4|node_c1|3|node_c2|5|node_c3}}

!colspan=4|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|nodea|5a|nodea}}

!1
{{CDD|nodes|2|node}}

!3
{{CDD|nodes|split2|node}}

!Alt

align=center

| [10]

|Order-4 dodecahedral (doehon)
{{CDD|nodes|split2|node|5|node_1}} ↔ {{CDD|node_h0|4|node|3|node|5|node_1}}

| (4)
40px
(5.5.5)

| -

| -

|

|100px

|120px

align=center

| [11]

|rectified order-4 dodecahedral (riddoh)
{{CDD|nodes|split2|node_1|5|node}} ↔ {{CDD|node_h0|4|node|3|node_1|5|node}}

| (2)
40px
(3.5.3.5)

| -

| (2)
40px
(3.3.3.3)

|

|100px

|120px

align=center

| [12]

|rectified order-5 cubic (ripech)
{{CDD|nodes_11|split2|node|5|node}} ↔ {{CDD|node_h0|4|node_1|3|node|5|node}}

| (1)
40px
(3.3.3.3.3)

| -

| (5)
40px
(3.4.3.4)

|

|100px

|120px

align=center

| [15]

|bitruncated order-5 cubic (ciddoh)
{{CDD|nodes_11|split2|node_1|5|node}} ↔ {{CDD|node_h0|4|node_1|3|node_1|5|node}}

| (1)
40px
(5.6.6)

| -

| (2)
40px
(4.6.6)

|

|100px

|120px

align=center

| [14]

|truncated order-4 dodecahedral (tiddoh)
{{CDD|nodes|split2|node_1|5|node_1}} ↔ {{CDD|node_h0|4|node|3|node_1|5|node_1}}

| (2)
40px
(3.10.10)

| -

| (1)
40px
(3.3.3.3)

|

|100px

|120px

align=center

| [17]

|cantellated order-4 dodecahedral (sriddoh)
{{CDD|nodes_11|split2|node|5|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node|5|node_1}}

| (1)
40px
(3.4.5.4)

| (2)
40px
(4.4.4)

| (1)
40px
(3.4.3.4)

|

|100px

|120px

align=center

| [20]

|cantitruncated order-4 dodecahedral (griddoh)
{{CDD|nodes_11|split2|node_1|5|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node_1|5|node_1}}

| (1)
40px
(4.6.10)

| (1)
40px
(4.4.4)

| (1)
40px
(4.6.6)

|

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|snub rectified order-4 dodecahedral
{{CDD|nodes_hh|split2|node_h|5|node_h}} ↔ {{CDD|node_h0|4|node_h|3|node_h|5|node_h}}

|(2)
40px
(3.3.3.3.5)

|(1)
40px
(3.3.3)

|(2)
40px
(3.3.3.3.3)

|(4)
40px
+(3.3.3)

100px
Irr. tridiminished icosahedron

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

There are 9 forms, generated by ring permutations of the Coxeter group: {{CDD|label4|branch|3ab|branch}}

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

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=5|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|branch}}

!1
{{CDD|nodeb|3b|branch}}

!2
{{CDD|label4|branch|3b|nodeb}}

!3
{{CDD|label4|branch|3a|nodea}}

!Alt

align=center

!38

|tetrahedral-cubic (gadtatdic)
{{CDD|label4|branch_10r|3ab|branch}}
{(3,3,3,4)}

| (4)
40px
(3.3.3)

| -

| (4)
40px
(4.4.4)

| (6)
40px
(3.4.3.4)

|

|100px

|125px

align=center

!39

|tetrahedral-octahedral (gacocaddit)
{{CDD|label4|branch|3ab|branch_10l}}
{(3,3,4,3)}

| (12)
40px
(3.3.3.3)

| (8)
40px
(3.3.3)

| -

| (8)
40px
(3.3.3.3)

|

|100px

|125px

align=center

!40

|cyclotruncated tetrahedral-cubic (cytitch)
{{CDD|label4|branch_10r|3ab|branch_10l}}
ct{(3,3,3,4)}

| (3)
40px
(3.6.6)

(1)
40px
(3.3.3)

| (1)
40px
(4.4.4)

| (3)
40px
(4.6.6)

|

|100px

|125px

align=center

!41

|cyclotruncated cube-tetrahedron (cyticth)
{{CDD|label4|branch_11|3ab|branch}}
ct{(4,3,3,3)}

| (1)
40px
(3.3.3)

| (1)
40px
(3.3.3)

| (3)
40px
(3.8.8)

| (3)
40px
(3.8.8)

|

|100px

|125px

align=center

!42

|cyclotruncated octahedral-tetrahedral (cytoth)
{{CDD|label4|branch|3ab|branch_11}}
ct{(3,3,4,3)}

| (4)
40px
(3.6.6)

| (4)
40px
(3.6.6)

| (1)
40px
(3.3.3.3)

| (1)
40px
(3.3.3.3)

|

|100px

|125px

align=center

!43

|rectified tetrahedral-cubic (ritch)
{{CDD|label4|branch_01r|3ab|branch_10l}}
r{(3,3,3,4)}

| (1)
40px
(3.3.3.3)

| (2)
40px
(3.4.3.4)

| (1)
40px
(3.4.3.4)

| (2)
40px
(3.4.4.4)

|

|100px

|125px

align=center

!44

|truncated tetrahedral-cubic (titch)
{{CDD|label4|branch_11|3ab|branch_10l}}
t{(3,3,3,4)}

| (1)
40px
(3.6.6)

| (1)
40px
(3.4.3.4)

| (1)
40px
(3.8.8)

| (2)
40px
(4.6.8)

|

|100px

|125px

align=center

!45

|truncated tetrahedral-octahedral (titdoh)
{{CDD|label4|branch_10r|3ab|branch_11}}
t{(3,3,4,3)}

| (2)
40px
(4.6.6)

| (1)
40px
(3.6.6)

| (1)
40px
(3.4.4.4)

| (1)
40px
(4.6.6)

|

|100px

|125px

align=center

!46

|omnitruncated tetrahedral-cubic (otitch)
{{CDD|label4|branch_11|3ab|branch_11}}
tr{(3,3,3,4)}

| (1)
40px
(4.6.6)

| (1)
40px
(4.6.6)

| (1)
40px
(4.6.8)

| (1)
40px
(4.6.8)

|

|100px

|125px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub tetrahedral-cubic
{{CDD|label4|branch_hh|3ab|branch_hh}}
sr{(3,3,3,4)}

|(1)
40px
(3.3.3.3.3)

|(1)
40px
(3.3.3.3.3)

|(1)
40px
(3.3.3.3.4)

|(1)
40px
(3.3.3.3.4)

|(4)
40px
+(3.3.3)

|100px

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

There are 9 forms, generated by ring permutations of the Coxeter group: {{CDD|label5|branch|3ab|branch}}

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

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=4|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|branch}}

!1
{{CDD|nodeb|3b|branch}}

!2
{{CDD|label5|branch|3b|nodeb}}

!3
{{CDD|label5|branch|3a|nodea}}

align=center

!47

|tetrahedral-dodecahedral
{{CDD|label5|branch_10r|3ab|branch}}

| (4)
40px
(3.3.3)

| -

| (4)
40px
(5.5.5)

| (6)
40px
(3.5.3.5)

|100px

|120px

align=center

!48

|tetrahedral-icosahedral
{{CDD|label5|branch|3ab|branch_10l}}

| (30)
40px
(3.3.3.3)

| (20)
40px
(3.3.3)

| -

| (12)
40px
(3.3.3.3.3)

|100px

|120px

align=center

!49

|cyclotruncated tetrahedral-dodecahedral
{{CDD|label5|branch_10r|3ab|branch_10l}}

| (3)
40px
(3.6.6)

| (1)
40px
(3.3.3)

| (1)
40px
(5.5.5)

| (3)
40px
(5.6.6)

|100px

|120px

align=center

!52

|rectified tetrahedral-dodecahedral
{{CDD|label5|branch_01r|3ab|branch_10l}}

| (1)
40px
(3.3.3.3)

| (2)
40px
(3.4.3.4)

| (1)
40px
(3.5.3.5)

| (2)
40px
(3.4.5.4)

|100px

|120px

align=center

!53

|truncated tetrahedral-dodecahedral
{{CDD|label5|branch_11|3ab|branch_10l}}

| (1)
40px
(3.6.6)

| (1)
40px
(3.4.3.4)

| (1)
40px
(3.10.10)

| (2)
40px
(4.6.10)

|100px

|120px

align=center

!54

|truncated tetrahedral-icosahedral
{{CDD|label5|branch_10r|3ab|branch_11}}

| (2)
40px
(4.6.6)

| (1)
40px
(3.6.6)

| (1)
40px
(3.4.5.4)

| (1)
40px
(5.6.6)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
{{CDD|label5|branch_c1|3ab|branch_c2}}

!colspan=3|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0,1
{{CDD|nodea|3a|branch}}

!2,3
{{CDD|label5|branch|3b|nodeb}}

!Alt

align=center

!50

|cyclotruncated dodecahedral-tetrahedral
{{CDD|label5|branch_11|3ab|branch}}

| (2)
40px
(3.3.3)

| (6)
40px
(3.10.10)

|

|100px

|120px

align=center

!51

|cyclotruncated tetrahedral-icosahedral
{{CDD|label5|branch|3ab|branch_11}}

| (10)
40px
(3.6.6)

| (2)
40px
(3.3.3.3.3)

|

|100px

|120px

align=center

!55

|omnitruncated tetrahedral-dodecahedral
{{CDD|label5|branch_11|3ab|branch_11}}

| (2)
40px
(4.6.6)

| (2)
40px
(4.6.10)

|

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub tetrahedral-dodecahedral
{{CDD|label5|branch_hh|3ab|branch_hh}}

|(2)
40px
(3.3.3.3.3)

|(2)
40px
(3.3.3.3.5)

|(4)
40px
+(3.3.3)

100px

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

There are 6 forms, generated by ring permutations of the Coxeter group: {{CDD|label4|branch|3ab|branch|label4}}. There are 4 extended symmetries possible based on the symmetry of the rings: {{CDD|label4|branch_c1-2|3ab|branch_c1-2|label4}}, {{CDD|label4|branch_c1|3ab|branch_c2|label4}}, {{CDD|label4|branch_c1-2|3ab|branch_c2-1|label4}}, and {{CDD|label4|branch_c1|3ab|branch_c1|label4}}.

This symmetry family is also related to a radical subgroup, index 6, {{CDD|branch_c1-2|4a4b|branch|labels}} ↔ {{CDD|node_c1|splitplit1u|branch3u_c2|3a3buc-cross|branch3u_c1|splitplit2u|node_c2}}, 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}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=4|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Pictures

0
{{CDD|nodea|3a|branch|label4}}

!1
{{CDD|nodeb|3b|branch|label4}}

!2
{{CDD|label4|branch|3b|nodeb}}

!3
{{CDD|label4|branch|3a|nodea}}

align=center

!56

|cubic-octahedral (cohon)
{{CDD|label4|branch_10r|3ab|branch|label4}}

| (6)
40px
(3.3.3.3)

| -

| (8)
40px
(4.4.4)

| (12)
40px
(3.4.3.4)

|100px

|120px

align=center

!60

|truncated cubic-octahedral (tucoh)
{{CDD|label4|branch_11|3ab|branch_10l|label4}}

| (1)
40px
(4.6.6)

| (1)
40px
(3.4.4.4)

| (1)
40px
(3.8.8)

| (2)
40px
(4.6.8)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
{{CDD|label4|branch_c1-2|3ab|branch_c1-2|label4}}

!colspan=3|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0,3
{{CDD|nodea|3a|branch|label4}}

!1,2
{{CDD|nodeb|3b|branch|label4}}

!Alt

align=center

!57

|cyclotruncated octahedral-cubic (cytoch)
{{CDD|label4|branch_10r|3ab|branch_10l|label4}}

| (6)
40px
(4.6.6)

| (2)
40px
(4.4.4)

|

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|cyclosnub octahedral-cubic
{{CDD|label4|branch_h0r|3ab|branch_h0l|label4}}

|(4)
40px
(3.3.3.3.3)

|(2)
40px
(3.3.3)

|(4)
40px
+(3.3.3.3)

|100px

|

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
{{CDD|label4|branch_c1|3ab|branch_c2|label4}}

!colspan=2|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0,1
{{CDD|nodea|3a|branch|label4}}

!2,3
{{CDD|label4|branch|3b|nodeb}}

align=center

!58

|cyclotruncated cubic-octahedral (cytacoh)
{{CDD|label4|branch_11|3ab|branch|label4}}

| (2)
40px
(3.3.3.3)

| (6)
40px
(3.8.8)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
{{CDD|label4|branch_c1-2|3ab|branch_c2-1|label4}}

!colspan=2|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0,2
{{CDD|nodea|3a|branch|label4}}

!1,3
{{CDD|nodeb|3b|branch|label4}}

align=center

!59

|rectified cubic-octahedral (racoh)
{{CDD|label4|branch_01r|3ab|branch_10l|label4}}

| (2)
40px
(3.4.3.4)

| (4)
40px
(3.4.4.4)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram
{{CDD|label4|branch_c1|3ab|branch_c1|label4}}

!colspan=2|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0,1,2,3
{{CDD|nodea|3a|branch|label4}}

!Alt

align=center

!61

|omnitruncated cubic-octahedral (otacoh)
{{CDD|label4|branch_11|3ab|branch_11|label4}}

| (4)
40px
(4.6.8)

|

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub cubic-octahedral
{{CDD|label4|branch_hh|3ab|branch_hh|label4}}

|(4)
40px
(3.3.3.3.4)

|(4)
40px
+(3.3.3)

|100px

|

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

There are 9 forms, generated by ring permutations of the Coxeter group: {{CDD||label5|branch|3ab|branch|label4}}

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

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=4|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|branch|label4}}

!1
{{CDD|nodeb|3b|branch|label4}}

!2
{{CDD|label5|branch|3b|nodeb}}

!3
{{CDD|label5|branch|3a|nodea}}

align=center

!62

|octahedral-dodecahedral
{{CDD|label5|branch_10r|3ab|branch|label4}}

| (6)
40px
(3.3.3.3)

| -

| (8)
40px
(5.5.5)

| (1)
40px
(3.5.3.5)

|100px

|120px

align=center

!63

|cubic-icosahedral
{{CDD|label5|branch|3ab|branch_10l|label4}}

| (30)
40px
(3.4.3.4)

| (20)
40px
(4.4.4)

| -

| (12)
40px
(3.3.3.3.3)

|100px

|120px

align=center

!64

|cyclotruncated octahedral-dodecahedral
{{CDD|label5|branch_10r|3ab|branch_10l|label4}}

| (3)
40px
(4.6.6)

| (1)
40px
(4.4.4)

| (1)
40px
(5.5.5)

| (3)
40px
(5.6.6)

|100px

|120px

align=center

!67

|rectified octahedral-dodecahedral
{{CDD|label5|branch_01r|3ab|branch_10l|label4}}

| (1)
40px
(3.4.3.4)

| (2)
40px
(3.4.4.4)

| (1)
40px
(3.5.3.5)

| (2)
40px
(3.4.5.4)

|100px

|120px

align=center

!68

|truncated octahedral-dodecahedral
{{CDD|label5|branch_11|3ab|branch_10l|label4}}

| (1)
40px
(4.6.6)

| (1)
40px
(3.4.4.4)

| (1)
40px
(3.10.10)

| (2)
40px
(4.6.10)

|100px

|120px

align=center

!69

|truncated cubic-dodecahedral
{{CDD|label5|branch_10r|3ab|branch_11|label4}}

| (2)
40px
(4.6.8)

| (1)
40px
(3.8.8)

| (1)
40px
(3.4.5.4)

| (1)
40px
(5.6.6)

|100px

|120px

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=3|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0,1
{{CDD|nodea|3a|branch|label4}}

!2,3
{{CDD|label5|branch|3b|nodeb}}

!Alt

align=center

!65

|cyclotruncated dodecahedral-octahedral
{{CDD|label5|branch_11|3ab|branch|label4}}

| (2)
40px
(3.3.3.3)

| (8)
40px
(3.10.10)

|

|100px

|120px

align=center

!66

|cyclotruncated cubic-icosahedral
{{CDD|label5|branch|3ab|branch_11|label4}}

| (10)
40px
(3.8.8)

| (2)
40px
(3.3.3.3.3)

|

|100px

|120px

align=center

!70

|omnitruncated octahedral-dodecahedral
{{CDD|label5|branch_11|3ab|branch_11|label4}}

| (2)
40px
(4.6.8)

| (2)
40px
(4.6.10)

|

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub octahedral-dodecahedral
{{CDD|label5|branch_hh|3ab|branch_hh|label4}}

|(2)
40px
(3.3.3.3.4)

|(2)
40px
(3.3.3.3.5)

|(4)
40px
+(3.3.3)

|100px

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

There are 6 forms, generated by ring permutations of the Coxeter group: {{CDD||label5|branch|3ab|branch|label5}}. There are 4 extended symmetries possible based on the symmetry of the rings: {{CDD|label5|branch_c1-2|3ab|branch_c1-2|label5}}, {{CDD|label5|branch_c1|3ab|branch_c2|label5}}, {{CDD|label5|branch_c1-2|3ab|branch_c2-1|label5}}, and {{CDD|label5|branch_c1|3ab|branch_c1|label5}}.

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

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

!colspan=5|Cells by location
(and count around each vertex)

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|branch|label5}}

!1
{{CDD|nodeb|3b|branch|label5}}

!2
{{CDD|label5|branch|3b|nodeb}}

!3
{{CDD|label5|branch|3a|nodea}}

!Alt

align=center

!71

|dodecahedral-icosahedral
{{CDD|label5|branch_10r|3ab|branch|label5}}

| (12)
40px
(3.3.3.3.3)

| -

| (20)
40px
(5.5.5)

| (30)
40px
(3.5.3.5)

|

|100px

|120px

align=center

!72

|cyclotruncated icosahedral-dodecahedral
{{CDD|label5|branch_10r|3ab|branch_10l|label5}}

| (3)
40px
(5.6.6)

| (1)
40px
(5.5.5)

| (1)
40px
(5.5.5)

| (3)
40px
(5.6.6)

|

|100px

|120px

align=center

!73

|cyclotruncated dodecahedral-icosahedral
{{CDD|label5|branch_11|3ab|branch|label5}}

| (1)
40px
(3.3.3.3.3)

| (1)
40px
(3.3.3.3.3)

| (3)
40px
(3.10.10)

| (3)
40px
(3.10.10)

|

|100px

|120px

align=center

!74

|rectified dodecahedral-icosahedral
{{CDD|label5|branch_01r|3ab|branch_10l|label5}}

| (1)
40px
(3.5.3.5)

| (2)
40px
(3.4.5.4)

| (1)
40px
(3.5.3.5)

| (2)
40px
(3.4.5.4)

|

|100px

|120px

align=center

!75

|truncated dodecahedral-icosahedral
{{CDD|label5|branch_11|3ab|branch_10l|label5}}

| (1)
40px
(5.6.6)

| (1)
40px
(3.4.5.4)

| (1)
40px
(3.10.10)

| (2)
40px
(4.6.10)

|

|100px

|120px

align=center

!76

|omnitruncated dodecahedral-icosahedral
{{CDD|label5|branch_11|3ab|branch_11|label5}}

| (1)
40px
(4.6.10)

| (1)
40px
(4.6.10)

| (1)
40px
(4.6.10)

| (1)
40px
(4.6.10)

|

|100px

|120px

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub dodecahedral-icosahedral
{{CDD|label5|branch_hh|3ab|branch_hh|label5}}

| (1)
40px
(3.3.3.3.5)

| (1)
40px
(3.3.3.3.5)

| (1)
40px
(3.3.3.3.5)

| (1)
40px
(3.3.3.3.5)

| (4)
40px
+(3.3.3)

| 100px

= 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}.{{Cite web | url=https://www.bendwavy.org/klitzing/dimensions/hyperbolic.htm | title=Hyperbolic Tesselations}}

In 1997 Wendy Krieger discovered an infinite series of uniform hyperbolic honeycombs with pseudoicosahedral vertex figures, made from 8 cubes and 12 p-gonal prisms at a vertex for any integer p. In the case p = 4, all cells are cubes and the result is the order-5 cubic honeycomb. The case p = 2 degenerates to the Euclidean cubic honeycomb.

Another four known ones are related to noncompact families. The tessellation {{CDD|node_1|4|node_1|3|node|8|node}} consists of truncated cubes {{CDD|node_1|4|node_1|3|node}} and infinite order-8 triangular tilings {{CDD|node_1|3|node|8|node}}. However the latter intersect the sphere at infinity orthogonally, having exactly the same curvature as the hyperbolic space, and can be replaced by mirror images of the remainder of the tessellation, resulting in a compact uniform honeycomb consisting only of the truncated cubes. (So they are analogous to the hemi-faces of spherical hemipolyhedra.){{Cite web | url=https://www.bendwavy.org/klitzing/incmats/x4x3o8o.htm | title=x4x3o8o}} Something similar can be done with the tessellation {{CDD|nodes_11|split2-43|node|8|node}} consisting of small rhombicuboctahedra {{CDD|node_1|4|node|3|node_1}}, infinite order-8 triangular tilings {{CDD|node_1|3|node|8|node}}, and infinite order-8 square tilings {{CDD|node_1|4|node|8|node}}. The order-8 square tilings already intersect the sphere at infinity orthogonally, and if the order-8 triangular tilings are augmented with a set of triangular prisms, the surface passing through their centre points also intersects the sphere at infinity orthogonally. After replacing with mirror images, the result is a compact honeycomb containing the small rhombicuboctahedra and the triangular prisms.{{Cite web | url=https://www.bendwavy.org/klitzing/incmats/lt-o8o4xb3x.htm | title=lt-o8o4xb3x}} Two more such constructions were discovered in 2023. The first one arises from the fact that {{CDD|node_1|3|node_1|4|node|6|node}} and {{CDD|node|3|node_1|4|node|6|node}} have the same circumradius; the former has truncated octahedra {{CDD|node_1|3|node_1|4|node}} and order-6 square tilings {{CDD|node_1|4|node|6|node}}, while the latter has cuboctahedra {{CDD|node|3|node_1|4|node}} and order-6 square tilings {{CDD|node_1|4|node|6|node}}. A compact uniform honeycomb is taken by discarding the order-6 square tilings they have in common, using only the truncated octahedra and cuboctahedra. The second one arises from a similar construction involving {{CDD|nodes_11|split2-53|node|4|node}} (which has small rhombicosidodecahedra {{CDD|node_1|5|node|3|node_1}}, octahedra {{CDD|node_1|3|node|4|node}}, and order-4 pentagonal tilings {{CDD|node_1|5|node|4|node}}) and {{CDD|node_1|2|node_1|5|node|4|node}} (which is the prism of the order-4 pentagonal tiling, having pentagonal prisms {{CDD|node_1|2|node_1|5|node}} and order-4 pentagonal tilings {{CDD|node_1|5|node|4|node}}). These two likewise have the same circumradius, and a compact uniform honeycomb is taken by using only the finite cells of both, discarding the order-4 pentagonal tilings they have in common.{{Cite web | url=https://www.bendwavy.org/klitzing/dimensions/coxeter.htm#trip | title=Hyperbolic Tessellations – Triangular Prismatic Domains}}

Another non-Wythoffian was discovered in 2021. It has as vertex figure a snub cube with 8 vertices removed and contains two octahedra and eight snub cubes at each vertex. Subsequently Krieger found a non-Wythoffian with a snub cube as the vertex figure, containing 32 tetrahedra and 6 octahedra at each vertex, and that the truncated and rectified versions of this honeycomb are still uniform. In 2022, Richard Klitzing generalised this construction to use any snub {{CDD|node_h|3|node_h|p|node_h}} as vertex figure: the result is compact for p=4 or 5 (with a snub cube or 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.

There are also other forms based on 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).

= 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.

class=wikitable

!Index

!Coxeter group

!Extended
symmetry

!colspan=2|Honeycombs

!Chiral
extended
symmetry

!colspan=2|Alternation honeycombs

align=center

!rowspan=2|H1

|rowspan=2|{\bar{BH}}_3
[4,3,5]
{{CDD|node|4|node|3|node|5|node}}

rowspan=2|[4,3,5]
{{CDD|node_c1|4|node_c2|3|node_c3|5|node_c4}}		rowspan=2| 15

|rowspan=2|{{CDD|node_1|5|node|3|node|4|node}} | {{CDD|node|5|node_1|3|node|4|node}} | {{CDD|node|5|node|3|node_1|4|node}} | {{CDD|node|5|node|3|node|4|node_1}} | {{CDD|node_1|5|node_1|3|node|4|node}}
{{CDD|node_1|5|node|3|node_1|4|node}} | {{CDD|node_1|5|node|3|node|4|node_1}} | {{CDD|node|5|node_1|3|node|4|node_1}} | {{CDD|node|5|node_1|3|node_1|4|node}} | {{CDD|node|5|node|3|node_1|4|node_1}}
{{CDD|node_1|5|node_1|3|node_1|4|node}} | {{CDD|node_1|5|node_1|3|node|4|node_1}} | {{CDD|node_1|5|node|3|node_1|4|node_1}} | {{CDD|node|5|node_1|3|node_1|4|node_1}} | {{CDD|node_1|5|node_1|3|node_1|4|node_1}}

|[1+,4,(3,5)+]

(2){{CDD|node|5|node|3|node|4|node_h1}} (= {{CDD|node|5|node|split1|nodes_10lu}})
{{CDD|node_h|5|node_h|3|node_h|4|node}}
align=center

|[4,3,5]+

(1){{CDD|node_h|5|node_h|3|node_h|4|node_h}}
align=center

!rowspan=2|H2

|rowspan=2|{\bar{J}}_3
[3,5,3]
{{CDD|node|3|node|5|node|3|node}}

[3,5,3]
{{CDD|node_c1|3|node_c2|5|node_c3|3|node_c4}}		6

| {{CDD|node_1|3|node|5|node|3|node}} | {{CDD|node|3|node_1|5|node|3|node}} | {{CDD|node_1|3|node_1|5|node|3|node}} | {{CDD|node_1|3|node|5|node_1|3|node}} | {{CDD|node_1|3|node_1|5|node_1|3|node}} | {{CDD|node_1|3|node_1|5|node|3|node_1}}

|colspan=3|

align=center[2+[3,5,3]]
{{CDD|node_c1|3|node_c2|5|node_c2|3|node_c1}}		5

| {{CDD|node_1|3|node|5|node|3|node_1}} | {{CDD|node|3|node_1|5|node_1|3|node}} | {{CDD|node_1|3|node_1|5|node_1|3|node_1}}

[2+[3,5,3]]+(1)

| {{CDD|node_h|3|node_h|5|node_h|3|node_h}}

align=center

!rowspan=2|H3

|rowspan=2|{\bar{DH}}_3
[5,31,1]
{{CDD|node|5|node|split1|nodes}}

[5,31,1]
{{CDD|node_c3|5|node_c4|split1|nodeab_c1-2}}		4

|{{CDD|node|5|node|split1|nodes_10lu}} | {{CDD|node_1|5|node|split1|nodes_10lu}} | {{CDD|node|5|node_1|split1|nodes_10lu}} | {{CDD|node_1|5|node_1|split1|nodes_10lu}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center

| [1[5,31,1]]=[5,3,4]
{{CDD|node_c1|5|node_c2|split1|nodeab_c3}} ↔ {{CDD|node_c1|5|node_c2|3|node_c3|4|node_h0}}

(7)

|{{CDD|node_1|5|node|split1|nodes}} | {{CDD|node|5|node_1|split1|nodes}} | {{CDD|node_1|5|node_1|split1|nodes}} | {{CDD|node|5|node|split1|nodes_11}} | {{CDD|node_1|5|node|split1|nodes_11}} | {{CDD|node|5|node_1|split1|nodes_11}} | {{CDD|node_1|5|node_1|split1|nodes_11}}

|[1[5,31,1]]+
=[5,3,4]+

(1)

|{{CDD|node_h|5|node_h|split1|nodes_hh}}

align=center

!rowspan=2|H4

|rowspan=2|{\widehat{AB}}_3
[(4,3,3,3)]
{{CDD|label4|branch|3ab|branch}}

[(4,3,3,3)]6

|{{CDD|label4|branch_10r|3ab|branch}} | {{CDD|label4|branch|3ab|branch_10l}} | {{CDD|label4|branch_01r|3ab|branch_10l}} | {{CDD|label4|branch_10r|3ab|branch_10l}} | {{CDD|label4|branch_11|3ab|branch_10l}} | {{CDD|label4|branch_10r|3ab|branch_11}}

|colspan=3|

align=center[2+[(4,3,3,3)]]
{{CDD|label4|branch_c1|3ab|branch_c2}}		3

| {{CDD|label4|branch_11|3ab|branch}} | {{CDD|label4|branch|3ab|branch_11}} | {{CDD|label4|branch_11|3ab|branch_11}}

[2+[(4,3,3,3)]]+(1)

| {{CDD|label4|branch_hh|3ab|branch_hh}}

align=center

!rowspan=2|H5

|rowspan=2|{\bar{K}}_3
[5,3,5]
{{CDD|node|5|node|3|node|5|node}}

[5,3,5]
{{CDD|node_c1|5|node_c2|3|node_c3|5|node_c4}}		6

| {{CDD|node_1|5|node|3|node|5|node}} | {{CDD|node|5|node_1|3|node|5|node}} | {{CDD|node_1|5|node_1|3|node|5|node}} | {{CDD|node_1|5|node|3|node_1|5|node}} | {{CDD|node_1|5|node_1|3|node_1|5|node}} | {{CDD|node_1|5|node_1|3|node|5|node_1}}

|colspan=3|

align=center[2+[5,3,5]]
{{CDD|branch_c1|5a5b|nodeab_c2}}		3

| {{CDD|node_1|5|node|3|node|5|node_1}} | {{CDD|node|5|node_1|3|node_1|5|node}} | {{CDD|node_1|5|node_1|3|node_1|5|node_1}}

[2+[5,3,5]]+(1)

| {{CDD|node_h|5|node_h|3|node_h|5|node_h}}

align=center

!rowspan=2|H6

|rowspan=2|{\widehat{AH}}_3
[(5,3,3,3)]
{{CDD|label5|branch|3ab|branch}}

[(5,3,3,3)]6

|{{CDD|label5|branch_10r|3ab|branch}} | {{CDD|label5|branch|3ab|branch_10l}} | {{CDD|label5|branch_01r|3ab|branch_10l}} | {{CDD|label5|branch_10r|3ab|branch_10l}} | {{CDD|label5|branch_11|3ab|branch_10l}} | {{CDD|label5|branch_10r|3ab|branch_11}}

|colspan=3|

align=center[2+[(5,3,3,3)]]
{{CDD|label5|branch_c1|3ab|branch_c2}}		3

| {{CDD|label5|branch_11|3ab|branch}} | {{CDD|label5|branch|3ab|branch_11}} | {{CDD|label5|branch_11|3ab|branch_11}}

[2+[(5,3,3,3)]]+(1)

| {{CDD|label5|branch_hh|3ab|branch_hh}}

align=center

!rowspan=5|H7

|rowspan=5|{\widehat{BB}}_3
[(3,4)[2]]
{{CDD|label4|branch|3ab|branch|label4}}

[(3,4)[2]]2{{CDD|label4|branch_10r|3ab|branch|label4}} | {{CDD|label4|branch_11|3ab|branch_10l|label4}}

|colspan=3|

align=center[2+[(3,4)[2]]]
{{CDD|label4|branch_c1-2|3ab|branch_c2-1|label4}}		1

| {{CDD|label4|branch_01r|3ab|branch_10l|label4}}

|colspan=3|

align=center[2+[(3,4)[2]]]
{{CDD|label4|branch_c1|3ab|branch_c2|label4}}		1

|{{CDD|label4|branch_11|3ab|branch|label4}}

|colspan=3|

align=center[2+[(3,4)[2]]]
{{CDD|label4|branch_c1-2|3ab|branch_c1-2|label4}}		1

|{{CDD|label4|branch_10r|3ab|branch_10l|label4}}

[2+[(3+,4)[2]]](1)

|{{CDD|label4|branch_h0r|3ab|branch_h0l|label4}}

align=center[(2,2)+[(3,4)[2]]]
{{CDD|label4|branch_c1|3ab|branch_c1|label4}}		1

| {{CDD|label4|branch_11|3ab|branch_11|label4}}

[(2,2)+[(3,4)[2]]]+(1)

| {{CDD|label4|branch_hh|3ab|branch_hh|label4}}

align=center

!rowspan=2|H8

|rowspan=2|{\widehat{BH}}_3
[(5,3,4,3)]
{{CDD|label4|branch|3ab|branch|label5}}

[(5,3,4,3)]6

|{{CDD|label5|branch_10r|3ab|branch|label4}} | {{CDD|label5|branch|3ab|branch_10l|label4}} | {{CDD|label5|branch_01r|3ab|branch_10l|label4}} | {{CDD|label5|branch_10r|3ab|branch_10l|label4}} | {{CDD|label5|branch_11|3ab|branch_10l|label4}} | {{CDD|label5|branch_10r|3ab|branch_11|label4}}

|colspan=3|

align=center[2+[(5,3,4,3)]]
{{CDD|label4|branch_c1|3ab|branch_c2|label5}}		3

| {{CDD|label5|branch_11|3ab|branch|label4}} | {{CDD|label5|branch|3ab|branch_11|label4}} | {{CDD|label5|branch_11|3ab|branch_11|label4}}

[2+[(5,3,4,3)]]+(1)

| {{CDD|label5|branch_hh|3ab|branch_hh|label4}}

align=center

!rowspan=5|H9

|rowspan=5|{\widehat{HH}}_3
[(3,5)[2]]
{{CDD|label5|branch|3ab|branch|label5}}

[(3,5)[2]]2

|{{CDD|label5|branch_10r|3ab|branch|label5}} | {{CDD|label5|branch_11|3ab|branch_10l|label5}}

|colspan=3|

align=center[2+[(3,5)[2]]]
{{CDD|label5|branch_c1-2|3ab|branch_c2-1|label5}}		1

|{{CDD|label5|branch_01r|3ab|branch_10l|label5}}

|colspan=3|

align=center[2+[(3,5)[2]]]
{{CDD|label5|branch_c1|3ab|branch_c2|label5}}		1

|{{CDD|label5|branch_11|3ab|branch|label5}}

|colspan=3|

align=center[2+[(3,5)[2]]]
{{CDD|label5|branch_c1-2|3ab|branch_c1-2|label5}}		1

|{{CDD|label5|branch_10r|3ab|branch_10l|label5}}

|colspan=3|

align=center[(2,2)+[(3,5)[2]]]
{{CDD|label5|branch_c1|3ab|branch_c1|label5}}		1

| {{CDD|label5|branch_11|3ab|branch_11|label5}}

[(2,2)+[(3,5)[2]]]+(1)

| {{CDD|label5|branch_hh|3ab|branch_hh|label5}}

See also

{{Commons category|Uniform tilings of hyperbolic 3-space}}

Notes

{{reflist}}

References

  • J. Humphreys (1990), Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29
  • H.S.M. Coxeter (1954), [https://web.archive.org/web/20190225204136/http://pdfs.semanticscholar.org/67fa/5c924fc515d05640308fe68f1d0974a3705c.pdf "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, 3rd. ed., Dover Publications, 1973. {{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, II
  • A. Felikson (2002), [https://arxiv.org/abs/math/0212010 "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 [http://cms.math.ca/cjm/a145822] {{Webarchive|url=https://web.archive.org/web/20150402131943/http://cms.math.ca/cjm/a145822 |date=2015-04-02 }}
  • N. W. Johnson (2018), Geometries and Transformations, Chapters 11–13
  • N. 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 [https://link.springer.com/article/10.1007%2FBF01238563]
  • N. W. Johnson, R. Kellerhals, J.G. Ratcliffe, S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups H3: p130. [http://www.sciencedirect.com/science/article/pii/S0024379501004773]
  • {{KlitzingPolytopes|hyperbolic.htm#3D-compact|Hyperbolic honeycombs|H3 compact}}

Category:Honeycombs (geometry)