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[[File:Hyperbolic orthogonal dodecahedral honeycomb.png|thumb|The [[Order-4 dodecahedral honeycomb|{5,3,4} honeycomb]] in 3D hyperbolic space, viewed from the center of a [[Beltrami-Klein model]]]]
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In [[hyperbolic geometry]], a '''uniform honeycomb in hyperbolic space''' is a [[uniform polytope|uniform]] [[tessellation]] of [[uniform polyhedron|uniform polyhedral]] [[Cell (geometry)|cells]]. In 3-dimensional [[hyperbolic space]] there are nine [[Coxeter group]] families of compact [[convex uniform honeycomb]]s, generated as [[Wythoff construction]]s, and represented by [[permutation]]s of [[Coxeter-Dynkin diagram#Application with uniform polytopes|rings]] of the [[Coxeter–Dynkin diagram]]s for each family.
 
== Hyperbolic uniform honeycomb families ==
The nine compact [[Coxeter group]]s are listed here with their [[Coxeter–Dynkin diagram]]s,<ref>Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [http://books.google.com/books?id=ODfjmOeNLMUC&lpg=PP1&ots=AX5SYxPQ9S&pg=PA141]</ref>  
in order of the relative volumes of their [[Fundamental domain|fundamental simplex domains]].<ref>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. One known example is 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>Felikson, 2002</ref>
!Witt<br>symbol
!Coxeter<br>symbol
!Coxeter<br>graph
!Honeycombs
|-
!H<sub>1</sub>
|0.0358850633
|<math>{\bar{BH}}_3</math> || [5,3,4]|| {{CDD|node|5|node|3|node|4|node}} || 15 forms
|-
!H<sub>2</sub>
|0.0390502856
|<math>{\bar{J}}_3</math> || [3,5,3]|| {{CDD|node|3|node|5|node|3|node}} || 9 forms
|-
!H<sub>3</sub>
|0.0717701267
|<math>{\bar{DH}}_3</math> || [5,3<sup>1,1</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)]|| {{CDD|label4|branch|3ab|branch}} || 9 forms
|-
!H<sub>5</sub>
|0.0933255395
|<math>{\bar{K}}_3</math> || [5,3,5]|| {{CDD|node|5|node|3|node|5|node}} || 9 forms
|-
!H<sub>6</sub>
|0.2052887885
|<math>{\widehat{AH}}_3</math> || [(5,3,3,3)]|| {{CDD|label5|branch|3ab|branch}} || 9 forms
|-
!H<sub>7</sub>
|0.2222287320
|<math>{\widehat{BB}}_3</math> || [(4,3)<sup>[2]</sup>]|| {{CDD|label4|branch|3ab|branch|label4}} || 6 forms
|-
!H<sub>8</sub>
|0.3586534401
|<math>{\widehat{BH}}_3</math> || [(3,4,3,5)]|| {{CDD|label5|branch|3ab|branch|label4}} || 9 forms
|-
!H<sub>9</sub>
|0.5021308905
|<math>{\widehat{HH}}_3</math> || [(5,3)<sup>[2]</sup>]|| {{CDD|label5|branch|3ab|branch|label5}} || 6 forms
|}
 
=== Paracompact hyperbolic uniform honeycombs ===
{{see|paracompact uniform honeycombs}}
There are also 23 [[Coxeter diagram#Paracompact (Koszul simplex groups)|paracompact Coxeter groups]] of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded [[Facet (geometry)|facets]] or [[vertex figure]], including [[ideal vertex|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 [[Coxeter diagram#Vinberg polytopes with rank n.2B2 for n dimensional space|Vinberg polytope]] fundamental domains, including these [[triangular bipyramid]] [[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|infin|node|3|node|3|node|infin|node}}, {{CDD|node|infin|node|3|node|4|node|infin|node}}, {{CDD|node|infin|node|4|node|4|node|infin|node}}, {{CDD|node|infin|node|3|node|5|node|infin|node}}, {{CDD|node|infin|node|3|node|6|node|infin|node}}
: {{CDD|labelinfin|branch|split2|node|3|node|infin|node}}, {{CDD|labelinfin|branch|split2|node|4|node|infin|node}}, {{CDD|labelinfin|branch|split2-43|node|3|node|infin|node}}, {{CDD|labelinfin|branch|split2-43|node|4|node|infin|node}}, {{CDD|labelinfin|branch|split2-44|node|4|node|infin|node}} ...
: {{CDD|labelinfin|branch|split2|node|split1|branch|labelinfin}}, {{CDD|labelinfin|branch|split2-43|node|split1|branch|labelinfin}}, {{CDD|labelinfin|branch|split2-53|node|split1|branch|labelinfin}}, {{CDD|labelinfin|branch|split2-44|node|split1|branch|labelinfin}}, {{CDD|labelinfin|branch|split2-43|node|split1-43|branch|labelinfin}} ...
|}
 
=== Compact Enumeration ===
 
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_10l}})<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_10l}} | {{CDD|node_1|5|node|split1|nodes_10l}} | {{CDD|node|5|node_1|split1|nodes_10l}} | {{CDD|node_1|5|node_1|split1|nodes_10l}}
|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}}
|}
 
=== [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]].<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/content/aus_journal_content_abs_2005_16_2.html]</ref>
 
The bitruncated and runcinated forms (5 and 6) contain the faces of two [[regular skew polyhedron]]s: {4,10|3} and {10,4|3}.
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin]]<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
!
|- 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
![[Vertex figure]]
!picture
|- align=center BGCOLOR="#f0e0e0"
!1
|(Regular) [[order-3 icosahedral honeycomb|icosahedral]]<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.png|100px]]
|[[File:Hyperb icosahedral hc.png|100px]]
|- align=center BGCOLOR="#f0e0e0"
!2
|[[rectified icosahedral honeycomb|rectified icosahedral]]<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]]
|[[Image:Rectified icosahedral honeycomb.png|100px]]
|- align=center BGCOLOR="#f0e0e0"
!3
|[[truncated icosahedral honeycomb|truncated icosahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
!4
|[[Cantellated icosahedral honeycomb|cantellated icosahedral]]<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]]
|- align=center BGCOLOR="#e0f0e0"
!5
|[[Runcinated icosahedral honeycomb|Runcinated icosahedral]]<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]]
|- align=center BGCOLOR="#e0f0e0"
!6
|[[Bitruncated icosahedral honeycomb|bitruncated icosahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
!7
|[[cantitruncated icosahedral honeycomb|cantitruncated icosahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
!8
|[[runcitruncated icosahedral honeycomb|runcitruncated icosahedral]]<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]]
|- align=center BGCOLOR="#e0f0e0"
!9
|[[omnitruncated icosahedral honeycomb|omnitruncated icosahedral]]<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]]
|- align=center BGCOLOR="#e0f0f0"
|Nonuniform
|Alternated omnitruncated icosahedral<br>{{CDD|node_h|3|node_h|5|node_h|3|node_h}}<br>ht<sub>0,1,2,3</sub>{3,5,3}
|[[Image:Snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
|[[Image:octahedron.png|40px]]<br>[[octahedron|(3.3.3.3]]
|[[Image:octahedron.png|40px]]<br>[[octahedron|(3.3.3.3)]]
|[[Image:Snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
|[[Image:tetrahedron.png|40px]]<br>+[[tetrahedron|(3.3.3)]]
|[[File:snub icosahedral honeycomb verf.png|100px]]
|- align=center BGCOLOR="#e0f0f0"
|[77]
|[[partially truncated icosahedral honeycomb|partially truncated icosahedral]]<br>pt{3,5,3}
|(4)<br>[[Image:dodecahedron.png|40px]]<br>[[dodecahedron|(5.5.5)]]
|
|
|
|[[Image:Pentagonal antiprism.png|40px]]<br>[[Pentagonal antiprism|(3.3.3.5)]]
|[[Image:Partial truncation order-3 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}}
 
{| class=wikitable
!rowspan=2|#
!rowspan=2|Name of honeycomb<br>[[Coxeter–Dynkin 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="#f0e0e0" align=center
|10
|(Regular) [[order-4 dodecahedral honeycomb|order-4 dodecahedral]]<br>{{CDD|node_1|5|node|3|node|4|node}}
| -
| -
| -
|(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]]
|[[Image:Hyperbolic orthogonal dodecahedral honeycomb.png|100px]]
|- BGCOLOR="#f0e0e0" align=center
|11
|[[Rectified order-4 dodecahedral honeycomb|Rectified order-4 dodecahedral]]<br>{{CDD|node|5|node_1|3|node|4|node}}
|(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]]
|[[Image:Rectified order 4 dodecahedral honeycomb.png|100px]]
|- BGCOLOR="#e0e0f0" align=center
|12
|[[Rectified order-5 cubic honeycomb|Rectified order-5 cubic]]<br>{{CDD|node|5|node|3|node_1|4|node}}
|(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]]
|
|- BGCOLOR="#e0e0f0" align=center
|13
|(Regular) [[order-5 cubic honeycomb|order-5 cubic]]<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.png|100px]]
|[[File:Hyperb gcubic hc.png|100px]]
|- BGCOLOR="#f0e0e0" align=center
|14
|[[Truncated order-4 dodecahedral honeycomb|Truncated order-4 dodecahedral]]<br>{{CDD|node_1|5|node_1|3|node|4|node}}
|(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]]
|
|- BGCOLOR="#e0f0e0" align=center
|15
|[[Bitruncated order-5 cubic honeycomb|Bitruncated order-5 cubic]]<br>{{CDD|node|5|node_1|3|node_1|4|node}}
|(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]]
|
|- BGCOLOR="#e0e0f0" align=center
|16
|[[Truncated order-5 cubic honeycomb|Truncated order-5 cubic]]<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]]
|
|- BGCOLOR="#f0e0e0" align=center
|17
|[[Cantellated order-4 dodecahedral honeycomb|Cantellated order-4 dodecahedral]]<br>{{CDD|node_1|5|node|3|node_1|4|node}}
|(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]]
|
|- BGCOLOR="#e0e0f0" align=center
|18
|[[Cantellated order-5 cubic honeycomb|Cantellated order-5 cubic]]<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]]
|
|- BGCOLOR="#e0f0e0" align=center
|19
|[[Runcinated order-5 cubic honeycomb|Runcinated order-5 cubic]]<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]]
|
|- BGCOLOR="#f0e0e0" align=center
|20
|[[Cantitruncated order-4 dodecahedral honeycomb|Cantitruncated order-4 dodecahedral]]<br>{{CDD|node_1|5|node_1|3|node_1|4|node}}
|(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]]
|
|- BGCOLOR="#e0e0f0" align=center
|21
|[[Cantitruncated order-5 cubic honeycomb|Cantitruncated order-5 cubic]]<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]]
|
|- BGCOLOR="#f0e0e0" align=center
|22
|[[Runcitruncated order-4 dodecahedral honeycomb|Runcitruncated order-4 dodecahedral]]<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]]
|
|- BGCOLOR="#e0e0f0" align=center
|23
|[[Runcitruncated order-5 cubic honeycomb|Runcitruncated order-5 cubic]]<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]]
|
|- BGCOLOR="#e0f0e0" align=center
|24
|[[Omnitruncated order-5 cubic honeycomb|Omnitruncated order-5 cubic]]<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]]
|
|- BGCOLOR="#d0f0f0" align=center
|[34]
|[[alternated order-5 cubic honeycomb|alternated order-5 cubic]]<br>{{CDD|node|5|node|3|node|4|node_h1}} = {{CDD|node|5|node|split1|nodes_10l}}
|(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
|Nonuniform
|Alternated cantitruncated<br>order-4 dodecahedral honeycomb<br>{{CDD|node_h|5|node_h|3|node_h|4|node}}
|{{CDD|node_h|3|node_h|4|node}}<br>[[File:Uniform polyhedron-43-h01.svg|40px]]<br>[[icosahedron|(3.3.3.3.3)]]
|{{CDD|node_h|2|node_h|4|node}}<br>[[File:Tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| -
|{{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)]]
|[[File:Alternated cantitruncated order-4 dodecahedral honeycomb verf.png|100px]]<br>Irr. [[tridiminished icosahedron]]
|
|- BGCOLOR="#d0f0f0" align=center
|Nonuniform
|Alternated omnitruncated order-5 cubic<br>{{CDD|node_h|5|node_h|3|node_h|4|node_h}}
|{{CDD|node_h|3|node_h|4|node_h}}<br>[[File:snub hexahedron.png|40px]]<br>[[snub cube|(3.3.3.3.4)]]
|{{CDD|node_h|2|node_h|4|node_h}}<br>[[File:square antiprism.png|40px]]<br>[[square antiprism|(3.3.3.4)]]
|{{CDD|node_h|5|node_h|2|node_h}}<br>[[File:pentagonal antiprism.png|40px]]<br>[[pentagonal antiprism|(3.3.3.5)]]
|{{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)]]
|[[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]]s: {4,6|5} and {6,4|5}.
 
{| class=wikitable
!rowspan=2|#
!rowspan=2|Name of honeycomb<br>[[Coxeter–Dynkin 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
|- align=center BGCOLOR="#f0e0e0"
|25
|(Regular) [[order-5 dodecahedral honeycomb|Order-5 dodecahedral]]<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]]
|[[Image:Order 5 dodecahedral honeycomb.png|100px]]
|- align=center BGCOLOR="#f0e0e0"
|26
|[[rectified order-5 dodecahedral honeycomb|rectified order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
|27
|[[truncated order-5 dodecahedral honeycomb|truncated order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
|28
|[[Cantellated order-5 dodecahedral honeycomb|cantellated order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#e0f0e0"
|29
|[[Runcinated order-5 dodecahedral honeycomb|Runcinated order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#e0f0e0"
|30
|[[Bitruncated order-5 dodecahedral honeycomb|bitruncated order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
|31
|[[cantitruncated order-5 dodecahedral honeycomb|cantitruncated order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#f0e0e0"
|32
|[[runcitruncated order-5 dodecahedral honeycomb|runcitruncated order-5 dodecahedral]]<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]]
|- align=center BGCOLOR="#e0f0e0"
|33
|[[omnitruncated order-5 dodecahedral honeycomb|omnitruncated order-5 dodecahedral]]<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]]
|- BGCOLOR="#d0f0f0" align=center
|Nonuniform
|Alternated omnitruncated 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}
|{{CDD|node_h|3|node_h|5|node_h}}<br>[[File:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
|{{CDD|node_h|2|node_h|5|node_h}}<br>[[File:pentagonal antiprism.png|40px]]<br>[[pentagonal antiprism|(3.3.3.5)]]
|{{CDD|node_h|5|node_h|2|node_h}}<br>[[File:pentagonal antiprism.png|40px]]<br>[[pentagonal antiprism|(3.3.3.5)]]
|{{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)]]
|[[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}}
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram]]
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[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 honeycomb|alternated order-5 cubic]]<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 honeycomb|Cantic order-5 cubic]]<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]]
|- align=center
|36
|[[runcic order-5 cubic honeycomb|runcic order-5 cubic]]<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]]
|- align=center
|37
|[[runcicantic order-5 cubic honeycomb|runcicantic order-5 cubic]]<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]]
|}
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram]]<BR>{{CDD|nodeab_c1|split2|node_c2|5|node_c3}} = {{CDD|node|4|node_c1|3|node_c2|5|node_c3}}
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[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]]<br>{{CDD|nodes|split2|node|5|node_1}} = {{CDD|node|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:Hyperbolic orthogonal dodecahedral honeycomb.png|100px]]
|- align=center
| [11]
|[[rectified order-4 dodecahedral honeycomb|rectified order-4 dodecahedral]]<br>{{CDD|nodes|split2|node_1|5|node}} = {{CDD|node|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]]
|- align=center
| [12]
|[[rectified order-5 cubic honeycomb|rectified order-5 cubic]]<br>{{CDD|nodes_11|split2|node|5|node}} = {{CDD|node|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]]
|- align=center
| [15]
|[[bitruncated order-5 cubic honeycomb|bitruncated order-5 cubic]]<br>{{CDD|nodes_11|split2|node_1|5|node}} = {{CDD|node|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]]
|- align=center
| [14]
|[[truncated order-4 dodecahedral honeycomb|truncated order-4 dodecahedral]]<br>{{CDD|nodes|split2|node_1|5|node_1}} = {{CDD|node|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]]
|- align=center
| [17]
|[[cantellated order-4 dodecahedral honeycomb|cantellated order-4 dodecahedral]]<br>{{CDD|nodes_11|split2|node|5|node_1}} = {{CDD|node|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]]
|- align=center
| [20]
|[[cantitruncated order-4 dodecahedral honeycomb|cantitruncated order-4 dodecahedral]]<br>{{CDD|nodes_11|split2|node_1|5|node_1}} = {{CDD|node|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]]
|- align=center
|Nonuniform
|Alternated cantitruncated<br>order-4 dodecahedral<br>{{CDD|nodes_hh|split2|node_h|5|node_h}} = {{CDD|node|4|node_h|3|node_h|5|node_h}}
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[File:Uniform polyhedron-33-t0.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| [[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| [[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]]
|}
 
=== [(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]]s: {8,6|3} and {6,8|3}.
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin<br>diagram]]
!colspan=5|Cells by location<br>(and count around each vertex)
!rowspan=2|[[vertex figure]]
|-
!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
|{{CDD|label4|branch_10r|3ab|branch}}
| (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]]
|- align=center
|39
|{{CDD|label4|branch|3ab|branch_10l}}
| (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]]
|- align=center
|40
|{{CDD|label4|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: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]]
|- align=center
|41
|{{CDD|label4|branch_11|3ab|branch}}
| (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]]
|- align=center
|42
|{{CDD|label4|branch|3ab|branch_11}}
| (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]]
|- align=center
|43
|{{CDD|label4|branch_01r|3ab|branch_10l}}
| (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]]
|- align=center
|44
|{{CDD|label4|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 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]]
|- align=center
|45
|{{CDD|label4|branch_10r|3ab|branch_11}}
| (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]]
|- align=center
|46
|{{CDD|label4|branch_11|3ab|branch_11}}
| (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]]
|- align=center
|Nonuniform
|{{CDD|label4|branch_hh|3ab|branch_hh}}
| [[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| [[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| [[Image:snub hexahedron.png|40px]]<br>[[snub cube|(3.3.3.3.4)]]
| [[Image:snub hexahedron.png|40px]]<br>[[snub cube|(3.3.3.3.4)]]
| [[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]]s: {10,6|3} and {6,10|3}.
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin<br>diagram]]
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[vertex figure]]
|-
!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
|{{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]]
|- align=center
|48
|{{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]]
|- align=center
|49
|{{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]]
|- align=center
|52
|{{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]]
|- align=center
|53
|{{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]]
|- align=center
|54
|{{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]]
|}
 
{|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]]
|-
!0,1<br>{{CDD|nodea|3a|branch}}
!2,3<br>{{CDD|label5|branch|3b|nodeb}}
!Alt
|- align=center
|50
|{{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]]
|- align=center
|51
|{{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]]
|- align=center
|55
|{{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]]
|- align=center
|Nonuniform
|{{CDD|label5|branch_hh|3ab|branch_hh}}
| [[File:Uniform polyhedron-33-s012.png|40px]]<br>[[Icosahedron|(3.3.3.3.3)]]
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[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}}
 
The truncated forms (57 and 58) contain the faces of two [[regular skew polyhedron]]s: {6,6|4} and {8,8|3}.
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin<br>diagram]]
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[vertex figure]]
|-
!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
|{{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]]
|- align=center
|60
|{{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 icuboctahedron|(4.6.8)]]
|[[File:Uniform t012 4343 honeycomb verf.png|100px]]
|}
 
{|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]]
|-
!0,3<br>{{CDD|nodea|3a|branch|label4}}
!1,2<br>{{CDD|nodeb|3b|branch|label4}}
!Alt
|- align=center
|57
|{{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]]
|- align=center
|Nonuniform
|{{CDD|label4|branch_h0r|3ab|branch_h0l|label4}}
| [[File:Uniform polyhedron-43-h01.png|40px]]<br>[[icosahedron|(3.3.3.3.3)]]
| [[Image:tetrahedron.png|40px]]<br>[[Tetrahedron|(3.3.3)]]
| [[Image:octahedron.png|40px]]<br>+[[Octahedron|(3.3.3.3)]]
| [[File:Alternated truncated 3434 honeycomb verf.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]]
|-
!0,1<br>{{CDD|nodea|3a|branch|label4}}
!2,3<br>{{CDD|label4|branch|3b|nodeb}}
|- align=center
|58
|{{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]]
|}
 
{|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]]
|-
!0,2<br>{{CDD|nodea|3a|branch|label4}}
!1,3<br>{{CDD|nodeb|3b|branch|label4}}
|- align=center
|59
|{{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]]
|}
 
{|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]]
|-
!0,1,2,3<br>{{CDD|nodea|3a|branch|label4}}
!Alt
|- align=center
|61
|{{CDD|label4|branch_11|3ab|branch_11|label4}}
| (4)<br>[[Image:great rhombicuboctahedron.png|40px]]<br>[[truncated icuboctahedron|(4.6.8)]]
|
|[[File:Uniform t0123 4343 honeycomb verf.png|100px]]
|- align=center
|Nonuniform
|{{CDD|label4|branch_hh|3ab|branch_hh|label4}}
| [[Image:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
| [[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]]s: {10,6|3} and {6,10|3}.
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin<br>diagram]]
!colspan=4|Cells by location<br>(and count around each vertex)
!rowspan=2|[[vertex figure]]
|-
!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
|{{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]]
|- align=center
|63
|{{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]]
|- align=center
|64
|{{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]]
|- align=center
|67
|{{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]]
|- align=center
|68
|{{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]]
|- align=center
|69
|{{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]]
|}
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin<br>diagram]]
!colspan=3|Cells by location<br>(and count around each vertex)
!rowspan=2|[[vertex figure]]
|-
!0,1<br>{{CDD|nodea|3a|branch|label4}}
!2,3<br>{{CDD|label5|branch|3b|nodeb}}
!Alt
|- align=center
|65
|{{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]]
|- align=center
|66
|{{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]]
|- align=center
|70
|{{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]]
|- align=center
|Nonuniform
|{{CDD|label5|branch_hh|3ab|branch_hh|label4}}
| [[File:snub hexahedron.png|40px]]<br>[[Snub cube|(3.3.3.3.4)]]
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[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}}
 
The truncated forms (72 and 73) contain the faces of two [[regular skew polyhedron]]s: {6,6|5} and {10,10|3}.
 
{|class="wikitable"
!rowspan=2|#
!rowspan=2|Honeycomb name<br>[[Coxeter–Dynkin diagram|Coxeter–Dynkin<br>diagram]]
!colspan=5|Cells by location<br>(and count around each vertex)
!rowspan=2|[[vertex figure]]
|-
!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
|{{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]]
|- align=center
|72
|{{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]]
|- align=center
|73
|{{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]]
|- align=center
|74
|{{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]]
|- align=center
|75
|{{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]]
|- align=center
|76
|{{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]]
|- align=center
|Nonuniform
|{{CDD|label5|branch_hh|3ab|branch_hh|label5}}
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[Image:snub dodecahedron cw.png|40px]]<br>[[snub dodecahedron|(3.3.3.3.5)]]
| [[Image:tetrahedron.png|40px]]<br>+[[tetrahedron|(3.3.3)]]
| [[File:Snub 5353 honeycomb verf.png|100px]]
|}
 
== See also ==
{{Commons category|Uniform tilings of hyperbolic 3-space}}
* [[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, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space])
*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)
* [[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]
* [http://arxiv.org/pdf/math/0212010v1.pdf Coxeter Decompositions of Hyperbolic Tetrahedra], [[arXiv]]/[[PDF]], A. Felikson, December 2002
* C. W. L. Garner, ''Regular Skew Polyhedra in Hyperbolic Three-Space'' Canad. J. Math. 19, 1179–1186, 1967. [[PDF]] [http://cms.math.ca/cjm/a145822]
*[[Norman Johnson (mathematician)|Norman Johnson]], ''Geometries and Transformations'', Chapters 11,12,13, preprint 2011
*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 [http://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]
 
[[Category:Honeycombs (geometry)]]

Latest revision as of 02:01, 24 February 2014

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