View source for 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|hyperbolic geometry]], a '''uniform honeycomb in hyperbolic space''' is a uniform [[Tessellation|tessellation]] of [[Uniform polyhedron|uniform polyhedral]] cells. In 3-dimensional [[Hyperbolic space|hyperbolic space]] there are nine [[Coxeter group]] families of compact [[Convex uniform honeycomb|convex uniform honeycomb]]s, generated as [[Wythoff construction]]s, and represented by [[Permutation|permutation]]s of rings of the Coxeter diagrams for each family.

{{Clear}}
{| class="wikitable" align=right
|+ Four compact regular hyperbolic honeycombs
|- bgcolor="#d0e0ff" align=center
|[[File:H3 534 CC center.png|200px]]<BR>[[Order-4 dodecahedral honeycomb]]<br>{{math|{5,3,4} }}
|[[File:H3 535 CC center.png|200px]]<BR>[[Order-5 dodecahedral honeycomb]]<br>{{math|{5,3,5} }}
|- bgcolor="#d0e0ff" align=center
|[[File:H3 435 CC center.png|200px]]<BR>[[Order-5 cubic honeycomb]]<br>{{math|{4,3,5} }}
|[[File:H3 353 CC center.png|200px]]<BR>[[Icosahedral honeycomb]]<br>{{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 group]]s, 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 group]]s are listed here with their Coxeter diagrams,<ref>Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [https://books.google.com/books?id=ODfjmOeNLMUC&pg=PA141]</ref> 
in order of the relative volumes of their [[Fundamental domain|fundamental simplex domains]].<ref name="Felikson, 2002">Felikson, 2002</ref>

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: [5,3<sup>1,1</sup>] ↔ [5,3,4,1<sup>+</sup>].

{| class=wikitable
!Indexed
!Fundamental<br>simplex<br>volume<ref name="Felikson, 2002"/>
!Witt<br>symbol
!Coxeter<br>notation
![[Commutator subgroup|Commutator<BR>subgroup]]
!Coxeter<BR>diagram
!Honeycombs
|-
!H<sub>1</sub>
|0.0358850633
|<math>{\bar{BH}}_3</math> || [5,3,4] || [(5,3)<sup>+</sup>,4,1<sup>+</sup>]<BR>= [5,3<sup>1,1</sup>]<sup>+</sup>|| {{CDD|node|5|node|3|node|4|node}} || 15 forms, 2 regular
|-
!H<sub>2</sub>
|0.0390502856
|<math>{\bar{J}}_3</math> || [3,5,3]|| [3,5,3]<sup>+</sup>|| {{CDD|node|3|node|5|node|3|node}} || 9 forms, 1 regular
|-
!H<sub>3</sub>
|0.0717701267
|<math>{\bar{DH}}_3</math> || [5,3<sup>1,1</sup>]|| [5,3<sup>1,1</sup>]<sup>+</sup>|| {{CDD|node|5|node|split1|nodes}} || 11 forms (7 overlap with [5,3,4] family, 4 are unique)
|-
!H<sub>4</sub>
|0.0857701820
|<math>{\widehat{AB}}_3</math> || [(4,3,3,3)]|| [(4,3,3,3)]<sup>+</sup>|| {{CDD|label4|branch|3ab|branch}} || 9 forms
|-
!H<sub>5</sub>
|0.0933255395
|<math>{\bar{K}}_3</math> || [5,3,5]|| [5,3,5]<sup>+</sup>|| {{CDD|node|5|node|3|node|5|node}} || 9 forms, 1 regular
|-
!H<sub>6</sub>
|0.2052887885
|<math>{\widehat{AH}}_3</math> || [(5,3,3,3)]|| [(5,3,3,3)]<sup>+</sup>|| {{CDD|label5|branch|3ab|branch}} || 9 forms
|-
!H<sub>7</sub>
|0.2222287320
|<math>{\widehat{BB}}_3</math> || [(4,3)<sup>[2]</sup>]|| [(4,3<sup>+</sup>,4,3<sup>+</sup>)]|| {{CDD|label4|branch|3ab|branch|label4}} || 6 forms
|-
!H<sub>8</sub>
|0.3586534401
|<math>{\widehat{BH}}_3</math> || [(3,4,3,5)]|| [(3,4,3,5)]<sup>+</sup>|| {{CDD|label5|branch|3ab|branch|label4}} || 9 forms
|-
!H<sub>9</sub>
|0.5021308905
|<math>{\widehat{HH}}_3</math> || [(5,3)<sup>[2]</sup>]|| [(5,3)<sup>[2]</sup>]<sup>+</sup>|| {{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<sup>*</sup>)], represented by Coxeter diagrams {{CDD|branch_c1-2|4a4b|branch|labels}} an index 6 subgroup with a [[Trigonal trapezohedron|trigonal trapezohedron]] [[Fundamental domain|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)<sup>*</sup>], 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 [[Facet (geometry)|facets]] or [[Vertex figure|vertex figure]], including ideal vertices at infinity.

{| class=wikitable
|+ 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|triangular bipyramid]] [[Fundamental domain|fundamental domain]]s (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.
{| class=wikitable
!Dimension
!Rank
!Graphs
|-
!H<sup>3</sup>
!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 [[Wythoff construction|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|tetrahedrally diminished dodecahedron]].<ref>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]</ref> Another is constructed with 2 antipodal vertices removed.<ref name=klitzing/>

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

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=4|Cell counts/vertex<br>and positions in honeycomb
!rowspan=2|[[Vertex figure]]
!rowspan=2|Picture
|- align=center
!0<br>{{CDD|node_n2|5|node_n3|3|node_n4}}
!1<br>{{CDD|node_n1|2|2|node_n3|3|node_n4}}
!2<br>{{CDD|node_n1|3|node_n2|2|node_n4}}
!3<br>{{CDD|node_n1|3|node_n2|5|node_n3}}
|- align=center BGCOLOR="#f0e0e0"
!1
|[[Icosahedral honeycomb|icosahedral]] (ikhon)<br>{{CDD|node_1|3|node|5|node|3|node}}<br>t<sub>0</sub>{3,5,3}
|&nbsp;
|&nbsp;
|&nbsp;
|(12)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|[[Image:Order-3 icosahedral honeycomb verf.svg|100px]]
|[[File:H3 353 CC center.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!2
|rectified icosahedral (rih)<br>{{CDD|node|3|node_1|5|node|3|node}}<br>t<sub>1</sub>{3,5,3}
|(2)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|&nbsp;
|&nbsp;
|(3)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[Image:Rectified icosahedral honeycomb verf.png|100px]]
|[[File:H3 353 CC center 0100.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!3
|truncated icosahedral (tih)<br>{{CDD|node_1|3|node_1|5|node|3|node}}<br>t<sub>0,1</sub>{3,5,3}
|(1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|&nbsp;
|&nbsp;
|(3)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[Image:Truncated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-0011 center ultrawide.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!4
|cantellated icosahedral (srih)<br>{{CDD|node_1|3|node|5|node_1|3|node}}<br>t<sub>0,2</sub>{3,5,3}
|(1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|(2)<br>[[Image:triangular prism.png|40px]]<br>[[Triangular prism|(4.4.3)]]
|&nbsp;
|(2)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.5.4.5)]]
|[[Image:Cantellated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-1010 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0e0"
!5
|runcinated icosahedral (spiddih)<br>{{CDD|node_1|3|node|5|node|3|node_1}}<br>t<sub>0,3</sub>{3,5,3}
|(1)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(5)<br>[[Image:triangular prism.png|40px]]<br>[[Triangular prism|(4.4.3)]]
|(5)<br>[[Image:triangular prism.png|40px]]<br>[[Triangular prism|(4.4.3)]]
|(1)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|[[Image:Runcinated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-1001 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0e0"
!6
|bitruncated icosahedral (dih)<br>{{CDD|node|3|node_1|5|node_1|3|node}}<br>t<sub>1,2</sub>{3,5,3}
|(2)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|&nbsp;
|&nbsp;
|(2)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|[[Image:Bitruncated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-0110 center ultrawide.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!7
|cantitruncated icosahedral (grih)<br>{{CDD|node_1|3|node_1|5|node_1|3|node}}<br>t<sub>0,1,2</sub>{3,5,3}
|(1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|(1)<br>[[Image:triangular prism.png|40px]]<br>[[Triangular prism|(4.4.3)]]
|&nbsp;
|(2)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[Image:Cantitruncated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-1110 center ultrawide.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!8
|runcitruncated icosahedral (prih)<br>{{CDD|node_1|3|node_1|5|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,5,3}
|(1)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.5.4.5)]]
|(1)<br>[[Image:triangular prism.png|40px]]<br>[[Triangular prism|(4.4.3)]]
|(2)<br>[[Image:hexagonal prism.png|40px]]<br>[[Hexagonal prism|(4.4.6)]]
|(1)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[Image:Runcitruncated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-1101 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0e0"
!9
|omnitruncated icosahedral (gipiddih)<br>{{CDD|node_1|3|node_1|5|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,5,3}
|(1)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|(1)<br>[[Image:hexagonal prism.png|40px]]<br>[[Hexagonal prism|(4.4.6)]]
|(1)<br>[[Image:hexagonal prism.png|40px]]<br>[[Hexagonal prism|(4.4.6)]]
|(1)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[Image:Omnitruncated icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-1111 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<br>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<br>symbols
!colspan=5|Cell counts/vertex<br>and positions in honeycomb
!rowspan=2|[[Vertex figure]]
!rowspan=2|Picture
|- align=center
!0<br>{{CDD|node_n2|5|node_n3|3|node_n4}}
!1<br>{{CDD|node_n1|2|2|node_n3|3|node_n4}}
!2<br>{{CDD|node_n1|3|node_n2|2|node_n4}}
!3<br>{{CDD|node_n1|3|node_n2|5|node_n3}}
!Alt
|- align=center BGCOLOR="#e0f0f0"
|[77]
|partially diminished icosahedral<br>pd{3,5,3}<ref>{{Cite web | url=http://www.bendwavy.org/klitzing/incmats/pt353.htm | title=Pd{3,5,3}}}</ref>
|
|
|
|(12)<BR>[[Image:Pentagonal antiprism.png|40px]]<br>[[Pentagonal antiprism|(3.3.3.5)]]
|(4)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|[[Image:Partial truncation order-3 icosahedral honeycomb verf.png|100px]]
|[[File:H3 353-pd center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|[78]
|semi-partially diminished icosahedral<br>spd{3,5,3}<ref name=klitzing>{{Cite web | url=http://www.bendwavy.org/klitzing/incmats/spt353.htm | title=Spd{3,5,3}}}</ref>
|
|
|
|(6)<BR>[[Image:Pentagonal antiprism.png|40px]]<br>[[Pentagonal antiprism|(3.3.3.5)]]<br>(6)<BR>40px<br>[[Icosahedron|(3.3.3.3.3)]]
|(2)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|
|
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|omnisnub icosahedral (snih)<br>{{CDD|node_h|3|node_h|5|node_h|3|node_h}}<br>ht<sub>0,1,2,3</sub>{3,5,3}
|(1)<BR>[[Image:Snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(1)<BR>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3]]
|(1)<BR>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
|(1)<BR>[[Image:Snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(4)<BR>[[Image:tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:snub icosahedral honeycomb verf.png|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 [5,3<sup>1,1</sup>] by a half symmetry [5,3,4,1<sup>+</sup>], 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}}.

{| class=wikitable
!rowspan=2|#
!rowspan=2|Name of honeycomb<br>Coxeter diagram
!colspan=4|Cells by location and count per vertex
!rowspan=2|[[Vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|node_n2|3|node_n3|4|node_n4}}
!1<br>{{CDD|node_n1|2|node_n3|4|node_n4}}
!2<br>{{CDD|node_n1|5|node_n2|2|node_n4}}
!3<br>{{CDD|node_n1|5|node_n2|3|node_n3}}
|- BGCOLOR="#f0e0e0" align=center
!10
|[[Order-4 dodecahedral honeycomb|order-4 dodecahedral]] (doehon)<br>{{CDD|node_1|5|node|3|node|4|node}} ↔ {{CDD|node_1|5|node|split1|nodes}}
| -
| -
| -
|(8)<br>{{CDD|node_1|5|node|3|node}}<br>[[File:Dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|[[File:Order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534 CC center.png|100px]]
|- BGCOLOR="#f0e0e0" align=center
!11
|rectified order-4 dodecahedral (riddoh)<br>{{CDD|node|5|node_1|3|node|4|node}} ↔ {{CDD|node|5|node_1|split1|nodes}}
|(2)<br>{{CDD|node_1|3|node|4|node}}<br>[[File:Octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| -
| -
|(4)<br>{{CDD|node|5|node_1|3|node}}<br>[[File:Icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[File:Rectified order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534 CC center 0100.png|120px]]
|- BGCOLOR="#e0e0f0" align=center
!12
|rectified order-5 cubic (ripech)<br>{{CDD|node|5|node|3|node_1|4|node}} ↔ {{CDD|node|5|node|split1|nodes_11}}
|(5)<br>{{CDD|node|3|node_1|4|node}}<br>[[File:Cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| -
| -
|(2)<br>{{CDD|node|5|node|3|node_1}}<br>[[File:Icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|[[File:Rectified order-5 cubic honeycomb verf.png|100px]]
| [[File:H3 435 CC center 0100.png|120px]]
|- BGCOLOR="#e0e0f0" align=center
!13
|[[Order-5 cubic honeycomb|order-5 cubic]] (pechon)<br>{{CDD|node|5|node|3|node|4|node_1}}
|(20)<br>{{CDD|node|3|node|4|node_1}}<br>[[File:Hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
| -
| -
| -
|[[File:Order-5 cubic honeycomb verf.svg|100px]]
|[[File:H3 435 CC center.png|120px]]
|- BGCOLOR="#f0e0e0" align=center
!14
|[[Truncated order-4 dodecahedral honeycomb|truncated order-4 dodecahedral]] (tiddoh)<br>{{CDD|node_1|5|node_1|3|node|4|node}} ↔ {{CDD|node_1|5|node_1|split1|nodes}}
|(1)<br>{{CDD|node_1|3|node|4|node}}<br>[[File:Octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| -
| -
|(4)<br>{{CDD|node_1|5|node_1|3|node}}<br>[[File:Truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|[[File:Truncated order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 435-0011 center ultrawide.png|120px]]
|- BGCOLOR="#e0f0e0" align=center
!15
|bitruncated order-5 cubic (ciddoh)<br>{{CDD|node|5|node_1|3|node_1|4|node}} ↔ {{CDD|node|5|node_1|split1|nodes_11}}
|(2)<br>{{CDD|node_1|3|node_1|4|node}}<br>[[File:Truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| -
| -
|(2)<br>{{CDD|node|5|node_1|3|node_1}}<br>[[File:Truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Bitruncated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-0110 center ultrawide.png|120px]]
|- BGCOLOR="#e0e0f0" align=center
!16
|[[Truncated order-5 cubic honeycomb|truncated order-5 cubic]] (tipech)<br>{{CDD|node|5|node|3|node_1|4|node_1}}
|(5)<br>{{CDD|node|3|node_1|4|node_1}}<br>[[File:Truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
| -
| -
|(1)<br>{{CDD|node|5|node|3|node_1}}<br>[[File:Icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|[[File:Truncated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-0011 center ultrawide.png|120px]]
|- BGCOLOR="#f0e0e0" align=center
!17
|cantellated order-4 dodecahedral (sriddoh)<br>{{CDD|node_1|5|node|3|node_1|4|node}} ↔ {{CDD|node_1|5|node|split1|nodes_11}}
|(1)<br>{{CDD|node|3|node_1|4|node}}<br>[[File:Cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|(2)<br>{{CDD|node_1|2|node_1|4|node}}<br>[[File:Tetragonal prism.png|40px]]<br>[[Cube|(4.4.4)]]
| -
|(2)<br>{{CDD|node_1|5|node|3|node_1}}<br>[[File:Small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[File:Cantellated order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534-1010 center ultrawide.png|120px]]
|- BGCOLOR="#e0e0f0" align=center
!18
|cantellated order-5 cubic (sripech)<br>{{CDD|node|5|node_1|3|node|4|node_1}}
|(2)<br>{{CDD|node_1|3|node|4|node_1}}<br>[[File:Small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
| -
|(2)<br>{{CDD|node|5|node_1|2|node_1}}<br>[[File:Pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(1)<br>{{CDD|node|5|node_1|3|node}}<br>[[File:Icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[File:Cantellated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-0101 center ultrawide.png|120px]]
|- BGCOLOR="#e0f0e0" align=center
!19
|runcinated order-5 cubic (sidpicdoh)<br>{{CDD|node_1|5|node|3|node|4|node_1}}
|(1)<br>{{CDD|node|3|node|4|node_1}}<br>[[File:Hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
|(3)<br>{{CDD|node_1|2|node|4|node_1}}<br>[[File:Tetragonal prism.png|40px]]<br>[[Cube|(4.4.4)]]
|(3)<br>{{CDD|node_1|5|node|2|node_1}}<br>[[File:Pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(1)<br>{{CDD|node_1|5|node|3|node}}<br>[[File:Dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|[[File:Runcinated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-1001 center ultrawide.png|120px]]
|- BGCOLOR="#f0e0e0" align=center
!20
|cantitruncated order-4 dodecahedral (griddoh)<br>{{CDD|node_1|5|node_1|3|node_1|4|node}} ↔ {{CDD|node_1|5|node_1|split1|nodes_11}}
|(1)<br>{{CDD|node_1|3|node_1|4|node}}<br>[[File:Truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
|(1)<br>{{CDD|node_1|2|node_1|4|node}}<br>[[File:Tetragonal prism.png|40px]]<br>[[Cube|(4.4.4)]]
| -
|(2)<br>{{CDD|node_1|5|node_1|3|node_1}}<br>[[File:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Cantitruncated order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534-1110 center ultrawide.png|120px]]
|- BGCOLOR="#e0e0f0" align=center
!21
|cantitruncated order-5 cubic (gripech)<br>{{CDD|node|5|node_1|3|node_1|4|node_1}}
|(2)<br>{{CDD|node_1|3|node_1|4|node_1}}<br>[[File:Great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
| -
|(1)<br>{{CDD|node|5|node_1|2|node_1}}<br>[[File:Pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(1)<br>{{CDD|node|5|node_1|3|node_1}}<br>[[File:Truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Cantitruncated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-0111 center ultrawide.png|120px]]
|- BGCOLOR="#f0e0e0" align=center
!22
|runcitruncated order-4 dodecahedral (pripech)<br>{{CDD|node_1|5|node_1|3|node|4|node_1}}
|(1)<br>{{CDD|node_1|3|node|4|node_1}}<br>[[File:Small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
|(1)<br>{{CDD|node_1|2|node|4|node_1}}<br>[[File:Tetragonal prism.png|40px]]<br>[[Cube|(4.4.4)]]
|(2)<br>{{CDD|node_1|5|node_1|2|node_1}}<br>[[File:Decagonal prism.png|40px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>{{CDD|node_1|5|node_1|3|node}}<br>[[File:Truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|[[File:Runcitruncated order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534-1101 center ultrawide.png|120px]]
|- BGCOLOR="#e0e0f0" align=center
!23
|runcitruncated order-5 cubic (priddoh)<br>{{CDD|node_1|5|node|3|node_1|4|node_1}}
|(1)<br>{{CDD|node|3|node_1|4|node_1}}<br>[[File:Truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
|(2)<br>{{CDD|node_1|2|node_1|4|node_1}}<br>[[File:Octagonal prism.png|40px]]<br>[[Octagonal prism|(4.4.8)]]
|(1)<br>{{CDD|node_1|5|node|2|node_1}}<br>[[File:Pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(1)<br>{{CDD|node_1|5|node|3|node_1}}<br>[[File:Small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[File:Runcitruncated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-1011 center ultrawide.png|120px]]
|- BGCOLOR="#e0f0e0" align=center
!24
|omnitruncated order-5 cubic (gidpicdoh)<br>{{CDD|node_1|5|node_1|3|node_1|4|node_1}}
|(1)<br>{{CDD|node_1|3|node_1|4|node_1}}<br>[[File:Great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
|(1)<br>{{CDD|node_1|2|node_1|4|node_1}}<br>[[File:Octagonal prism.png|40px]]<br>[[Octagonal prism|(4.4.8)]]
|(1)<br>{{CDD|node_1|5|node_1|2|node_1}}<br>[[File:Decagonal prism.png|40px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>{{CDD|node_1|5|node_1|3|node_1}}<br>[[File:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Omnitruncated order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534-1111 center ultrawide.png|120px]]
|}

{| class=wikitable
!rowspan=2|#
!rowspan=2|Name of honeycomb<br>Coxeter diagram
!colspan=5|Cells by location and count per vertex
!rowspan=2|[[Vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|node_n2|3|node_n3|4|node_n4}}
!1<br>{{CDD|node_n1|2|node_n3|4|node_n4}}
!2<br>{{CDD|node_n1|5|node_n2|2|node_n4}}
!3<br>{{CDD|node_n1|5|node_n2|3|node_n3}}
!Alt
|- BGCOLOR="#d0f0f0" align=center
|[34]
|alternated order-5 cubic (apech)<br>{{CDD|node|5|node|3|node|4|node_h1}} ↔ {{CDD|node|5|node|split1|nodes_10lu}}
|(20)<br>{{CDD|node|3|node|4|node_h1}}<br>[[File:Tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
|&nbsp;
|&nbsp;
|(12)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|
|[[Image:alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:Alternated order 5 cubic honeycomb.png|100px]]

|- BGCOLOR="#d0f0f0" align=center
|[35]
|cantic order-5 cubic (tapech)<br>{{CDD|node|5|node_1|3|node|4|node_h1}} ↔ {{CDD|node|5|node_1|split1|nodes_10lu}}
|(1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| -
|(2)<br>[[Image:Truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|(2)<br>[[Image:Truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|
|[[Image:Truncated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 5311-0110 center ultrawide.png|120px]]
|
|- BGCOLOR="#d0f0f0" align=center
|[36]
|runcic order-5 cubic (birapech)<br>{{CDD|node_1|5|node|3|node|4|node_h1}} ↔ {{CDD|node_1|5|node|split1|nodes_10lu}}
| (1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| -
| (3)<br>[[Image:Small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (1)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
|
|[[Image:Runcinated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 5311-1010 center ultrawide.png|120px]]
|
|- BGCOLOR="#d0f0f0" align=center
|[37]
|runcicantic order-5 cubic (bitapech)<br>{{CDD|node_1|5|node_1|3|node|4|node_h1}} ↔ {{CDD|node_1|5|node_1|split1|nodes_10lu}}
|(1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| -
|(2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|(1)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|
|[[Image:Runcitruncated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 5311-1110 center ultrawide.png|120px]]
|
|- BGCOLOR="#d0f0f0" align=center
|Nonuniform
|[[Snub rectified order-4 dodecahedral honeycomb|snub rectified order-4 dodecahedral]]<br>{{CDD|node_h|5|node_h|3|node_h|4|node}}
|(1)<BR>{{CDD|node_h|3|node_h|4|node}}<br>[[File:Uniform polyhedron-43-h01.svg|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(1)<BR>{{CDD|node_h|2x|node_h|4|node}}<br>[[File:Tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| -
|(2)<BR>{{CDD|node_h|5|node_h|3|node_h}}<br>[[File:Snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(4)<BR>[[File:Tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:Alternated cantitruncated order-4 dodecahedral honeycomb verf.png|100px]]<br>Irr. [[Tridiminished icosahedron|tridiminished icosahedron]]
|
|- BGCOLOR="#d0f0f0" align=center
|Nonuniform
|runcic snub rectified order-4 dodecahedral<br>{{CDD|node_h|5|node_h|3|node_h|4|node_1}}
|{{CDD|node_h|3|node_h|4|node_1}}<br>[[File:Rhombicuboctahedron uniform edge coloring.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
|{{CDD|node_h|2x|node_h|4|node_1}}<br>[[File:Cube rotorotational symmetry.png|40px]]<br>[[Cube|(4.4.4.4)]]
| -
|{{CDD|node_h|5|node_h|3|node_h}}<br>[[File:Snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|[[File:Tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|
|
|- BGCOLOR="#d0f0f0" align=center
|Nonuniform
|omnisnub order-5 cubic<br>{{CDD|node_h|5|node_h|3|node_h|4|node_h}}
|(1)<BR>{{CDD|node_h|3|node_h|4|node_h}}<br>[[File:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
|(1)<BR>{{CDD|node_h|2x|node_h|4|node_h}}<br>[[File:square antiprism.png|40px]]<br>[[Square antiprism|(3.3.3.4)]]
|(1)<BR>{{CDD|node_h|5|node_h|2x|node_h}}<br>[[File:pentagonal antiprism.png|40px]]<br>[[Pentagonal antiprism|(3.3.3.5)]]
|(1)<BR>{{CDD|node_h|5|node_h|3|node_h}}<br>[[File:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(4)<BR>[[File:Tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:Snub order-4 dodecahedral honeycomb verf.png|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 polyhedron|regular skew polyhedron]]s: {4,6|5} and {6,4|5}.

{| class=wikitable
!rowspan=2|#
!rowspan=2|Name of honeycomb<br>Coxeter diagram
!colspan=4|Cells by location and count per vertex
!rowspan=2|[[Vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|node|3|node|5|node}}
!1<br>{{CDD|node|2|node|5|node}}
!2<br>{{CDD|node|5|node|2|node}}
!3<br>{{CDD|node|5|node|3|node}}
|- align=center BGCOLOR="#f0e0e0"
!25
|(Regular) [[Order-5 dodecahedral honeycomb|Order-5 dodecahedral]] (pedhon)<br>{{CDD|node_1|5|node|3|node|5|node}}<br>t<sub>0</sub>{5,3,5}
|&nbsp;
|&nbsp;
|&nbsp;
|(20)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|[[Image:Order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535 CC center.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!26
|rectified order-5 dodecahedral (ripped)<br>{{CDD|node|5|node_1|3|node|5|node}}<br>t<sub>1</sub>{5,3,5}
|(2)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|&nbsp;
|&nbsp;
|(5)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[Image:Rectified order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535 CC center 0100.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!27
|[[Truncated order-5 dodecahedral honeycomb|truncated order-5 dodecahedral]] (tipped)<br>{{CDD|node_1|5|node_1|3|node|5|node}}<br>t<sub>0,1</sub>{5,3,5}
|(1)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|&nbsp;
|&nbsp;
|(5)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|[[Image:Truncated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-0011 center ultrawide.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!28
|cantellated order-5 dodecahedral (sripped)<br>{{CDD|node_1|5|node|3|node_1|5|node}}<br>t<sub>0,2</sub>{5,3,5}
|(1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|(2)<br>[[Image:pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|&nbsp;
|(2)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.5.4.5)]]
|[[Image:Cantellated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-1010 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0e0"
!29
|Runcinated order-5 dodecahedral (spidded)<br>{{CDD|node_1|5|node|3|node|5|node_1}}<br>t<sub>0,3</sub>{5,3,5}
|(1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|(3)<br>[[Image:pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(3)<br>[[Image:pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
|[[Image:Runcinated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-1001 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0e0"
!30
|bitruncated order-5 dodecahedral (diddoh)<br>{{CDD|node|5|node_1|3|node_1|5|node}}<br>t<sub>1,2</sub>{5,3,5}
|(2)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|&nbsp;
|&nbsp;
|(2)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[Image:Bitruncated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-0110 center ultrawide.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!31
|cantitruncated order-5 dodecahedral (gripped)<br>{{CDD|node_1|5|node_1|3|node_1|5|node}}<br>t<sub>0,1,2</sub>{5,3,5}
|(1)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|(1)<br>[[Image:pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|&nbsp;
|(2)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[Image:Cantitruncated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-1110 center ultrawide.png|120px]]
|- align=center BGCOLOR="#f0e0e0"
!32
|runcitruncated order-5 dodecahedral (pripped)<br>{{CDD|node_1|5|node_1|3|node|5|node_1}}<br>t<sub>0,1,3</sub>{5,3,5}
|(1)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.5.4.5)]]
|(1)<br>[[Image:pentagonal prism.png|40px]]<br>[[Pentagonal prism|(4.4.5)]]
|(2)<br>[[Image:decagonal prism.png|40px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|[[Image:Runcitruncated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-1101 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0e0"
!33
|omnitruncated order-5 dodecahedral (gipidded)<br>{{CDD|node_1|5|node_1|3|node_1|5|node_1}}<br>t<sub>0,1,2,3</sub>{5,3,5}
|(1)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|(1)<br>[[Image:decagonal prism.png|40px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>[[Image:decagonal prism.png|40px]]<br>[[Decagonal prism|(4.4.10)]]
|(1)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[Image:Omnitruncated order-5 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 535-1111 center ultrawide.png|120px]]
|}

{| class=wikitable
!rowspan=2|#
!rowspan=2|Name of honeycomb<br>Coxeter diagram
!colspan=5|Cells by location and count per vertex
!rowspan=2|[[Vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|node|3|node|5|node}}
!1<br>{{CDD|node|2|node|5|node}}
!2<br>{{CDD|node|5|node|2|node}}
!3<br>{{CDD|node|5|node|3|node}}
!Alt
|- BGCOLOR="#d0f0f0" align=center
|Nonuniform
|omnisnub order-5 dodecahedral<br>{{CDD|node_h|5|node_h|3|node_h|5|node_h}}<br>ht<sub>0,1,2,3</sub>{5,3,5}
|(1)<BR>{{CDD|node_h|3|node_h|5|node_h}}<br>[[File:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(1)<BR>{{CDD|node_h|2x|node_h|5|node_h}}<br>[[File:pentagonal antiprism.png|40px]]<br>[[Pentagonal antiprism|(3.3.3.5)]]
|(1)<BR>{{CDD|node_h|5|node_h|2x|node_h}}<br>[[File:pentagonal antiprism.png|40px]]<br>[[Pentagonal antiprism|(3.3.3.5)]]
|(1)<BR>{{CDD|node_h|5|node_h|3|node_h}}<br>[[File:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(4)<BR>[[File:Tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:Snub order-5 dodecahedral honeycomb verf.png|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]]: [5,3<sup>1,1</sup>] 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<br>Coxeter diagram
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|nodea|3a|nodea|5a|nodea}}
!1<br>{{CDD|nodes|2|node}}
!0'<br>{{CDD|nodea|3a|nodea|5a|nodea}}
!3<br>{{CDD|nodes|split2|node}}
|- align=center
!34
|alternated order-5 cubic (apech)<br>{{CDD|nodes_10ru|split2|node|5|node}} ↔ {{CDD|node_h1|4|node|3|node|5|node}}
| -
| -
|(12)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(20)<br>[[Image:Tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
|[[Image:alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:Alternated order 5 cubic honeycomb.png|100px]]
|- align=center
!35
|cantic order-5 cubic (tapech)<br>{{CDD|nodes_10ru|split2|node_1|5|node}} ↔ {{CDD|node_h1|4|node|3|node_1|5|node}}
|(1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| -
|(2)<br>[[Image:Truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|(2)<br>[[Image:Truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|[[Image:Truncated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 5311-0110 center ultrawide.png|120px]]
|- align=center
!36
|runcic order-5 cubic (birapech)<br>{{CDD|nodes_10ru|split2|node|5|node_1}} ↔ {{CDD|node_h1|4|node|3|node|5|node_1}}
| (1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| -
| (3)<br>[[Image:Small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (1)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
|[[Image:Runcinated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 5311-1010 center ultrawide.png|120px]]
|- align=center
!37
|runcicantic order-5 cubic (bitapech)<br>{{CDD|nodes_10ru|split2|node_1|5|node_1}} ↔ {{CDD|node_h1|4|node|3|node_1|5|node_1}}
|(1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| -
|(2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|(1)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|[[Image:Runcitruncated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 5311-1110 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<BR>{{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<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|nodea|3a|nodea|5a|nodea}}
!1<br>{{CDD|nodes|2|node}}
!3<br>{{CDD|nodes|split2|node}}
!Alt
|- align=center
| [10]
|[[Order-4 dodecahedral honeycomb|Order-4 dodecahedral]] (doehon)<br>{{CDD|nodes|split2|node|5|node_1}} ↔ {{CDD|node_h0|4|node|3|node|5|node_1}}
| (4)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| -
| -
|
|[[File:Order-4 dodecahedral honeycomb verf.png|100px]]
|[[File:H3 534 CC center.png|120px]]
|- align=center
| [11]
|rectified order-4 dodecahedral (riddoh)<br>{{CDD|nodes|split2|node_1|5|node}} ↔ {{CDD|node_h0|4|node|3|node_1|5|node}}
| (2)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| -
| (2)<br>[[Image:Uniform polyhedron-33-t1.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
|
|[[Image:Rectified alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534 CC center 0100.png|120px]]
|- align=center
| [12]
|rectified order-5 cubic (ripech)<br>{{CDD|nodes_11|split2|node|5|node}} ↔ {{CDD|node_h0|4|node_1|3|node|5|node}}
| (1)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| -
| (5)<br>[[Image:Uniform polyhedron-33-t02.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|
|[[Image:Cantellated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 435 CC center 0100.png|120px]]
|- align=center
| [15]
|bitruncated order-5 cubic (ciddoh)<br>{{CDD|nodes_11|split2|node_1|5|node}} ↔ {{CDD|node_h0|4|node_1|3|node_1|5|node}}
| (1)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
| -
| (2)<br>[[Image:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
|
|[[Image:Cantitruncated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-0110 center ultrawide.png|120px]]
|- align=center
| [14]
|[[Truncated order-4 dodecahedral honeycomb|truncated order-4 dodecahedral]] (tiddoh)<br>{{CDD|nodes|split2|node_1|5|node_1}} ↔ {{CDD|node_h0|4|node|3|node_1|5|node_1}}
| (2)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| -
| (1)<br>[[Image:Uniform polyhedron-33-t1.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
|
|[[Image:Bicantellated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 435-0011 center ultrawide.png|120px]]
|- align=center
| [17]
|cantellated order-4 dodecahedral (sriddoh)<br>{{CDD|nodes_11|split2|node|5|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node|5|node_1}}
| (1)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (2)<br>[[Image:Uniform polyhedron 222-t012.png|40px]]<br>[[Cube|(4.4.4)]]
| (1)<br>[[Image:Uniform polyhedron-33-t02.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|
|[[Image:Runcicantellated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-1010 center ultrawide.png|120px]]
|- align=center
| [20]
|cantitruncated order-4 dodecahedral (griddoh)<br>{{CDD|nodes_11|split2|node_1|5|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node_1|5|node_1}}
| (1)<br>[[Image:great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
| (1)<br>[[Image:Uniform polyhedron 222-t012.png|40px]]<br>[[Cube|(4.4.4)]]
| (1)<br>[[Image:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
|
|[[Image:Omnitruncated alternated order-5 cubic honeycomb verf.png|100px]]
|[[File:H3 534-1110 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|[[Snub rectified order-4 dodecahedral honeycomb|snub rectified order-4 dodecahedral]]<br>{{CDD|nodes_hh|split2|node_h|5|node_h}} ↔ {{CDD|node_h0|4|node_h|3|node_h|5|node_h}}
|(2)<BR>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(1)<BR>[[File:Uniform polyhedron-33-t0.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
|(2)<BR>[[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(4)<BR>[[File:Uniform polyhedron-33-t2.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
||[[File:Alternated cantitruncated order-4 dodecahedral honeycomb verf.png|100px]]<br>Irr. [[Tridiminished icosahedron|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 polyhedron|regular skew polyhedron]]s: {8,6|3} and {6,8|3}.

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram
!colspan=5|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|nodea|3a|branch}}
!1<br>{{CDD|nodeb|3b|branch}}
!2<br>{{CDD|label4|branch|3b|nodeb}}
!3<br>{{CDD|label4|branch|3a|nodea}}
!Alt
|- align=center
!38
|[[Tetrahedral-cubic honeycomb|tetrahedral-cubic]] (gadtatdic)<BR>{{CDD|label4|branch_10r|3ab|branch}}<BR>{(3,3,3,4)}
| (4)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| -
| (4)<br>[[Image:hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
| (6)<br>[[Image:cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|
|[[File:Uniform t0 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-1000 center ultrawide.png|125px]]
|- align=center
!39
|[[Hyperbolic tetrahedral-octahedral honeycomb|tetrahedral-octahedral]] (gacocaddit)<BR>{{CDD|label4|branch|3ab|branch_10l}}<BR>{(3,3,4,3)}
| (12)<br>[[File:Uniform polyhedron-33-t1.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (8)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| -
| (8)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
|
|[[File:Uniform t2 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-0100 center ultrawide.png|125px]]
|- align=center
!40
|cyclotruncated tetrahedral-cubic (cytitch)<BR>{{CDD|label4|branch_10r|3ab|branch_10l}}<BR>ct{(3,3,3,4)}
| (3)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]|| (1)<br>40px<br>[[Tetrahedron|(3.3.3)]]
| (1)<br>[[Image:hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
| (3)<br>[[Image:truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
|
|[[File:Uniform t12 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-0110 center ultrawide.png|125px]]
|- align=center
!41
|cyclotruncated cube-tetrahedron (cyticth)<BR>{{CDD|label4|branch_11|3ab|branch}}<BR>ct{(4,3,3,3)}
| (1)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| (1)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| (3)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
| (3)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
|
|[[File:Uniform t01 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-1100 center ultrawide.png|125px]]
|- align=center
!42
|cyclotruncated octahedral-tetrahedral (cytoth)<BR>{{CDD|label4|branch|3ab|branch_11}}<BR>ct{(3,3,4,3)}
| (4)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (4)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (1)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (1)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
|
|[[File:Uniform t23 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-0011 center ultrawide.png|125px]]
|- align=center
!43
|rectified tetrahedral-cubic (ritch)<BR>{{CDD|label4|branch_01r|3ab|branch_10l}}<BR>r{(3,3,3,4)}
| (1)<br>[[File:Uniform polyhedron-33-t1.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (2)<br>[[File:Uniform polyhedron-33-t02.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (1)<br>[[Image:cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (2)<br>[[Image:small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
|
|[[File:Uniform t02 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-1010 center ultrawide.png|125px]]
|- align=center
!44
|truncated tetrahedral-cubic (titch)<BR>{{CDD|label4|branch_11|3ab|branch_10l}}<BR>t{(3,3,3,4)}
| (1)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (1)<br>[[Image:Uniform polyhedron-33-t02.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (1)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
| (2)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
|
|[[File:Uniform t012 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-1110 center ultrawide.png|125px]]
|- align=center
!45
|truncated tetrahedral-octahedral (titdoh)<BR>{{CDD|label4|branch_10r|3ab|branch_11}}<BR>t{(3,3,4,3)}
| (2)<br>[[File:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[Image:truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (1)<br>[[Image:small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
| (1)<br>[[Image:truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
|
|[[File:Uniform t123 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-0111 center ultrawide.png|125px]]
|- align=center
!46
|omnitruncated tetrahedral-cubic (otitch)<BR>{{CDD|label4|branch_11|3ab|branch_11}}<BR>tr{(3,3,3,4)}
| (1)<br>[[File:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[File:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
| (1)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
|
|[[File:Uniform t0123 4333 honeycomb verf.png|100px]]
|[[File:H3 4333-1111 center ultrawide.png|125px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|omnisnub tetrahedral-cubic<BR>{{CDD|label4|branch_hh|3ab|branch_hh}}<BR>sr{(3,3,3,4)}
|(1)<BR>[[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(1)<BR>[[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(1)<BR>[[Image:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
|(1)<BR>[[Image:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
|(4)<BR>[[Image:tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:Snub 4333 honeycomb verf.png|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 polyhedron|regular skew polyhedron]]s: {10,6|3} and {6,10|3}.

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|nodea|3a|branch}}
!1<br>{{CDD|nodeb|3b|branch}}
!2<br>{{CDD|label5|branch|3b|nodeb}}
!3<br>{{CDD|label5|branch|3a|nodea}}
|- align=center
!47
|[[Tetrahedral-dodecahedral honeycomb|tetrahedral-dodecahedral]]<BR>{{CDD|label5|branch_10r|3ab|branch}}
| (4)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| -
| (4)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (6)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[File:Uniform t0 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-1000 center ultrawide.png|120px]]
|- align=center
!48
|[[Tetrahedral-icosahedral honeycomb|tetrahedral-icosahedral]]<BR>{{CDD|label5|branch|3ab|branch_10l}}
| (30)<br>[[Image:Uniform polyhedron-33-t1.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (20)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| -
| (12)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|[[File:Uniform t2 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-0010 center ultrawide.png|120px]]
|- align=center
!49
|cyclotruncated tetrahedral-dodecahedral<BR>{{CDD|label5|branch_10r|3ab|branch_10l}}
| (3)<br>[[Image:Truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (1)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| (1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (3)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Uniform t12 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-0110 center ultrawide.png|120px]]
|- align=center
!52
|rectified tetrahedral-dodecahedral<BR>{{CDD|label5|branch_01r|3ab|branch_10l}}
| (1)<br>[[Image:Uniform polyhedron-33-t1.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (2)<br>[[Image:Uniform polyhedron-33-t02.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| (2)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[File:Uniform t02 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-1010 center ultrawide.png|120px]]
|- align=center
!53
|truncated tetrahedral-dodecahedral<BR>{{CDD|label5|branch_11|3ab|branch_10l}}
| (1)<br>[[Image:Truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (1)<br>[[Image:Uniform polyhedron-33-t02.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| (2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Uniform t012 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-1110 center ultrawide.png|120px]]
|- align=center
!54
|truncated tetrahedral-icosahedral<BR>{{CDD|label5|branch_10r|3ab|branch_11}}
| (2)<br>[[Image:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[Image:Truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (1)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (1)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Uniform t123 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-0111 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<BR>{{CDD|label5|branch_c1|3ab|branch_c2}}
!colspan=3|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0,1<br>{{CDD|nodea|3a|branch}}
!2,3<br>{{CDD|label5|branch|3b|nodeb}}
!Alt
|- align=center
!50
|cyclotruncated dodecahedral-tetrahedral<BR>{{CDD|label5|branch_11|3ab|branch}}
| (2)<br>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| (6)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|
|[[File:Uniform t01 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-1100 center ultrawide.png|120px]]
|- align=center
!51
|cyclotruncated tetrahedral-icosahedral<BR>{{CDD|label5|branch|3ab|branch_11}}
| (10)<br>[[Image:Truncated tetrahedron.png|40px]]<br>[[Truncated tetrahedron|(3.6.6)]]
| (2)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|
|[[File:Uniform t23 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-0011 center ultrawide.png|120px]]
|- align=center
!55
|omnitruncated tetrahedral-dodecahedral<BR>{{CDD|label5|branch_11|3ab|branch_11}}
| (2)<br>[[Image:Uniform polyhedron-33-t012.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|
|[[File:Uniform t0123 5333 honeycomb verf.png|100px]]
|[[File:H3 5333-1111 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|omnisnub tetrahedral-dodecahedral<BR>{{CDD|label5|branch_hh|3ab|branch_hh}}
|(2)<BR>[[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(2)<BR>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(4)<BR>[[Image:tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|| [[File:Snub 5333 honeycomb verf.png|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<sup>*</sup>)], and represents a [[Trigonal trapezohedron|trigonal trapezohedron]] [[Fundamental domain|fundamental domain]].

The truncated forms (57 and 58) contain the faces of two [[Regular skew polyhedron|regular skew polyhedron]]s: {6,6|4} and {8,8|3}.
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Pictures
|-
!0<br>{{CDD|nodea|3a|branch|label4}}
!1<br>{{CDD|nodeb|3b|branch|label4}}
!2<br>{{CDD|label4|branch|3b|nodeb}}
!3<br>{{CDD|label4|branch|3a|nodea}}
|- align=center
!56
|[[Cubic-octahedral honeycomb|cubic-octahedral]] (cohon)<BR>{{CDD|label4|branch_10r|3ab|branch|label4}}
| (6)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| -
| (8)<br>[[Image:hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
| (12)<br>[[Image:cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|[[File:Uniform t0 4343 honeycomb verf.png|100px]]
|[[File:H3 4343-1000 center ultrawide.png|120px]]
|- align=center
!60
|[[Truncated cubic-octahedral honeycomb|truncated cubic-octahedral]] (tucoh)<BR>{{CDD|label4|branch_11|3ab|branch_10l|label4}}
| (1)<br>[[Image:truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[Image:small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
| (1)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
| (2)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
|[[File:Uniform t012 4343 honeycomb verf.png|100px]]
|[[File:H3 4343-1110 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<BR>{{CDD|label4|branch_c1-2|3ab|branch_c1-2|label4}}
!colspan=3|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0,3<br>{{CDD|nodea|3a|branch|label4}}
!1,2<br>{{CDD|nodeb|3b|branch|label4}}
!Alt
|- align=center
!57
|cyclotruncated octahedral-cubic (cytoch)<BR>{{CDD|label4|branch_10r|3ab|branch_10l|label4}} 
| (6)<br>[[Image:truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (2)<br>[[Image:hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
|
|[[File:Uniform t12 4343 honeycomb verf.png|100px]]
|[[File:H3 4343-0110 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|cyclosnub octahedral-cubic<BR>{{CDD|label4|branch_h0r|3ab|branch_h0l|label4}}
|(4)<BR>[[File:Uniform polyhedron-43-h01.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|(2)<BR>[[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
|(4)<BR>[[Image:octahedron.png|40px]]<br>+[[Octahedron|(3.3.3.3)]]
|[[File:Cyclosnub cubic-octahedral honeycomb vertex figure.png|100px]]
|
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<BR>{{CDD|label4|branch_c1|3ab|branch_c2|label4}}
!colspan=2|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0,1<br>{{CDD|nodea|3a|branch|label4}}
!2,3<br>{{CDD|label4|branch|3b|nodeb}}
|- align=center
!58
|cyclotruncated cubic-octahedral (cytacoh)<BR>{{CDD|label4|branch_11|3ab|branch|label4}}
| (2)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (6)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
|[[File:Uniform t01 4343 honeycomb verf.png|100px]]
|[[File:H3 4343-0110 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<BR>{{CDD|label4|branch_c1-2|3ab|branch_c2-1|label4}}
!colspan=2|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0,2<br>{{CDD|nodea|3a|branch|label4}}
!1,3<br>{{CDD|nodeb|3b|branch|label4}}
|- align=center
!59
|rectified cubic-octahedral (racoh)<BR>{{CDD|label4|branch_01r|3ab|branch_10l|label4}}
| (2)<br>[[Image:cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (4)<br>[[Image:small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
|[[File:Uniform t02 4343 honeycomb verf.png|100px]]
|[[File:H3 4343-1010 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram<BR>{{CDD|label4|branch_c1|3ab|branch_c1|label4}}
!colspan=2|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0,1,2,3<br>{{CDD|nodea|3a|branch|label4}}
!Alt
|- align=center
!61
|omnitruncated cubic-octahedral (otacoh)<BR>{{CDD|label4|branch_11|3ab|branch_11|label4}}
| (4)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
|
|[[File:Uniform t0123 4343 honeycomb verf.png|100px]]
|[[File:H3 4343-1111 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|omnisnub cubic-octahedral<BR>{{CDD|label4|branch_hh|3ab|branch_hh|label4}}
|(4)<BR>[[Image:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
|(4)<BR>[[Image:tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:Snub 4343 honeycomb verf.png|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 polyhedron|regular skew polyhedron]]s: {10,6|3} and {6,10|3}.

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|nodea|3a|branch|label4}}
!1<br>{{CDD|nodeb|3b|branch|label4}}
!2<br>{{CDD|label5|branch|3b|nodeb}}
!3<br>{{CDD|label5|branch|3a|nodea}}
|- align=center
!62
|[[Octahedral-dodecahedral honeycomb|octahedral-dodecahedral]]<BR>{{CDD|label5|branch_10r|3ab|branch|label4}}
| (6)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| -
| (8)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[File:Uniform t0 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-0010 center ultrawide.png|120px]]
|- align=center
!63
|[[Cubic-icosahedral honeycomb|cubic-icosahedral]]<BR>{{CDD|label5|branch|3ab|branch_10l|label4}}
| (30)<br>[[Image:cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (20)<br>[[Image:hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
| -
| (12)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|[[File:Uniform t2 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-1000 center ultrawide.png|120px]]
|- align=center
!64
|cyclotruncated octahedral-dodecahedral<BR>{{CDD|label5|branch_10r|3ab|branch_10l|label4}}
| (3)<br>[[Image:truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[Image:hexahedron.png|40px]]<br>[[Cube|(4.4.4)]]
| (1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (3)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Uniform t12 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-0110 center ultrawide.png|120px]]
|- align=center
!67
|rectified octahedral-dodecahedral<BR>{{CDD|label5|branch_01r|3ab|branch_10l|label4}}
| (1)<br>[[Image:cuboctahedron.png|40px]]<br>[[Cuboctahedron|(3.4.3.4)]]
| (2)<br>[[Image:Small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
| (1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| (2)<br>[[Image:Small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[File:Uniform t02 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-0101 center ultrawide.png|120px]]
|- align=center
!68
|truncated octahedral-dodecahedral<BR>{{CDD|label5|branch_11|3ab|branch_10l|label4}}
| (1)<br>[[Image:truncated octahedron.png|40px]]<br>[[Truncated octahedron|(4.6.6)]]
| (1)<br>[[Image:Small rhombicuboctahedron.png|40px]]<br>[[Rhombicuboctahedron|(3.4.4.4)]]
| (1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| (2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Uniform t012 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-1110 center ultrawide.png|120px]]
|- align=center
!69
|truncated cubic-dodecahedral<BR>{{CDD|label5|branch_10r|3ab|branch_11|label4}}
| (2)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
| (1)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
| (1)<br>[[Image:Small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (1)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Uniform t123 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-0111 center ultrawide.png|120px]]
|}

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram
!colspan=3|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0,1<br>{{CDD|nodea|3a|branch|label4}}
!2,3<br>{{CDD|label5|branch|3b|nodeb}}
!Alt
|- align=center
!65
|cyclotruncated dodecahedral-octahedral<BR>{{CDD|label5|branch_11|3ab|branch|label4}}
| (2)<br>[[Image:octahedron.png|40px]]<br>[[Octahedron|(3.3.3.3)]]
| (8)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|
|[[File:Uniform t01 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-1100 center ultrawide.png|120px]]
|- align=center
!66
|cyclotruncated cubic-icosahedral<BR>{{CDD|label5|branch|3ab|branch_11|label4}}
| (10)<br>[[Image:truncated hexahedron.png|40px]]<br>[[Truncated cube|(3.8.8)]]
| (2)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|
|[[File:Uniform t23 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-0011 center ultrawide.png|120px]]
|- align=center
!70
|omnitruncated octahedral-dodecahedral<BR>{{CDD|label5|branch_11|3ab|branch_11|label4}}
| (2)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
| (2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|
|[[File:Uniform t0123 5343 honeycomb verf.png|100px]]
|[[File:H3 4353-1111 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|omnisnub octahedral-dodecahedral<BR>{{CDD|label5|branch_hh|3ab|branch_hh|label4}}
|(2)<BR>[[File:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
|(2)<BR>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|(4)<BR>[[Image:tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
|[[File:Snub 5343 honeycomb verf.png|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 polyhedron|regular skew polyhedron]]s: {6,6|5} and {10,10|3}.

{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>Coxeter diagram
!colspan=5|Cells by location<br>(and count around each vertex)
!rowspan=2|[[Vertex figure|vertex figure]]
!rowspan=2|Picture
|-
!0<br>{{CDD|nodea|3a|branch|label5}}
!1<br>{{CDD|nodeb|3b|branch|label5}}
!2<br>{{CDD|label5|branch|3b|nodeb}}
!3<br>{{CDD|label5|branch|3a|nodea}}
!Alt
|- align=center
!71
|[[Dodecahedral-icosahedral honeycomb|dodecahedral-icosahedral]]<BR>{{CDD|label5|branch_10r|3ab|branch|label5}}
| (12)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| -
| (20)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (30)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|
|[[File:Uniform t0 5353 honeycomb verf.png|100px]]
|[[File:H3 5353-1000 center ultrawide.png|120px]]
|- align=center
!72
|cyclotruncated icosahedral-dodecahedral<BR>{{CDD|label5|branch_10r|3ab|branch_10l|label5}}
| (3)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
| (1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (1)<br>[[Image:dodecahedron.png|40px]]<br>[[Dodecahedron|(5.5.5)]]
| (3)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
|
|[[File:Uniform t12 5353 honeycomb verf.png|100px]]
|[[File:H3 5353-0110 center ultrawide.png|120px]]
|- align=center
!73
|cyclotruncated dodecahedral-icosahedral<BR>{{CDD|label5|branch_11|3ab|branch|label5}}
| (1)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| (1)<br>[[Image:icosahedron.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| (3)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| (3)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|
|[[File:Uniform t01 5353 honeycomb verf.png|100px]]
|[[File:H3 5353-1100 center ultrawide.png|120px]]
|- align=center
!74
|rectified dodecahedral-icosahedral<BR>{{CDD|label5|branch_01r|3ab|branch_10l|label5}}
| (1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| (2)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (1)<br>[[Image:icosidodecahedron.png|40px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
| (2)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
|
|[[File:Uniform t02 5353 honeycomb verf.png|100px]]
|[[File:H3 5353-1010 center ultrawide.png|120px]]
|- align=center
!75
|[[Truncated dodecahedral-icosahedral honeycomb|truncated dodecahedral-icosahedral]]<BR>{{CDD|label5|branch_11|3ab|branch_10l|label5}}
| (1)<br>[[Image:truncated icosahedron.png|40px]]<br>[[Truncated icosahedron|(5.6.6)]]
| (1)<br>[[Image:small rhombicosidodecahedron.png|40px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
| (1)<br>[[Image:truncated dodecahedron.png|40px]]<br>[[Truncated dodecahedron|(3.10.10)]]
| (2)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|
|[[File:Uniform t012 5353 honeycomb verf.png|100px]]
|[[File:H3 5353-1101 center ultrawide.png|120px]]
|- align=center
!76
|omnitruncated dodecahedral-icosahedral<BR>{{CDD|label5|branch_11|3ab|branch_11|label5}}
| (1)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
| (1)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
| (1)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
| (1)<br>[[Image:Great rhombicosidodecahedron.png|40px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|
|[[File:Uniform t0123 5353 honeycomb verf.png|100px]]
|[[File:H3 5353-1111 center ultrawide.png|120px]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|omnisnub dodecahedral-icosahedral<BR>{{CDD|label5|branch_hh|3ab|branch_hh|label5}}
| (1)<br>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
| (1)<br>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
| (1)<br>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
| (1)<br>[[Image:snub dodecahedron cw.png|40px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
| (4)<br>[[Image:tetrahedron.png|40px]]<br>+[[Tetrahedron|(3.3.3)]]
| [[File:Snub 5353 honeycomb verf.png|100px]]
|}

=== Other non-Wythoffians ===
There are several other known non-Wythoffian uniform compact hyperbolic honeycombs, and it is not known how many are left to be discovered. Two have been listed above as diminishings of the icosahedral honeycomb {3,5,3}.<ref name=klitzing2>{{Cite web | url=https://www.bendwavy.org/klitzing/dimensions/hyperbolic.htm | title=Hyperbolic Tesselations}}</ref>

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.<ref name=klitzing2/>

Another two known ones are related to ''noncompact'' families. The tessellation {{CDD|node_1|4|node_1|3|node|8|node}} consists of [[Truncated cube|truncated cube]]s {{CDD|node_1|4|node_1|3|node}} and infinite [[Order-8 triangular tiling|order-8 triangular tiling]]s {{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.)<ref name=klitzing2/><ref name=klitzing3>{{Cite web | url=https://www.bendwavy.org/klitzing/incmats/x4x3o8o.htm | title=x4x3o8o}}</ref> 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 tiling|order-8 triangular tiling]]s {{CDD|node_1|3|node|8|node}}, and infinite [[Order-8 square tiling|order-8 square tiling]]s {{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 prism|triangular prism]]s, 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.<ref name=klitzing4>{{Cite web | url=https://www.bendwavy.org/klitzing/incmats/lt-o8o4xb3x.htm | title=lt-o8o4xb3x}}</ref>

Another non-Wythoffian was discovered in 2021. It has as vertex figure a snub cube with 8 vertices removed and contains two [[Octahedron|octahedra]] and eight [[Snub cube|snub cube]]s at each vertex.<ref name=klitzing2/> 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 dodecahedron|snub dodecahedral]] vertex figure respectively), paracompact for p=6 (with a [[Snub trihexagonal tiling|snub trihexagonal tiling]] as the vertex figure), and hypercompact for p>6. Again, the truncated and rectified versions of these honeycombs are still uniform.<ref name=klitzing2/>

=== Summary enumeration of compact uniform honeycombs ===

This is the complete enumeration of the 76 Wythoffian uniform honeycombs. The [[Alternation (geometry)|alternations]] are listed for completeness, but most are non-uniform.

{| class=wikitable
!Index
!Coxeter group
![[Goursat tetrahedron#Compact hyperbolic 3-space solutions|Extended<br>symmetry]]
!colspan=2|Honeycombs
!Chiral<br>extended<br>symmetry
!colspan=2|Alternation honeycombs

|- align=center
!rowspan=2|H<sub>1</sub>
|rowspan=2|<math>{\bar{BH}}_3</math><br>[4,3,5]<br>{{CDD|node|4|node|3|node|5|node}} || rowspan=2|[4,3,5]<br>{{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}}<br>{{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}}<br>{{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<sup>+</sup>,4,(3,5)<sup>+</sup>]|| (2)||{{CDD|node|5|node|3|node|4|node_h1}} (= {{CDD|node|5|node|split1|nodes_10lu}})<BR>{{CDD|node_h|5|node_h|3|node_h|4|node}} 
|- align=center
|[4,3,5]<sup>+</sup>|| (1)||{{CDD|node_h|5|node_h|3|node_h|4|node_h}}
|- align=center
!rowspan=2|H<sub>2</sub>
|rowspan=2|<math>{\bar{J}}_3</math><br>[3,5,3]<br>{{CDD|node|3|node|5|node|3|node}} ||[3,5,3]<br>{{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<sup>+</sup>[3,5,3]]<br>{{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<sup>+</sup>[3,5,3]]<sup>+</sup> ||(1)
| {{CDD|node_h|3|node_h|5|node_h|3|node_h}}

|- align=center
!rowspan=2|H<sub>3</sub>
|rowspan=2|<math>{\bar{DH}}_3</math><br>[5,3<sup>1,1</sup>]<br>{{CDD|node|5|node|split1|nodes}} ||[5,3<sup>1,1</sup>]<br>{{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[5,3<sup>1,1</sup>]]=[5,3,4]<br>{{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[5,3<sup>1,1</sup>]]<sup>+</sup><br>=[5,3,4]<sup>+</sup>||(1)
|{{CDD|node_h|5|node_h|split1|nodes_hh}}

|- align=center
!rowspan=2|H<sub>4</sub>
|rowspan=2|<math>{\widehat{AB}}_3</math><br>[(4,3,3,3)]<br>{{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<sup>+</sup>[(4,3,3,3)]]<br>{{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<sup>+</sup>[(4,3,3,3)]]<sup>+</sup> ||(1)
| {{CDD|label4|branch_hh|3ab|branch_hh}}

|- align=center
!rowspan=2|H<sub>5</sub>
|rowspan=2|<math>{\bar{K}}_3</math><br>[5,3,5]<br>{{CDD|node|5|node|3|node|5|node}} ||[5,3,5]<br>{{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<sup>+</sup>[5,3,5]]<br>{{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<sup>+</sup>[5,3,5]]<sup>+</sup> ||(1)
| {{CDD|node_h|5|node_h|3|node_h|5|node_h}}

|- align=center
!rowspan=2|H<sub>6</sub>
|rowspan=2|<math>{\widehat{AH}}_3</math><br>[(5,3,3,3)]<br>{{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<sup>+</sup>[(5,3,3,3)]]<br>{{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<sup>+</sup>[(5,3,3,3)]]<sup>+</sup> ||(1)
| {{CDD|label5|branch_hh|3ab|branch_hh}}

|- align=center
!rowspan=5|H<sub>7</sub>
|rowspan=5|<math>{\widehat{BB}}_3</math><br>[(3,4)<sup>[2]</sup>]<br>{{CDD|label4|branch|3ab|branch|label4}} || [(3,4)<sup>[2]</sup>]|| 2||{{CDD|label4|branch_10r|3ab|branch|label4}} | {{CDD|label4|branch_11|3ab|branch_10l|label4}}
|colspan=3|
|- align=center
||[2<sup>+</sup>[(3,4)<sup>[2]</sup>]]<br>{{CDD|label4|branch_c1-2|3ab|branch_c2-1|label4}}||1
| {{CDD|label4|branch_01r|3ab|branch_10l|label4}}
|colspan=3|
|- align=center
||[2<sup>+</sup>[(3,4)<sup>[2]</sup>]]<br>{{CDD|label4|branch_c1|3ab|branch_c2|label4}}||1
|{{CDD|label4|branch_11|3ab|branch|label4}}
|colspan=3|
|- align=center
||[2<sup>+</sup>[(3,4)<sup>[2]</sup>]]<br>{{CDD|label4|branch_c1-2|3ab|branch_c1-2|label4}}||1
|{{CDD|label4|branch_10r|3ab|branch_10l|label4}}
||[2<sup>+</sup>[(3<sup>+</sup>,4)<sup>[2]</sup>]]||(1)
|{{CDD|label4|branch_h0r|3ab|branch_h0l|label4}}
|- align=center
||[(2,2)<sup>+</sup>[(3,4)<sup>[2]</sup>]]<br>{{CDD|label4|branch_c1|3ab|branch_c1|label4}}||1
| {{CDD|label4|branch_11|3ab|branch_11|label4}}
||[(2,2)<sup>+</sup>[(3,4)<sup>[2]</sup>]]<sup>+</sup>||(1)
| {{CDD|label4|branch_hh|3ab|branch_hh|label4}}

|- align=center
!rowspan=2|H<sub>8</sub>
|rowspan=2|<math>{\widehat{BH}}_3</math><br>[(5,3,4,3)]<br>{{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<sup>+</sup>[(5,3,4,3)]]<br>{{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<sup>+</sup>[(5,3,4,3)]]<sup>+</sup> ||(1)
| {{CDD|label5|branch_hh|3ab|branch_hh|label4}}

|- align=center
!rowspan=5|H<sub>9</sub>
|rowspan=5|<math>{\widehat{HH}}_3</math><br>[(3,5)<sup>[2]</sup>]<br>{{CDD|label5|branch|3ab|branch|label5}} || [(3,5)<sup>[2]</sup>]|| 2
|{{CDD|label5|branch_10r|3ab|branch|label5}} | {{CDD|label5|branch_11|3ab|branch_10l|label5}}
|colspan=3|
|- align=center
||[2<sup>+</sup>[(3,5)<sup>[2]</sup>]]<br>{{CDD|label5|branch_c1-2|3ab|branch_c2-1|label5}}||1
|{{CDD|label5|branch_01r|3ab|branch_10l|label5}} 
|colspan=3|
|- align=center
||[2<sup>+</sup>[(3,5)<sup>[2]</sup>]]<br>{{CDD|label5|branch_c1|3ab|branch_c2|label5}}||1
|{{CDD|label5|branch_11|3ab|branch|label5}}
|colspan=3|
|- align=center
||[2<sup>+</sup>[(3,5)<sup>[2]</sup>]]<br>{{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)<sup>+</sup>[(3,5)<sup>[2]</sup>]]<br>{{CDD|label5|branch_c1|3ab|branch_c1|label5}}||1
| {{CDD|label5|branch_11|3ab|branch_11|label5}}
||[(2,2)<sup>+</sup>[(3,5)<sup>[2]</sup>]]<sup>+</sup>||(1)
| {{CDD|label5|branch_hh|3ab|branch_hh|label5}}
|}

== See also ==
* [[Uniform tilings in hyperbolic plane]]
* List of regular polytopes#Tessellations of hyperbolic 3-space

== Notes ==
{{reflist}}

== References ==
* James E. Humphreys, ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990) 
* ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}} (Chapter 10, [https://web.archive.org/web/20190225204136/http://pdfs.semanticscholar.org/67fa/5c924fc515d05640308fe68f1d0974a3705c.pdf Regular Honeycombs in Hyperbolic Space])
*Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)
* [[Biography:Jeffrey Weeks (mathematician)|Jeffrey R. Weeks]] ''The Shape of Space, 2nd edition'' {{isbn|0-8247-0709-5}} (Chapters 16–17: Geometries on Three-manifolds I, II) [http://www.maa.org/reviews/shapeofspace.html]
* [https://arxiv.org/abs/math/0212010 Coxeter Decompositions of Hyperbolic Tetrahedra], [[ArXiv|arXiv]]/[[PDF]], A. Felikson, December 2002
* 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 }}
*[[Biography:Norman Johnson (mathematician)|Norman Johnson]], ''Geometries and Transformations'' (2018), Chapters 11,12,13
*N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, ''The size of a hyperbolic Coxeter simplex'', Transformation Groups 1999, 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'' H<sup>3</sup>: p130. [http://www.sciencedirect.com/science/article/pii/S0024379501004773]
* {{KlitzingPolytopes|hyperbolic.htm#3D-compact|Hyperbolic honeycombs|H3 compact}}

[[Category:Honeycombs (geometry)]]

{{Sourceattribution|Uniform honeycombs in hyperbolic space}}