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Paracompact uniform honeycombs#Regular paracompact honeycombs

{{Short description|Tessellation of convex uniform polyhedron cells}}

class=wikitable align=right width=400

|+ Example paracompact regular honeycombs

align=center

|100px
{3,3,6}
{{CDD|node 1|3|node|3|node|6|node}}

|100px
{6,3,3}
{{CDD|node 1|6|node|3|node|3|node}}

|100px
{4,3,6}
{{CDD|node 1|4|node|3|node|6|node}}

|100px
{6,3,4}
{{CDD|node 1|6|node|3|node|4|node}}

align=center

|100px
{5,3,6}
{{CDD|node 1|5|node|3|node|6|node}}

|100px
{6,3,5}
{{CDD|node 1|6|node|3|node|5|node}}

|100px
{6,3,6}
{{CDD|node 1|6|node|3|node|6|node}}

|100px
{3,6,3}
{{CDD|node 1|3|node|6|node|3|node}}

align=center

|100px
{4,4,3}
{{CDD|node 1|4|node|4|node|3|node}}

|100px
{3,4,4}
{{CDD|node 1|3|node|4|node|4|node}}

|100px
{4,4,4}
{{CDD|node 1|4|node|4|node|4|node}}

In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in 2-dimensions.

Regular paracompact honeycombs

Of the uniform paracompact H3 honeycombs, 11 are regular, meaning that their group of symmetries acts transitively on their flags. These have Schläfli symbol {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}, and are shown below. Four have finite Ideal polyhedral cells: {3,3,6}, {4,3,6}, {3,4,4}, and {5,3,6}.

{{Regular_paracompact_H3_honeycombs}}

class="wikitable"
Name

!Schläfli
Symbol
{p,q,r}

!Coxeter
{{CDD|node|p|node|q|node|r|node}}

!Cell
type
{p,q}

!Face
type
{p}

!Edge
figure
{r}

!Vertex
figure
{q,r}

!Dual

!Coxeter
group

BGCOLOR="#ffe0e0" align=center

|Order-6 tetrahedral honeycomb

{3,3,6}{{CDD|node_1|3|node|3|node|6|node}}{3,3}{3}{6}{3,6}{6,3,3}rowspan=2 BGCOLOR="#ffe0ff"|[6,3,3]
BGCOLOR="#e0e0ff" align=center

|Hexagonal tiling honeycomb

{6,3,3}{{CDD|node_1|6|node|3|node|3|node}}{6,3}{6}{3}{3,3}{3,3,6}
BGCOLOR="#ffe0e0" align=center

|Order-4 octahedral honeycomb

{3,4,4}{{CDD|node_1|3|node|4|node|4|node}}{3,4}{3}{4}{4,4}{4,4,3}rowspan=2 BGCOLOR="#ffe0ff"|[4,4,3]
BGCOLOR="#e0e0ff" align=center

|Square tiling honeycomb

{4,4,3}{{CDD|node_1|4|node|4|node|3|node}}{4,4}{4}{3}{4,3}{3,4,4}
BGCOLOR="#e0e0e0" align=center

|Triangular tiling honeycomb

{3,6,3}{{CDD|node_1|3|node|6|node|3|node}}{3,6}{3}{3}{6,3}Self-dual[3,6,3]
BGCOLOR="#ffe0e0" align=center

|Order-6 cubic honeycomb

{4,3,6}{{CDD|node_1|4|node|3|node|6|node}}{4,3}{4}{4}{3,6}{6,3,4}rowspan=2 BGCOLOR="#ffe0ff"|[6,3,4]
BGCOLOR="#e0e0ff" align=center

|Order-4 hexagonal tiling honeycomb

{6,3,4}{{CDD|node_1|6|node|3|node|4|node}}{6,3}{6}{4}{3,4}{4,3,6}
BGCOLOR="#e0e0e0" align=center

|Order-4 square tiling honeycomb

{4,4,4}{{CDD|node_1|4|node|4|node|4|node}}{4,4}{4}{4}{4,4}Self-dual[4,4,4]
BGCOLOR="#ffe0e0" align=center

|Order-6 dodecahedral honeycomb

{5,3,6}{{CDD|node_1|5|node|3|node|6|node}}{5,3}{5}{5}{3,6}{6,3,5}rowspan=2 BGCOLOR="#ffe0ff"|[6,3,5]
BGCOLOR="#e0e0ff" align=center

|Order-5 hexagonal tiling honeycomb

{6,3,5}{{CDD|node_1|6|node|3|node|5|node}}{6,3}{6}{5}{3,5}{5,3,6}
BGCOLOR="#e0e0e0" align=center

|Order-6 hexagonal tiling honeycomb

{6,3,6}{{CDD|node_1|6|node|3|node|6|node}}{6,3}{6}{6}{3,6}Self-dual[6,3,6]

Coxeter groups of paracompact uniform honeycombs

class=wikitable align=right width=300

|250px

|160px

colspan=2|These graphs show subgroup relations of paracompact hyperbolic Coxeter groups. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry.

This is a complete enumeration of the 151 unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains (rank 4 paracompact coxeter groups). The honeycombs are indexed here for cross-referencing duplicate forms, with brackets around the nonprimary constructions.

The alternations are listed, but are either repeats or don't generate uniform solutions. Single-hole alternations represent a mirror removal operation. If an end-node is removed, another simplex (tetrahedral) family is generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex groups, and their uniform honeycombs have not been systematically explored. These nonsimplectic (pyramidal) Coxeter groups are not enumerated on this page, except as special cases of half groups of the tetrahedral ones. Seven uniform honeycombs that arise here as alternations have been numbered 152 to 158, after the 151 Wythoffian forms not requiring alternation for their construction.

class="wikitable sortable"

|+ Tetrahedral hyperbolic paracompact group summary

!colspan=2|Coxeter group

!Simplex
volume

!Commutator subgroup

!Unique honeycomb count

align=center

|[6,3,3]

{{CDD|node|6|node|3|node|3|node}}0.0422892336[1+,6,(3,3)+] = [3,3[3]]+15
align=center

|[4,4,3]

{{CDD|node|4|node|4|node|3|node}}0.0763304662[1+,4,1+,4,3+]15
align=center

|[3,3[3]]

{{CDD|node|3|node|split1|branch}}0.0845784672[3,3[3]]+4
align=center

|[6,3,4]

{{CDD|node|6|node|3|node|4|node}}0.1057230840[1+,6,3+,4,1+] = [3[]x[]]+15
align=center

|[3,41,1]

{{CDD|node|3|node|split1-44|nodes}}0.1526609324[3+,41+,1+]4
align=center

|[3,6,3]

{{CDD|node|3|node|6|node|3|node}}0.1691569344[3+,6,3+]8
align=center

|[6,3,5]

{{CDD|node|6|node|3|node|5|node}}0.1715016613[1+,6,(3,5)+] = [5,3[3]]+15
align=center

|[6,31,1]

{{CDD|node|6|node|split1|nodes}}0.2114461680[1+,6,(31,1)+] = [3[]x[]]+4
align=center

|[4,3[3]]

{{CDD|node|4|node|split1|branch}}0.2114461680[1+,4,3[3]]+ = [3[]x[]]+4
align=center

|[4,4,4]

{{CDD|node|4|node|4|node|4|node}}0.2289913985[4+,4+,4+]+6
align=center

|[6,3,6]

{{CDD|node|6|node|3|node|6|node}}0.2537354016[1+,6,3+,6,1+] = [3[3,3]]+8
align=center

|[(4,4,3,3)]

{{CDD|node|split1-44|nodes|split2|node}}0.3053218647[(4,1+,4,(3,3)+)]4
align=center

|[5,3[3]]

{{CDD|node|5|node|split1|branch}}0.3430033226[5,3[3]]+4
align=center

|[(6,3,3,3)]

{{CDD|label6|branch|3ab|branch|2}}0.3641071004[(6,3,3,3)]+9
align=center

|[3[]x[]]

{{CDD|node|split1|branch|split2|node}}0.4228923360[3[]x[]]+1
align=center

|[41,1,1]

{{CDD|node|4|node|split1-44|nodes}}0.4579827971[1+,41+,1+,1+]0
align=center

|[6,3[3]]

{{CDD|node|6|node|split1|branch}}0.5074708032[1+,6,3[3]] = [3[3,3]]+2
align=center

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

{{CDD|label6|branch|3ab|branch|label4}}0.5258402692[(6,3+,4,3+)]9
align=center

|[(4,4,4,3)]

{{CDD|label4|branch|4-4|branch}}0.5562821156[(4,1+,4,1+,4,3+)]9
align=center

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

{{CDD|label6|branch|3ab|branch|label5}}0.6729858045[(6,3,5,3)]+9
align=center

|[(6,3,6,3)]

{{CDD|label6|branch|3ab|branch|label6}}0.8457846720[(6,3+,6,3+)]5
align=center

|[(4,4,4,4)]

{{CDD|label4|branch|4-4|branch|label4}}0.9159655942[(4+,4+,4+,4+)]1
align=center

|[3[3,3]]

{{CDD|branch|splitcross|branch}}1.014916064[3[3,3]]+0

The complete list of nonsimplectic (non-tetrahedral) paracompact Coxeter groups was published by P. Tumarkin in 2003.[https://arxiv.org/abs/math/0301133.pdf P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)] The smallest paracompact form in H3 can be represented by {{CDD|node|ultra|node|3|node|3|node|ultra|node}} or {{CDD|node|split1|nodes|2a2b-cross|nodes}}, or [∞,3,3,∞] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1+,4] : {{CDD|node_c1|3|node_c2|4|node_h0|4|node_c3}} = {{CDD|node_c1|split1|nodeab_c2|2a2b-cross|nodeab_c3}}. The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramid is {{CDD|node|ultra|node|4|node|4|node|ultra|node}} or {{CDD|node|split1-44|nodes|2a2b-cross|nodes}}, constructed as [4,4,1+,4] = [∞,4,4,∞] : {{CDD|node_c1|4|node_c2|4|node_h0|4|node_c3}} = {{CDD|node_c1|split1-44|nodeab_c2|2a2b-cross|nodeab_c3}}.

Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: [(3,3,4,1+,4)] = [((3,∞,3)),((3,∞,3))] or {{CDD|branchu|split2|node|split1|branchu}}, [(3,4,4,1+,4)] = [((4,∞,3)),((3,∞,4))] or {{CDD|branchu|split2-43|node|split1-43|branchu}}, [(4,4,4,1+,4)] = [((4,∞,4)),((4,∞,4))] or {{CDD|branchu|split2-44|node|split1-44|branchu}}. {{CDD|labelh|node|split1-44|nodeab_c1-2|split2|node_c3}} = {{CDD|labelinfin|branch_c1-2|split2|node_c3|split1|branch_c1-2|labelinfin}}, {{CDD|labelh|node|split1-44|nodeab_c1-2|split2-43|node_c3}} = {{CDD|labelinfin|branch_c1-2|split2-43|node_c3|split1-43|branch_c1-2|labelinfin}}, {{CDD|labelh|node|split1-44|nodeab_c1-2|split2-44|node_c3}} = {{CDD|labelinfin|branch_c1-2|split2-44|node_c3|split1-44|branch_c1-2|labelinfin}}.

Another nonsimplectic half groups is {{CDD|nodeab_c1-2|split2-44|node_h0|4|node_c3}} ↔ {{CDD|node_c3|split1-uu|nodeab_c1-2|2a2b-cross|nodeab_c1-2|split2-uu|node_c3}}.

A radical nonsimplectic subgroup is {{CDD|label4|branch_c1-2|4a4b|branch|labels}} ↔ {{CDD|node_c1|splitplit1u-44|branch3u_c2|4a4buc-cross|branch3u_c1|splitplit2u-44|node_c2}}, which can be doubled into a triangular prism domain as {{CDD|node_c1|splitplit1u-44|branch3u_c2|4a4buc-cross|branch3u_c3|splitplit2u-44|node_c4}} ↔ {{CDD|branchu_c1-4|4a4b|branch_c2-3|split2-44|node|labelh}}.

class=wikitable

|+ Pyramidal hyperbolic paracompact group summary

Dimension

!Rank

!Graphs

H3

!5

|

{{CDD|node|split1|nodes|2a2b-cross|nodes}} | {{CDD|node|split1-43|nodes|2a2b-cross|nodes}} | {{CDD|node|split1-44|nodes|2a2b-cross|nodes}} | {{CDD|node|split1-53|nodes|2a2b-cross|nodes}} | {{CDD|node|split1-63|nodes|2a2b-cross|nodes}}
{{CDD|branchu|split2|node|3|node|ultra|node}} | {{CDD|branchu|split2|node|4|node|ultra|node}} | {{CDD|branchu|split2-43|node|3|node|ultra|node}} | {{CDD|branchu|split2-43|node|4|node|ultra|node}} | {{CDD|branchu|split2-44|node|3|node|ultra|node}} | {{CDD|branchu|split2-44|node|4|node|ultra|node}}
{{CDD|branchu|split2-53|node|3|node|ultra|node}} | {{CDD|branchu|split2-54|node|3|node|ultra|node}} | {{CDD|branchu|split2-55|node|3|node|ultra|node}} | {{CDD|branchu|split2-63|node|3|node|ultra|node}} | {{CDD|branchu|split2-64|node|3|node|ultra|node}} | {{CDD|branchu|split2-65|node|3|node|ultra|node}} | {{CDD|branchu|split2-66|node|3|node|ultra|node}}
{{CDD|branchu|split2|node|split1|branchu}} | {{CDD|branchu|split2-43|node|split1|branchu}} | {{CDD|branchu|split2-53|node|split1|branchu}} | {{CDD|branchu|split2-44|node|split1|branchu}} | {{CDD|branchu|split2-43|node|split1-43|branchu}} | {{CDD|branchu|split2-44|node|split1-43|branchu}} | {{CDD|branchu|split2-44|node|split1-44|branchu}} | {{CDD|branchu|split2-54|node|split1|branchu}} | {{CDD|branchu|split2-55|node|split1|branchu}} | {{CDD|branchu|split2-63|node|split1|branchu}} | {{CDD|branchu|split2-64|node|split1|branchu}} | {{CDD|branchu|split2-65|node|split1|branchu}} | {{CDD|branchu|split2-66|node|split1|branchu}}

Linear graphs

= [6,3,3] family =

class= "wikitable"
rowspan="2" |#

! rowspan="2" |Honeycomb name
Coxeter diagram: {{CDD|node_n1|6|node_n2|3|node_n3|3|node_n4}}
Schläfli symbol

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

! rowspan="2" |Vertex figure

! rowspan="2" |Picture

1
{{CDD|node_n2|3|node_n3|3|node_n4}}

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

!3
{{CDD|node_n1|6|node_n2|2|node_n4}}

!4
{{CDD|node_n1|6|node_n2|3|node_n3}}

align=center BGCOLOR="#f0e0e0"

!1

| hexagonal (hexah)
{{CDD|node 1|6|node|3|node|3|node}}
{6,3,3}

| -

| -

| -

|(4)
40px
(6.6.6)

|80px {{CDD|node_1|3|node|3|node}}
Tetrahedron

|120px

align=center BGCOLOR="#f0e0e0"

!2

| rectified hexagonal (rihexah)
{{CDD|node|6|node 1|3|node|3|node}}
t1{6,3,3} or r{6,3,3}

|(2)
40px
(3.3.3)

| -

| -

|(3)
40px
(3.6.3.6)

|80px {{CDD|node_1|2|node_1|3|node}}
Triangular prism

|120px

align=center BGCOLOR="#e0e0f0"

!3

| rectified order-6 tetrahedral (rath)
{{CDD|node|6|node|3|node_1|3|node}}
t1{3,3,6} or r{3,3,6}

|(6)
40px
(3.3.3.3)

| -

| -

|(2)
40px
(3.3.3.3.3.3)

|80px {{CDD|node|6|node_1|2|node_1}}
Hexagonal prism

|120px

align=center BGCOLOR="#e0e0f0"

!4

|order-6 tetrahedral (thon)
{{CDD|node|6|node|3|node|3|node_1}}
{3,3,6}

|(∞)
40px
(3.3.3)

| -

| -

| -

|40px {{CDD|node|6|node|3|node_1}}
Triangular tiling

|120px

align=center BGCOLOR="#f0e0e0"

!5

| truncated hexagonal (thexah)
{{CDD|node 1|6|node 1|3|node|3|node}}
t0,1{6,3,3} or t{6,3,3}

|(1)
40px
(3.3.3)

| -

| -

|(3)
40px
(3.12.12)

| 80px
Triangular pyramid

|120px

align=center BGCOLOR="#f0e0e0"

!6

|cantellated hexagonal (srihexah)
{{CDD|node_1|6|node|3|node_1|3|node}}
t0,2{6,3,3} or rr{6,3,3}

|(1)
40px
3.3.3.3

|(2)
40px
(4.4.3)

| -

|(2)
40px
(3.4.6.4)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!7

|runcinated hexagonal (sidpithexah)
{{CDD|node_1|6|node|3|node|3|node_1}}
t0,3{6,3,3}

|(1)
40px
(3.3.3)

|(3)
40px
(4.4.3)

|(3)
40px
(4.4.6)

|(1)
40px
(6.6.6)

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!8

|cantellated order-6 tetrahedral (srath)
{{CDD|node|6|node_1|3|node|3|node_1}}
t0,2{3,3,6} or rr{3,3,6}

|(1)
40px
(3.4.3.4)

| -

|(2)
40px
(4.4.6)

|(2)
40px
(3.6.3.6)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!9

|bitruncated hexagonal (tehexah)
{{CDD|node|6|node_1|3|node_1|3|node}}
t1,2{6,3,3} or 2t{6,3,3}

|(2)
40px
(3.6.6)

| -

| -

|(2)
40px
(6.6.6)

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!10

|truncated order-6 tetrahedral (tath)
{{CDD|node|6|node|3|node_1|3|node_1}}
t0,1{3,3,6} or t{3,3,6}

|(6)
40px
(3.6.6)

| -

| -

|(1)
40px
(3.3.3.3.3.3)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!11

|cantitruncated hexagonal (grihexah)
{{CDD|node_1|6|node_1|3|node_1|3|node}}
t0,1,2{6,3,3} or tr{6,3,3}

|(1)
40px
(3.6.6)

|(1)
40px
(4.4.3)

| -

|(2)
40px
(4.6.12)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!12

|runcitruncated hexagonal (prath)
{{CDD|node_1|6|node_1|3|node|3|node_1}}
t0,1,3{6,3,3}

|(1)
40px
(3.4.3.4)

|(2)
40px
(4.4.3)

|(1)
40px
(4.4.12)

|(1)
40px
(3.12.12)

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!13

|runcitruncated order-6 tetrahedral (prihexah)
{{CDD|node_1|6|node|3|node_1|3|node_1}}
t0,1,3{3,3,6}

|(1)
40px
(3.6.6)

|(1)
40px
(4.4.6)

|(2)
40px
(4.4.6)

|(1)
40px
(3.4.6.4)

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!14

|cantitruncated order-6 tetrahedral (grath)
{{CDD|node|6|node_1|3|node_1|3|node_1}}
t0,1,2{3,3,6} or tr{3,3,6}

|(2)
40px
(4.6.6)

| -

|(1)
40px
(4.4.6)

|(1)
40px
(6.6.6)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!15

|omnitruncated hexagonal (gidpithexah)
{{CDD|node_1|6|node_1|3|node_1|3|node_1}}
t0,1,2,3{6,3,3}

|(1)
40px
(4.6.6)

|(1)
40px
(4.4.6)

|(1)
40px
(4.4.12)

|(1)
40px
(4.6.12)

|80px

|120px

class= "wikitable"

|+ Alternated forms

rowspan="2" |#

! rowspan="2" |Honeycomb name
Coxeter diagram: {{CDD|node_n1|6|node_n2|3|node_n3|3|node_n4}}
Schläfli symbol

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

! rowspan="2" |Vertex figure

! rowspan="2" |Picture

1
{{CDD|node_n2|3|node_n3|3|node_n4}}

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

!3
{{CDD|node_n1|6|node_n2|2|node_n4}}

!4
{{CDD|node_n1|6|node_n2|3|node_n3}}

!Alt

align=center BGCOLOR="#e0f0f0"

|[137]

|alternated hexagonal (ahexah)
({{CDD|node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}}) = {{CDD|branch_hh|splitcross|branch_hh}}

| -

| -

|

|(4)
40px
(3.3.3.3.3.3)

|(4)
40px
(3.3.3)

|40px {{CDD|node_1|3|node_1|3|node}}
(3.6.6)

|

align=center BGCOLOR="#e0f0f0"

|[138]

|cantic hexagonal (tahexah)
{{CDD|node_h1|6|node|3|node_1|3|node}} ↔ {{CDD|branch_10ru|split2|node_1|3|node}}

|(1)
40px
(3.3.3.3)

| -

|

|(2)
40px
(3.6.3.6)

|(2)
40px
(3.6.6)

|80px

|

align=center BGCOLOR="#e0f0f0"

|[139]

|runcic hexagonal (birahexah)
{{CDD|node_h1|6|node|3|node|3|node_1}} ↔ {{CDD|branch_10ru|split2|node|3|node_1}}

|(1)
40px
(4.4.4)

|(1)
40px
(4.4.3)

|

|(1)
40px
(3.3.3.3.3.3)

|(3)
40px
(3.4.3.4)

|80px

|

align=center BGCOLOR="#e0f0f0"

|[140]

|runcicantic hexagonal (bitahexah)
{{CDD|node_h1|6|node|3|node_1|3|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|3|node_1}}

|(1)
40px
(3.6.6)

|(1)
40px
(4.4.3)

|

|(1)
40px
(3.6.3.6)

|(2)
40px
(4.6.6)

|80px

|

align=center BGCOLOR="#e0f0f0"

| Nonuniform

|snub rectified order-6 tetrahedral
{{CDD|node_h0|6|node_h|3|node_h|3|node_h}} ↔ {{CDD|branch_hh|split2|node_h|3|node_h}}
sr{3,3,6}

|40px

|

|

|40px

|40px
Irr. (3.3.3)

|80px

|

align=center BGCOLOR="#e0f0f0"

| Nonuniform

|cantic snub order-6 tetrahedral
{{CDD|node_1|6|node_h|3|node_h|3|node_h}}
sr3{3,3,6}

|

|

|

|

|

|

|

align=center BGCOLOR="#e0f0f0"

| Nonuniform

|omnisnub order-6 tetrahedral
{{CDD|node_h|6|node_h|3|node_h|3|node_h}}
ht0,1,2,3{6,3,3}

|40px

|

|

|40px

|40px
Irr. (3.3.3)

|

|

= [6,3,4] family =

There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or {{CDD|node|6|node|3|node|4|node}}

class=wikitable

!rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram
Schläfli symbol

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

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

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

!2
{{CDD|node_n1|6|node_n2|2|node_n4}}

!3
{{CDD|node_n1|6|node_n2|3|node_n3}}

BGCOLOR="#f0e0e0" align=center

!16

|(Regular) order-4 hexagonal (shexah)
{{CDD|node_1|6|node|3|node|4|node}}
{6,3,4}

| -

| -

| -

|(8)
{{CDD|node_1|6|node|3|node}}
40px
(6.6.6)

|80px {{CDD|node_1|3|node|4|node}}
(3.3.3.3)

|120px

BGCOLOR="#f0e0e0" align=center

!17

|rectified order-4 hexagonal (rishexah)
{{CDD|node|6|node_1|3|node|4|node}}
t1{6,3,4} or r{6,3,4}

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

| -

| -

|(4)
{{CDD|node|6|node_1|3|node}}
40px
(3.6.3.6)

|80px {{CDD|node_1|2|node_1|4|node}}
(4.4.4)

|120px

BGCOLOR="#e0e0f0" align=center

!18

|rectified order-6 cubic (rihach)
{{CDD|node|6|node|3|node_1|4|node}}
t1{4,3,6} or r{4,3,6}

|(6)
{{CDD|node|3|node_1|4|node}}
40px
(3.4.3.4)

| -

| -

|(2)
{{CDD|node|6|node|3|node_1}}
40px
(3.3.3.3.3.3)

|80px {{CDD|node|6|node_1|2|node_1}}
(6.4.4)

|120px

BGCOLOR="#e0e0f0" align=center

!19

|order-6 cubic (hachon)
{{CDD|node|6|node|3|node|4|node_1}}
{4,3,6}

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

| -

| -

| -

|40px {{CDD|node|6|node|3|node_1}}
(3.3.3.3.3.3)

|120px

BGCOLOR="#f0e0e0" align=center

!20

|truncated order-4 hexagonal (tishexah)
{{CDD|node_1|6|node_1|3|node|4|node}}
t0,1{6,3,4} or t{6,3,4}

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

| -

| -

|(4)
{{CDD|node_1|6|node_1|3|node}}
40px
(3.12.12)

|80px

|120px

BGCOLOR="#e0f0e0" align=center

!21

|bitruncated order-6 cubic (chexah)
{{CDD|node|6|node_1|3|node_1|4|node}}
t1,2{6,3,4} or 2t{6,3,4}

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

| -

| -

|(2)
{{CDD|node|6|node_1|3|node_1}}
40px
(6.6.6)

|80px

|120px

BGCOLOR="#e0e0f0" align=center

!22

|truncated order-6 cubic (thach)
{{CDD|node|6|node|3|node_1|4|node_1}}
t0,1{4,3,6} or t{4,3,6}

|(6)
{{CDD|node|3|node_1|4|node_1}}
40px
(3.8.8)

| -

| -

|(1)
{{CDD|node|6|node|3|node_1}}
40px
(3.3.3.3.3.3)

|80px

|120px

BGCOLOR="#f0e0e0" align=center

!23

|cantellated order-4 hexagonal (srishexah)
{{CDD|node_1|6|node|3|node_1|4|node}}
t0,2{6,3,4} or rr{6,3,4}

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

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

| -

|(2)
{{CDD|node_1|6|node|3|node_1}}
40px
(3.4.6.4)

|80px

|120px

BGCOLOR="#e0e0f0" align=center

!24

|cantellated order-6 cubic (srihach)
{{CDD|node|6|node_1|3|node|4|node_1}}
t0,2{4,3,6} or rr{4,3,6}

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

| -

|(2)
{{CDD|node|6|node_1|2|node_1}}
40px
(4.4.6)

|(1)
{{CDD|node|6|node_1|3|node}}
40px
(3.6.3.6)

|80px

|120px

BGCOLOR="#e0f0e0" align=center

!25

|runcinated order-6 cubic (sidpichexah)
{{CDD|node_1|6|node|3|node|4|node_1}}
t0,3{6,3,4}

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

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

|(3)
{{CDD|node_1|6|node|2|node_1}}
40px
(4.4.6)

|(1)
{{CDD|node_1|6|node|3|node}}
40px
(6.6.6)

|80px

|120px

BGCOLOR="#f0e0e0" align=center

!26

|cantitruncated order-4 hexagonal (grishexah)
{{CDD|node_1|6|node_1|3|node_1|4|node}}
t0,1,2{6,3,4} or tr{6,3,4}

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

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

| -

|(2)
{{CDD|node_1|6|node_1|3|node_1}}
40px
(4.6.12)

|80px

|120px

BGCOLOR="#e0e0f0" align=center

!27

|cantitruncated order-6 cubic (grihach)
{{CDD|node|6|node_1|3|node_1|4|node_1}}
t0,1,2{4,3,6} or tr{4,3,6}

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

| -

|(1)
{{CDD|node|6|node_1|2|node_1}}
40px
(4.4.6)

|(1)
{{CDD|node|6|node_1|3|node_1}}
40px
(6.6.6)

|80px

|120px

BGCOLOR="#f0e0e0" align=center

!28

|runcitruncated order-4 hexagonal (prihach)
{{CDD|node_1|6|node_1|3|node|4|node_1}}
t0,1,3{6,3,4}

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

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

|(2)
{{CDD|node_1|6|node_1|2|node_1}}
40px
(4.4.12)

|(1)
{{CDD|node_1|6|node_1|3|node}}
40px
(3.12.12)

|80px

|120px

BGCOLOR="#e0e0f0" align=center

!29

|runcitruncated order-6 cubic (prishexah)
{{CDD|node_1|6|node|3|node_1|4|node_1}}
t0,1,3{4,3,6}

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

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

|(1)
{{CDD|node_1|6|node|2|node_1}}
40px
(4.4.6)

|(1)
{{CDD|node_1|6|node|3|node_1}}
40px
(3.4.6.4)

|80px

|120px

BGCOLOR="#e0f0e0" align=center

!30

|omnitruncated order-6 cubic (gidpichexah)
{{CDD|node_1|6|node_1|3|node_1|4|node_1}}
t0,1,2,3{6,3,4}

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

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

|(1)
{{CDD|node_1|6|node_1|2|node_1}}
40px
(4.4.12)

|(1)
{{CDD|node_1|6|node_1|3|node_1}}
40px
(4.6.12)

|80px

|120px

class=wikitable

|+ Alternated forms

rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram
Schläfli symbol

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

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

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

!2
{{CDD|node_n1|6|node_n2|2|node_n4}}

!3
{{CDD|node_n1|6|node_n2|3|node_n3}}

!Alt

BGCOLOR="#e0f0f0" align=center

|[87]

|alternated order-6 cubic (ahach)
{{CDD|node_h1|4|node|3|node|6|node}} ↔ {{CDD|nodes_10ru|split2|node|6|node}}
h{4,3,6}

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

| 

| 

|

|40px {{CDD|node_1|3|node|6|node}}
(3.3.3.3.3.3)

|{{CDD|node|6|node_1|3|node}}
40px
(3.6.3.6)

|

align=center BGCOLOR="#e0f0f0"

|[88]

|cantic order-6 cubic (tachach)
{{CDD|node_h1|4|node|3|node_1|6|node}} ↔ {{CDD|nodes_10ru|split2|node_1|6|node}}
h2{4,3,6}

|(2)
40px
(3.6.6)

| -

| -

|(1)
40px
(3.6.3.6)

|(2)
40px
(6.6.6)

|80px

|

align=center BGCOLOR="#e0f0f0"

|[89]

|runcic order-6 cubic (birachach)
{{CDD|node_h1|4|node|3|node|6|node_1}} ↔ {{CDD|nodes_10ru|split2|node|6|node_1}}
h3{4,3,6}

| (1)
40px
(3.3.3)

| -

| -

| (1)
40px
(6.6.6)

| (3)
40px
(3.4.6.4)

|80px

|

align=center BGCOLOR="#e0f0f0"

|[90]

|runcicantic order-6 cubic (bitachach)
{{CDD|node_h1|4|node|3|node_1|6|node_1}} ↔ {{CDD|nodes_10ru|split2|node_1|6|node_1}}
h2,3{4,3,6}

| (1)
40px
(3.6.6)

| -

| -

| (1)
40px
(3.12.12)

| (2)
40px
(4.6.12)

|80px

|

BGCOLOR="#e0f0f0" align=center

|[141]

|alternated order-4 hexagonal (ashexah)
{{CDD|node_h1|6|node_g|3sg|node_g|4g|node_g}} ↔ {{CDD|branch_10ru|split2|node|4|node_h0}} ↔ {{CDD|node|split1|branch_10luru|split2|node}}
h{6,3,4}

| -

| -

|

|40px
(3.3.3.3.3.3)

|40px
(3.3.3.3)

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

|

BGCOLOR="#e0f0f0" align=center

|[142]

|cantic order-4 hexagonal (tashexah)
{{CDD|node_h1|6|node|3|node_1|4|node_h0}} ↔ {{CDD|branch_10ru|split2|node_1|4|node_h0}} ↔ {{CDD|node_1|split1|branch_10luru|split2|node_1}}
h1{6,3,4}

|(1)
40px
(3.4.3.4)

| -

|

|(2)
40px
(3.6.3.6)

|(2)
40px
(4.6.6)

|80px

|

BGCOLOR="#e0f0f0" align=center

|[143]

|runcic order-4 hexagonal (birashexah)
{{CDD|node_h1|6|node|3|node|4|node_1}} ↔ {{CDD|branch_10ru|split2|node|4|node_1}}
h3{6,3,4}

|(1)
40px
(4.4.4)

|(1)
40px
(4.4.3)

|

|(1)
40px
(3.3.3.3.3.3)

|(3)
40px
(3.4.4.4)

|80px

|

BGCOLOR="#e0f0f0" align=center

|[144]

|runcicantic order-4 hexagonal (bitashexah)
{{CDD|node_h1|6|node|3|node_1|4|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|4|node_1}}
h2,3{6,3,4}

|(1)
40px
(3.8.8)

|(1)
40px
(4.4.3)

|

| (1)
40px
(3.6.3.6)

|(2)
40px
(4.6.8)

|80px

|

BGCOLOR="#e0f0f0" align=center

|[151]

|quarter order-4 hexagonal (quishexah)
{{CDD|node_h1|6|node|3|node|4|node_h1}} ↔ {{CDD|node_1|split1|branch_10luru|split2|node}}
q{6,3,4}

|(3)
40px

|(1)
40px

| -

|(1)
40px

|(3)
40px

|80px

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|bisnub order-6 cubic
{{CDD|node_h0|6|node_h|3|node_h|4|node_h0}} ↔ {{CDD|node_h|split1|branch_hh|split2|node_h}}
2s{4,3,6}

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

| -

| -

|{{CDD|node|6|node_h|3|node_h}}
40px
(3.3.3.3.3.3)

|40px
+(3.3.3)

|80px

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|runcic bisnub order-6 cubic
{{CDD|node_1|6|node_h|3|node_h|4|node_1}}

|{{CDD|node_h|3|node_h|4|node_1}}

|{{CDD|node_1|2|node_h|4|node_1}}

|{{CDD|node_1|6|node_h|2|node_1}}

|{{CDD|node_1|6|node_h|3|node_h}}

|

|

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|snub rectified order-6 cubic
{{CDD|node_h0|6|node_h|3|node_h|4|node_h}} ↔ {{CDD|branch_hh|split2|node_h|4|node_h}}
sr{4,3,6}

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

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

|{{CDD|node|6|node_h|2x|node_h}}
40px
(3.3.3.3)

|{{CDD|node|6|node_h|3|node_h}}
40px
(3.3.3.3.6)

|40px
+(3.3.3)

|

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|runcic snub rectified order-6 cubic
{{CDD|node_1|6|node_h|3|node_h|4|node_h}}
sr3{4,3,6}

|{{CDD|node_h|3|node_h|4|node_h}}

|{{CDD|node_1|2|node_h|4|node_h}}

|{{CDD|node_1|6|node_h|2x|node_h}}

|{{CDD|node_1|6|node_h|3|node_h}}

|

|

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|snub rectified order-4 hexagonal
{{CDD|node_h|6|node_h|3|node_h|4|node_h0}} ↔ {{CDD|node_h|6|node_h|split1|nodes_hh}}
sr{6,3,4}

|{{CDD|node_h|3|node_h|4|node}}
40px
(3.3.3.3.3.3)

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

| -

|{{CDD|node_h|6|node_h|3|node_h}}
40px
(3.3.3.3.6)

|40px
+(3.3.3)

|

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|runcisnub rectified order-4 hexagonal
{{CDD|node_h|6|node_h|3|node_h|4|node_1}}
sr3{6,3,4}

|

|

|

|

|

|

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|omnisnub rectified order-6 cubic
{{CDD|node_h|6|node_h|3|node_h|4|node_h}}
ht0,1,2,3{6,3,4}

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

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

|{{CDD|node_h|6|node_h|2x|node_h}}
40px
(3.3.3.6)

|{{CDD|node_h|6|node_h|3|node_h}}
40px
(3.3.3.3.6)

|40px
+(3.3.3)

|

|

= [6,3,5] family =

class="wikitable"
rowspan="2" | #

! rowspan="2" | Honeycomb name
Coxeter diagram
Schläfli symbol

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

! rowspan="2" |Vertex figure

! rowspan="2" |Picture

align=center

!0
{{CDD|node_n2|3|node_n3|5|node_n5}}

!1
{{CDD|node_n1|2|node_n3|5|node_n5}}

!2
{{CDD|node_n1|6|node_n2|2|node_n5}}

!3
{{CDD|node_n1|6|node_n2|3|node_n3}}

align=center BGCOLOR="#f0e0e0"

! 31

| order-5 hexagonal (phexah)
{{CDD|node 1|6|node|3|node|5|node}}
{6,3,5}

| -

| -

| -

|(20)
50px
(6)3

|80px {{CDD|node 1|3|node|5|node}}
Icosahedron

|120px

align=center BGCOLOR="#f0e0e0"

!32

|rectified order-5 hexagonal (riphexah)
{{CDD|node|6|node_1|3|node|5|node}}
t1{6,3,5} or r{6,3,5}

|(2)
40px
(3.3.3.3.3)

| -

| -

|(5)
50px
(3.6)2

|80px {{CDD|node_1|2|node_1|5|node}}
(5.4.4)

|120px

align=center BGCOLOR="#e0e0f0"

!33

|rectified order-6 dodecahedral (rihed)
{{CDD|node|6|node|3|node_1|5|node}}
t1{5,3,6} or r{5,3,6}

|(5)
40px
(3.5.3.5)

| -

| -

|(2)
50px
(3)6

80px {{CDD|node|6|node_1|2|node_1}}
(6.4.4)

|120px

align=center BGCOLOR="#e0e0f0"

!34

|order-6 dodecahedral (hedhon)
{{CDD|node|6|node|3|node|5|node_1}}
{5,3,6}

|40px
(5.5.5)

| -

| -

| -

|(∞)
50px {{CDD|node|6|node|3|node_1}}
(3)6

|120px

align=center BGCOLOR="#f0e0e0"

!35

|truncated order-5 hexagonal (tiphexah)
{{CDD|node_1|6|node_1|3|node|5|node}}
t0,1{6,3,5} or t{6,3,5}

|(1)
40px
(3.3.3.3.3)

| -

| -

|(5)
50px
3.12.12

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!36

|cantellated order-5 hexagonal (sriphexah)
{{CDD|node_1|6|node|3|node_1|5|node}}
t0,2{6,3,5} or rr{6,3,5}

|(1)
40px
(3.5.3.5)

|(2)
40px
(5.4.4)

| -

|(2)
50px
3.4.6.4

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!37

|runcinated order-6 dodecahedral (sidpidohexah)
{{CDD|node_1|6|node|3|node|5|node_1}}
t0,3{6,3,5}

|(1)
40px
(5.5.5)

| -

|(6)
40px
(6.4.4)

|(1)
50px
(6)3

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!38

|cantellated order-6 dodecahedral (srihed)
{{CDD|node|6|node_1|3|node|5|node_1}}
t0,2{5,3,6} or rr{5,3,6}

|(2)
40px
(4.3.4.5)

| -

|(2)
40px
(6.4.4)

|(1)
50px
(3.6)2

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!39

|bitruncated order-6 dodecahedral (dohexah)
{{CDD|node|6|node_1|3|node_1|5|node}}
t1,2{6,3,5} or 2t{6,3,5}

|(2)
40px
(5.6.6)

| -

| -

|(2)
50px
(6)3

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!40

|truncated order-6 dodecahedral (thed)
{{CDD|node|6|node|3|node_1|5|node_1}}
t0,1{5,3,6} or t{5,3,6}

|(6)
40px
(3.10.10)

| -

| -

|(1)
50px
(3)6

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!41

|cantitruncated order-5 hexagonal (griphexah)
{{CDD|node_1|6|node_1|3|node_1|5|node}}
t0,1,2{6,3,5} or tr{6,3,5}

|(1)
40px
(5.6.6)

|(1)
40px
(5.4.4)

| -

|(2)
50px
4.6.10

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!42

|runcitruncated order-5 hexagonal (prihed)
{{CDD|node_1|6|node_1|3|node|5|node_1}}
t0,1,3{6,3,5}

|(1)
40px
(4.3.4.5)

|(1)
40px
(5.4.4)

|(2)
40px
(12.4.4)

|(1)
50px
3.12.12

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!43

|runcitruncated order-6 dodecahedral (priphaxh)
{{CDD|node_1|6|node|3|node_1|5|node_1}}
t0,1,3{5,3,6}

|(1)
40px
(3.10.10)

|(1)
40px
(10.4.4)

|(2)
40px
(6.4.4)

|(1)
50px
3.4.6.4

|80px

|120px

align=center BGCOLOR="#e0e0f0"

!44

|cantitruncated order-6 dodecahedral (grihed)
{{CDD|node|6|node_1|3|node_1|5|node_1}}
t0,1,2{5,3,6} or tr{5,3,6}

|(1)
40px
(4.6.10)

| -

|(2)
40px
(6.4.4)

|(1)
50px
(6)3

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!45

|omnitruncated order-6 dodecahedral (gidpidohaxh)
{{CDD|node_1|6|node_1|3|node_1|5|node_1}}
t0,1,2,3{6,3,5}

|(1)
40px
(4.6.10)

|(1)
40px
(10.4.4)

|(1)
40px
(12.4.4)

|(1)
50px
4.6.12

|80px

|120px

class="wikitable"

|+ Alternated forms

rowspan="2" | #

! rowspan="2" | Honeycomb name
Coxeter diagram
Schläfli symbol

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

! rowspan="2" |Vertex figure

! rowspan="2" |Picture

align=center

!0
{{CDD|node_n2|3|node_n3|5|node_n5}}

!1
{{CDD|node_n1|2|node_n3|5|node_n5}}

!2
{{CDD|node_n1|6|node_n2|2|node_n5}}

!3
{{CDD|node_n1|6|node_n2|3|node_n3}}

!Alt

align=center BGCOLOR="#e0f0f0"

| [145]

|alternated order-5 hexagonal (aphexah)
{{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}
h{6,3,5}

| -

| -

| -

|(20)
50px
(3)6

|(12)
40px
(3)5

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

|

align=center BGCOLOR="#e0f0f0"

|[146]

|cantic order-5 hexagonal (taphexah)
{{CDD|node_h1|6|node|3|node_1|5|node}} ↔ {{CDD|branch_10ru|split2|node_1|5|node}}
h2{6,3,5}

|(1)
40px
(3.5.3.5)

| -

|

|(2)
40px
(3.6.3.6)

|(2)
40px
(5.6.6)

|80px

|

align=center BGCOLOR="#e0f0f0"

|[147]

|runcic order-5 hexagonal (biraphexah)
{{CDD|node_h1|6|node|3|node|5|node_1}} ↔ {{CDD|branch_10ru|split2|node|5|node_1}}
h3{6,3,5}

|(1)
40px
(5.5.5)

|(1)
40px
(4.4.3)

|

|(1)
40px
(3.3.3.3.3.3)

|(3)
40px
(3.4.5.4)

|80px

|

align=center BGCOLOR="#e0f0f0"

|[148]

|runcicantic order-5 hexagonal (bitaphexah)
{{CDD|node_h1|6|node|3|node_1|5|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|5|node_1}}
h2,3{6,3,5}

|(1)
40px
(3.10.10)

|(1)
40px
(4.4.3)

|

| (1)
40px
(3.6.3.6)

|(2)
40px
(4.6.10)

|80px

|

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|snub rectified order-6 dodecahedral
{{CDD|node_h0|6|node_h|3|node_h|5|node_h}} ↔ {{CDD|branch_hh|split2|node_h|5|node_h}}
sr{5,3,6}

|40px
(3.3.5.3.5)
{{CDD|node_h|3|node_h|5|node_h}}

| -

|40px
(3.3.3.3)
{{CDD|node|6|node_h|2x|node_h}}

|40px
(3.3.3.3.3.3)
{{CDD|node|6|node_h|3|node_h}}

|40px
irr. tet

|

|

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub order-5 hexagonal
{{CDD|node_h|6|node_h|3|node_h|5|node_h}}
ht0,1,2,3{6,3,5}

|40px
(3.3.5.3.5)
{{CDD|node_h|3|node_h|5|node_h}}

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

|40px
(3.3.3.6)
{{CDD|node_h|6|node_h|2x|node_h}}

|40px
(3.3.6.3.6)
{{CDD|node_h|6|node_h|3|node_h}}

|40px
irr. tet

|

|

= [6,3,6] family =

There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or {{CDD|node|6|node|3|node|6|node}}

class=wikitable

!rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram
Schläfli symbol

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node_n2|3|node_n3|6|node_n4}}

!1
{{CDD|node_n1|2|node_n3|6|node_n4}}

!2
{{CDD|node_n1|6|node_n3|2|node_n4}}

!3
{{CDD|node_n1|6|node_n2|3|node_n3}}

align=center BGCOLOR="#f0e0e0"

!46

|order-6 hexagonal (hihexah)
{{CDD|node_1|6|node|3|node|6|node}}
{6,3,6}

| -

| -

| -

|(20)
40px {{CDD|node_1|3|node|6|node}}
(6.6.6)

|40px {{CDD|node_1|3|node|6|node}}
(3.3.3.3.3.3)

|120px

align=center BGCOLOR="#f0e0e0"

!47

|rectified order-6 hexagonal (rihihexah)
{{CDD|node|6|node_1|3|node|6|node}}
t1{6,3,6} or r{6,3,6}

|(2)
40px {{CDD|node_1|3|node|6|node}}
(3.3.3.3.3.3)

| -

| -

|(6)
40px
(3.6.3.6)

|80px {{CDD|node_1|2|node_1|6|node}}
(6.4.4)

|120px

align=center BGCOLOR="#f0e0e0"

!48

|truncated order-6 hexagonal (thihexah)
{{CDD|node_1|6|node_1|3|node|6|node}}
t0,1{6,3,6} or t{6,3,6}

|(1)
40px
(3.3.3.3.3.3)

| -

| -

|(6)
40px
(3.12.12)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!49

|cantellated order-6 hexagonal (srihihexah)
{{CDD|node_1|6|node|3|node_1|6|node}}
t0,2{6,3,6} or rr{6,3,6}

|(1)
40px
(3.6.3.6)

|(2)
40px
(4.4.6)

| -

|(2)
40px
(3.6.4.6)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!50

|Runcinated order-6 hexagonal (spiddihexah)
{{CDD|node_1|6|node|3|node|6|node_1}}
t0,3{6,3,6}

|(1)
40px
(6.6.6)

|(3)
40px
(4.4.6)

|(3)
40px
(4.4.6)

|(1)
40px
(6.6.6)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!51

|cantitruncated order-6 hexagonal (grihihexah)
{{CDD|node_1|6|node_1|3|node_1|6|node}}
t0,1,2{6,3,6} or tr{6,3,6}

|(1)
40px
(6.6.6)

|(1)
40px
(4.4.6)

| -

|(2)
40px
(4.6.12)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!52

|runcitruncated order-6 hexagonal (prihihexah)
{{CDD|node_1|6|node_1|3|node|6|node_1}}
t0,1,3{6,3,6}

|(1)
40px
(3.6.4.6)

|(1)
40px
(4.4.6)

|(2)
40px
(4.4.12)

|(1)
40px
(3.12.12)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!53

|omnitruncated order-6 hexagonal (gidpiddihexah)
{{CDD|node_1|6|node_1|3|node_1|6|node_1}}
t0,1,2,3{6,3,6}

|(1)
40px
(4.6.12)

|(1)
40px
(4.4.12)

|(1)
40px
(4.4.12)

|(1)
40px
(4.6.12)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

|[1]

|bitruncated order-6 hexagonal (hexah)
{{CDD|node_h0|6|node_1|3|node_1|6|node_h0}} ↔ {{CDD|node 1|6|node_g|3sg|node_g|3g|node_g}} ↔ {{CDD|branch_11|splitcross|branch_11}}
t1,2{6,3,6} or 2t{6,3,6}

|(2)
40px
(6.6.6)

| -

| -

|(2)
40px
(6.6.6)

|80px

|120px

class=wikitable

|+ Alternated forms

rowspan=2|#

!rowspan=2|Name of honeycomb
Coxeter diagram
Schläfli symbol

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node_n2|3|node_n3|6|node_n4}}

!1
{{CDD|node_n1|2|node_n3|6|node_n4}}

!2
{{CDD|node_n1|6|node_n3|2|node_n4}}

!3
{{CDD|node_n1|6|node_n2|3|node_n3}}

!Alt

align=center BGCOLOR="#e0f0f0"

|[47]

|rectified order-6 hexagonal (rihihexah)
{{CDD|node_h1|6|node|3|node|6|node_h1}} ↔ {{CDD|node|splitsplit1|branch4_11|splitsplit2|node}} ↔ {{CDD|node_h0|6|node_1|3|node|6|node_h0}}
q{6,3,6} = r{6,3,6}

|(2)
40px {{CDD|node_1|2|node_1|6|node}}
(3.3.3.3.3.3)

| -

| -

|(6)
40px
(3.6.3.6)

|

|80px {{CDD|node_1|2|node_1|6|node}}
(6.4.4)

|120px

align=center BGCOLOR="#e0f0f0"

| [54]

|triangular (trah)
({{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}) = {{CDD|node_1|3|node|6|node|3|node}}
h{6,3,6} = {3,6,3}

| -

| -

| -

|{{CDD|node_h|6|node|3|node}}
40px
(3.3.3.3.3.3)

|{{CDD|node_1|3|node|6|node}}
40px
(3.3.3.3.3.3)

|40px {{CDD|node_1|6|node|3|node}}
{6,3}

|120px

align=center BGCOLOR="#e0f0f0"

|[55]

|cantic order-6 hexagonal (ritrah)
( {{CDD|node_h1|6|node|3|node_1|6|node}} ↔ {{CDD|branch_10ru|split2|node_1|6|node}}) = {{CDD|node|3|node_1|6|node|3|node}}
h2{6,3,6} = r{3,6,3}

|(1)
40px
(3.6.3.6)

| -

|(2)
40px
(6.6.6)

|(2)
40px
(3.6.3.6)

|

|80px

|120px

align=center BGCOLOR="#e0f0f0"

|[149]

|runcic order-6 hexagonal
{{CDD|node_h1|6|node|3|node|6|node_1}} ↔ {{CDD|branch_10ru|split2|node|6|node_1}}
h3{6,3,6}

|(1)
40px
(6.6.6)

|(1)
40px
(4.4.3)

|(3)
40px
(3.4.6.4)

|(1)
40px
(3.3.3.3.3.3)

|

|80px

|

align=center BGCOLOR="#e0f0f0"

|[150]

|runcicantic order-6 hexagonal
{{CDD|node_h1|6|node|3|node_1|6|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|6|node_1}}
h2,3{6,3,6}

|(1)
40px
(3.12.12)

|(1)
40px
(4.4.3)

|(2)
40px
(4.6.12)

|(1)
40px
(3.6.3.6)

|

|80px

|

align=center BGCOLOR="#e0f0f0"

|[137]

|alternated hexagonal (ahexah)
({{CDD|node_h0|6|node_h|3|node_h|6|node_h0}} ↔ {{CDD|node h1|6|node_g|3sg|node_g|3g|node_g}} ↔ {{CDD|branch_hh|splitcross|branch_hh}}) = {{CDD|branch_10ru|split2|node|3|node}}
2s{6,3,6} = h{6,3,3}

|{{CDD|node_h|3|node_h|6|node}}
40px
(3.3.3.3.6)

| -

| -

|{{CDD|node|6|node_h|3|node_h}}
40px
(3.3.3.3.6)

|40px
+(3.3.3)

|40px {{CDD|node_1|3|node_1|3|node}}
(3.6.6)

|

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|snub rectified order-6 hexagonal
{{CDD|node_h|6|node_h|3|node_h|6|node}}
sr{6,3,6}

|{{CDD|node_h|3|node_h|6|node}}
40px
(3.3.3.3.3.3)

|{{CDD|node_h|2x|node_h|6|node}}
40px
(3.3.3.3)

| -

|{{CDD|node_h|6|node_h|3|node_h}}
40px
(3.3.3.3.6)

|40px
+(3.3.3)

|

|

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|alternated runcinated order-6 hexagonal
{{CDD|node_h|6|node|3|node|6|node_h}}
ht0,3{6,3,6}

|{{CDD|node|3|node|6|node_h}}
40px
(3.3.3.3.3.3)

|{{CDD|node|6|node_h|2x|node_h}}
40px
(3.3.3.3)

|{{CDD|node|6|node_h|2x|node_h}}
40px
(3.3.3.3)

|{{CDD|node_h|6|node|3|node}}
40px
(3.3.3.3.3.3)

|40px
+(3.3.3)

|

|

BGCOLOR="#e0f0f0" align=center

|Nonuniform

|omnisnub order-6 hexagonal
{{CDD|node_h|6|node_h|3|node_h|6|node_h}}
ht0,1,2,3{6,3,6}

|{{CDD|node_h|3|node_h|6|node_h}}
40px
(3.3.3.3.6)

|{{CDD|node_h|2x|node_h|6|node_h}}
40px
(3.3.3.6)

|{{CDD|node_h|6|node_h|2x|node_h}}
40px
(3.3.3.6)

|{{CDD|node_h|6|node_h|3|node_h}}
40px
(3.3.3.3.6)

|40px
+(3.3.3)

|

|

= [3,6,3] family =

There are 9 forms, generated by ring permutations of the Coxeter group: [3,6,3] or {{CDD|node|3|node|6|node|3|node}}

class="wikitable"

!rowspan=2|#

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

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

!0
{{CDD|node_n2|6|node_n3|3|node_n4}}

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

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

!3
{{CDD|node_n1|3|node_n2|6|node_n3}}

align=center BGCOLOR="#f0e0e0"

!54

|triangular (trah)
{{CDD|node_1|3|node|6|node|3|node}}
{3,6,3}

| -

| -

| -

|(∞)
40px
{3,6}

|40px {{CDD|node_1|6|node|3|node}}
{6,3}

|120px

align=center BGCOLOR="#f0e0e0"

!55

|rectified triangular (ritrah)
{{CDD|node|3|node_1|6|node|3|node}}
t1{3,6,3} or r{3,6,3}

|(2)
40px
(6)3

| -

| -

|(3)
40px
(3.6)2

|80px
(3.4.4)

|120px

align=center BGCOLOR="#f0e0e0"

!56

|cantellated triangular (sritrah)
{{CDD|node_1|3|node|6|node_1|3|node}}
t0,2{3,6,3} or rr{3,6,3}

|(1)
40px
(3.6)2

|(2)
40px
(4.4.3)

| -

|(2)
40px
(3.6.4.6)

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!57

|runcinated triangular (spidditrah)
{{CDD|node_1|3|node|6|node|3|node_1}}
t0,3{3,6,3}

|(1)
40px
(3)6

|(6)
40px
(4.4.3)

|(6)
40px
(4.4.3)

|(1)
40px
(3)6

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!58

|bitruncated triangular (ditrah)
{{CDD|node|3|node_1|6|node_1|3|node}}
t1,2{3,6,3} or 2t{3,6,3}

|(2)
40px
(3.12.12)

| -

| -

|(2)
40px
(3.12.12)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!59

|cantitruncated triangular (gritrah)
{{CDD|node_1|3|node_1|6|node_1|3|node}}
t0,1,2{3,6,3} or tr{3,6,3}

|(1)
40px
(3.12.12)

|(1)
40px
(4.4.3)

| -

|(2)
40px
(4.6.12)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

!60

|runcitruncated triangular (pritrah)
{{CDD|node_1|3|node_1|6|node|3|node_1}}
t0,1,3{3,6,3}

|(1)
40px
(3.6.4.6)

|(1)
40px
(4.4.3)

|(2)
40px
(4.4.6)

|(1)
40px
(6)3

|80px

|120px

align=center BGCOLOR="#e0f0e0"

!61

|omnitruncated triangular (gipidditrah)
{{CDD|node_1|3|node_1|6|node_1|3|node_1}}
t0,1,2,3{3,6,3}

|(1)
40px
(4.6.12)

|(1)
40px
(4.4.6)

|(1)
40px
(4.4.6)

|(1)
40px
(4.6.12)

|80px

|120px

align=center BGCOLOR="#f0e0e0"

|[1]

|truncated triangular (hexah)
{{CDD|node_1|3|node_1|6|node_g|3sg|node_g}} ↔ {{CDD|node_1|6|node_g|3sg|node_g|3g|node_g}} ↔ {{CDD|branch_11|splitcross|branch_11}}
t0,1{3,6,3} or t{3,6,3} = {6,3,3}

|(1)
40px
(6)3

| -

| -

|(3)
40px
(6)3

|80px {{CDD|node_1|3|node|3|node}}
{3,3}

|120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

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

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

!0
{{CDD|node_n2|6|node_n3|3|node_n4}}

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

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

!3
{{CDD|node_n1|3|node_n2|6|node_n3}}

!Alt

align=center BGCOLOR="#e0f0f0"

|[56]

|cantellated triangular (sritrah)
{{CDD|node_h|3|node_h|6|node_1|3|node}} = {{CDD|node_1|3|node|6|node_1|3|node}}
s2{3,6,3}

|(1)
40px
(3.6)2
{{CDD|node_h|6|node_1|3|node}}

| -

| -

|(2)
40px
(3.6.4.6)
{{CDD|node_h|3|node_h|6|node_1}}

|40px
(3.4.4)

|80px

|120px

align=center BGCOLOR="#e0f0f0"

|[60]

|runcitruncated triangular (pritrah)
{{CDD|node_h|3|node_h|6|node_1|3|node_1}} = {{CDD|node_1|3|node|6|node_1|3|node_1}}
s2,3{3,6,3}

|(1)
40px
(6)3
{{CDD|node_h|6|node_1|3|node_1}}

| -

|(1)
40px
(4.4.3)
{{CDD|node_h|3|node_h|2|node_1}}

|(1)
40px
(3.6.4.6)
{{CDD|node_h|3|node_h|6|node_1}}

|(2)
40px
(4.4.6)

|80px

|120px

align=center BGCOLOR="#e0f0f0"

|[137]

|alternated hexagonal (ahexah)
( {{CDD|node_h|3|node_h|6|node_g|3sg|node_g}} ↔ {{CDD|branch_hh|splitcross|branch_hh}} ) = ({{CDD|node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}})
s{3,6,3}

| 40px
(3)6
{{CDD|node_h|6|node|3|node}}

| -

| -

|40px
(3)6
{{CDD|node_h|3|node_h|6|node}}

|40px
+(3)3

|40px {{CDD|node_1|3|node_1|3|node}}
(3.6.6)

|

align=center BGCOLOR="#e0f0f0"

|Scaliform

|runcisnub triangular (pristrah)
{{CDD|node_h|3|node_h|6|node|3|node_1}}
s3{3,6,3}

|40px
r{6,3}
{{CDD|node_h|6|node|3|node_1}}

| -

|40px
(3.4.4)
{{CDD|node_h|3|node_h|2|node_1}}

|40px
(3)6
{{CDD|node_h|3|node_h|6|node}}

|40px
tricup

|

|

align=center BGCOLOR="#e0f0f0"

|Nonuniform

|omnisnub triangular tiling honeycomb (snatrah)
{{CDD|node_h|3|node_h|6|node_h|3|node_h}}
ht0,1,2,3{3,6,3}

|40px
(3.3.3.3.6)
{{CDD|node_h|6|node_h|3|node_h}}

|40px
(3)4
{{CDD|node_h|2x|node_h|3|node_h}}

|40px
(3)4
{{CDD|node_h|3|node_h|2x|node_h}}

|40px
(3.3.3.3.6)
{{CDD|node_h|3|node_h|6|node_h}}

|40px
+(3)3

|

|

= [4,4,3] family =

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

class="wikitable"

!rowspan=2|#

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

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

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

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

!2
{{CDD|node_n1|4|node_n2|2|node_n4}}

!3
{{CDD|node_n1|4|node_n2|4|node_n3}}

align=center BGCOLOR="#f0e0e0"

!62

square (squah)
{{CDD|node_1|4|node|4|node|3|node}} = {{CDD|node|4|node_1|4|node|4|node}}
{4,4,3}		---(6)
{{CDD|node_1|4|node|4|node}}
40px		80px {{CDD|node_1|4|node|3|node}}
Cube		120px
align=center BGCOLOR="#f0e0e0"

!63

rectified square (risquah)
{{CDD|node|4|node_1|4|node|3|node}} = {{CDD|node_1|4|node|4|node_1|4|node}}
t1{4,4,3} or r{4,4,3}		(2)
{{CDD|node_1|4|node|3|node}}
40px		--(3)
{{CDD|node|4|node_1|4|node}}
40px		80px
{{CDD|node_1|2|node_1|3|node}}
Triangular prism		120px
align=center BGCOLOR="#e0e0f0"

!64

rectified order-4 octahedral (rocth)
{{CDD|node|4|node|4|node_1|3|node}}
t1{3,4,4} or r{3,4,4}		(4)
{{CDD|node|4|node_1|3|node}}
40px		--(2)
{{CDD|node|4|node|4|node_1}}
40px		80px120px
align=center BGCOLOR="#e0e0f0"

!65

order-4 octahedral (octh)
{{CDD|node|4|node|4|node|3|node_1}}
{3,4,4}		(∞)
{{CDD|node|4|node|3|node_1}}
40px		---40px {{CDD|node|4|node|4|node_1}}120px
align=center BGCOLOR="#f0e0e0"

!66

truncated square (tisquah)
{{CDD|node_1|4|node_1|4|node|3|node}} = {{CDD|node_1|4|node_1|4|node_1|4|node}}
t0,1{4,4,3} or t{4,4,3}		(1)
{{CDD|node_1|4|node|3|node}}
40px		--(3)
{{CDD|node_1|4|node_1|4|node}}
40px		80px120px
align=center BGCOLOR="#e0e0f0"

!67

truncated order-4 octahedral (tocth)
{{CDD|node|4|node|4|node_1|3|node_1}}
t0,1{3,4,4} or t{3,4,4}		(4)
{{CDD|node|4|node_1|3|node_1}}
40px		--(1)
{{CDD|node|4|node|4|node_1}}
40px		80px120px
align=center BGCOLOR="#e0f0e0"

!68

bitruncated square (osquah)
{{CDD|node|4|node_1|4|node_1|3|node}}
t1,2{4,4,3} or 2t{4,4,3}		(2)
{{CDD|node_1|4|node_1|3|node}}
40px		--(2)
{{CDD|node|4|node_1|4|node_1}}
40px		80px120px
align=center BGCOLOR="#f0e0e0"

!69

cantellated square (srisquah)
{{CDD|node_1|4|node|4|node_1|3|node}}
t0,2{4,4,3} or rr{4,4,3}		(1)
{{CDD|node|4|node_1|3|node}}
40px		(2)
{{CDD|node_1|2|node_1|3|node}}
40px		-(2)
{{CDD|node_1|4|node|4|node_1}}
40px		80px120px
align=center BGCOLOR="#e0e0f0"

!70

cantellated order-4 octahedral (srocth)
{{CDD|node|4|node_1|4|node|3|node_1}}
t0,2{3,4,4} or rr{3,4,4}		(2)
{{CDD|node_1|4|node|3|node_1}}
40px		-(2)
{{CDD|node|4|node_1|2|node_1}}
40px		(1)
{{CDD|node|4|node_1|4|node}}
40px		80px120px
align=center BGCOLOR="#e0f0e0"

!71

runcinated square (sidposquah)
{{CDD|node_1|4|node|4|node|3|node_1}}
t0,3{4,4,3}		(1)
{{CDD|node|4|node|3|node_1}}
40px		(3)
{{CDD|node_1|2|node|3|node_1}}
40px		(3)
{{CDD|node_1|4|node|2|node_1}}
40px		(1)
{{CDD|node_1|4|node|4|node}}
40px		80px120px
align=center BGCOLOR="#f0e0e0"

!72

cantitruncated square (grisquah)
{{CDD|node_1|4|node_1|4|node_1|3|node}}
t0,1,2{4,4,3} or tr{4,4,3}		(1)
{{CDD|node_1|4|node_1|3|node}}
40px		(1)
{{CDD|node_1|2|node_1|3|node}}
40px		-(2)
{{CDD|node_1|4|node_1|4|node_1}}
40px		80px120px
align=center BGCOLOR="#e0e0f0"

!73

cantitruncated order-4 octahedral (grocth)
{{CDD|node|4|node_1|4|node_1|3|node_1}}
t0,1,2{3,4,4} or tr{3,4,4}		(2)
{{CDD|node_1|4|node_1|3|node_1}}
40px		-(1)
{{CDD|node|4|node_1|2|node_1}}
40px		(1)
{{CDD|node|4|node_1|4|node_1}}
40px		80px120px
align=center BGCOLOR="#f0e0e0"

!74

runcitruncated square (procth)
{{CDD|node_1|4|node_1|4|node|3|node_1}}
t0,1,3{4,4,3}		(1)
{{CDD|node_1|4|node|3|node_1}}
40px		(1)
{{CDD|node_1|2|node|3|node_1}}
40px		(2)
{{CDD|node_1|4|node_1|2|node_1}}
40px		(1)
{{CDD|node_1|4|node_1|4|node}}
40px		80px120px
align=center BGCOLOR="#e0e0f0"

!75

runcitruncated order-4 octahedral (prisquah)
{{CDD|node_1|4|node|4|node_1|3|node_1}}
t0,1,3{3,4,4}		(1)
{{CDD|node|4|node_1|3|node_1}}
40px		(2)
{{CDD|node_1|2|node_1|3|node_1}}
40px		(1)
{{CDD|node_1|4|node|2|node_1}}
40px		(1)
{{CDD|node_1|4|node|4|node_1}}
40px		80px120px
align=center BGCOLOR="#f0e0e0"

!76

omnitruncated square (gidposquah)
{{CDD|node_1|4|node_1|4|node_1|3|node_1}}
t0,1,2,3{4,4,3}		(1)
{{CDD|node_1|4|node_1|3|node_1}}
40px		(1)
{{CDD|node_1|2|node_1|3|node_1}}
40px		(1)
{{CDD|node_1|4|node_1|2|node_1}}
40px		(1)
{{CDD|node_1|4|node_1|4|node_1}}
40px		80px120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

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

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

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

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

!2
{{CDD|node_n1|4|node_n2|2|node_n4}}

!3
{{CDD|node_n1|4|node_n2|4|node_n3}}

!Alt

align=center BGCOLOR="#e0f0f0"

|[83]

alternated square
{{CDD|node_h1|4|node|4|node|3|node}} ↔ {{CDD|nodes_10ru|split2-44|node|3|node}}
h{4,4,3}		---(6)
{{CDD|node_h1|4|node|4|node}}
40px		(8)
{{CDD|node_1|4|node|3|node}}
40px		40px {{CDD|node|4|node_1|3|node}}
align=center BGCOLOR="#e0f0f0"

|[84]

cantic square
{{CDD|node_h1|4|node|4|node_1|3|node}} ↔ {{CDD|nodes_10ru|split2-44|node_1|3|node}}
h2{4,4,3}

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

| -

| -

|(2)
{{CDD|node_h1|4|node|3|node_1}}
40px

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

|80px

|

align=center BGCOLOR="#e0f0f0"

|[85]

runcic square
{{CDD|node_h1|4|node|4|node|3|node_1}} ↔ {{CDD|nodes_10ru|split2-44|node|3|node_1}}
h3{4,4,3}

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

| -

| -

|(1)
{{CDD|node_h1|4|node|4|node}}
40px.

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

|80px

|

align=center BGCOLOR="#e0f0f0"

|[86]

|runcicantic square
{{CDD|node_h1|4|node|4|node_1|3|node_1}} ↔ {{CDD|nodes_10ru|split2-44|node_1|3|node_1}}

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

| -

| -

|(1)
{{CDD|node_h1|4|node|4|node_1}}
40px

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

|80px

|

align=center BGCOLOR="#e0f0f0"

|[153]

alternated rectified square
{{CDD|node|4|node_h1|4|node|3|node}} ↔ {{CDD|nodes_10|2a2b-cross|nodes_10ru|split2|node}}
hr{4,4,3}		{{CDD|node_h1|4|node|3|node}}
40px		--{{CDD|node|4|node_h1|4|node}}
40px		{}x{3}
align=center BGCOLOR="#e0f0f0"

|157

{{CDD|node|4|node_h1|4|node|3|node_1}}{{CDD|node_h1|4|node|3|node_1}}
40px		--{{CDD|node|4|node_h1|4|node}}
40px		{}x{6}
align=center BGCOLOR="#e0f0f0"

|Scaliform

snub order-4 octahedral
{{CDD|node|4|node|4|node_h|3|node_h}} = {{CDD|nodes|split2-44|node_h|3|node_h}} = {{CDD|node|split1-44|nodes_hh|split2|node_h}}
s{3,4,4}		{{CDD|node|4|node_h|3|node_h}}
40px		--{{CDD|node|4|node|4|node_h}}
40px		{}v{4}
align=center BGCOLOR="#e0f0f0"

|Scaliform

runcisnub order-4 octahedral
{{CDD|node_1|4|node|4|node_h|3|node_h}}
s3{3,4,4}		{{CDD|node|4|node_h|3|node_h}}
40px		{{CDD|node_1|2|node_h|3|node_h}}
40px		{{CDD|node_1|4|node|2|node_h}}
40px		{{CDD|node_1|4|node|4|node_h}}
40px		cup-4
align=center BGCOLOR="#e0f0f0"

|152

snub square
{{CDD|node_h|4|node_h|4|node|3|node}} = {{CDD|node_h|4|node_h|4|node_h|4|node}}
s{4,4,3}		{{CDD|node_h|4|node|3|node}}
40px		--{{CDD|node_h|4|node_h|4|node}}
40px		{3,3}80px
align=center BGCOLOR="#e0f0f0"

|Nonuniform

snub rectified order-4 octahedral
{{CDD|node|4|node_h|4|node_h|3|node_h}}
sr{3,4,4}		{{CDD|node_h|4|node_h|3|node_h}}
40px		-{{CDD|node|4|node_h|2x|node_h}}
40px		{{CDD|node|4|node_h|4|node_h}}
40px		irr. {3,3}
align=center BGCOLOR="#e0f0f0"

|Nonuniform

alternated runcitruncated square
{{CDD|node_h|4|node|4|node_h|3|node_h}}
ht0,1,3{3,4,4}		{{CDD|node|4|node_h|3|node_h}}
40px		{{CDD|node_h|2x|node_h|3|node_h}}
40px		{{CDD|node|4|node_h|2x|node_h}}
40px		{{CDD|node_h|4|node|4|node_h}}
40px		irr. {}v{4}
align=center BGCOLOR="#e0f0f0"

|Nonuniform

omnisnub square
{{CDD|node_h|4|node_h|4|node_h|3|node_h}}
ht0,1,2,3{4,4,3}		{{CDD|node_h|4|node_h|3|node_h}}
40px		{{CDD|node_h|2x|node_h|3|node_h}}
40px		{{CDD|node_h|4|node_h|2x|node_h}}
40px		{{CDD|node_h|4|node_h|4|node_h}}
40px		irr. {3,3}

= [4,4,4] family =

There are 9 forms, generated by ring permutations of the Coxeter group: [4,4,4] or {{CDD|node|4|node|4|node|4|node}}.

class="wikitable"

!rowspan=2|#

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

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

!rowspan=2|Symmetry

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

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

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

!2
{{CDD|node_n1|4|node_n2|2|node_n4}}

!3
{{CDD|node_n1|4|node_n2|4|node_n3}}

align=center BGCOLOR="#f0e0e0"

!77

order-4 square (sisquah)
{{CDD|node_1|4|node|4|node|4|node}}
{4,4,4}		---{{CDD|node_1|4|node|4|node}}
40px		[4,4,4]{{CDD|node_1|4|node|4|node}}
40px
Cube		120px
align=center BGCOLOR="#f0e0e0"

!78

truncated order-4 square (tissish)
{{CDD|node_1|4|node_1|4|node|4|node}}
t0,1{4,4,4} or t{4,4,4}		{{CDD|node_1|4|node|4|node}}
40px		--{{CDD|node_1|4|node_1|4|node}}
40px		[4,4,4]80px120px
align=center BGCOLOR="#e0f0e0"

!79

bitruncated order-4 square (dish)
{{CDD|node|4|node_1|4|node_1|4|node}}
t1,2{4,4,4} or 2t{4,4,4}		{{CDD|node_1|4|node_1|4|node}}
40px		--{{CDD|node|4|node_1|4|node_1}}
40px		4,4,480px120px
align=center BGCOLOR="#e0f0e0"

!80

runcinated order-4 square (spiddish)
{{CDD|node_1|4|node|4|node|4|node_1}}
t0,3{4,4,4}		{{CDD|node|4|node|4|node_1}}
40px		{{CDD|node_1|2|node|4|node_1}}
40px		{{CDD|node_1|4|node|2|node_1}}
40px		{{CDD|node_1|4|node|4|node}}
40px		4,4,480px120px
align=center BGCOLOR="#f0e0e0"

!81

runcitruncated order-4 square (prissish)
{{CDD|node_1|4|node_1|4|node|4|node_1}}
t0,1,3{4,4,4}		{{CDD|node_1|4|node|4|node_1}}
40px		{{CDD|node_1|2|node|4|node_1}}
40px		{{CDD|node_1|4|node_1|2|node_1}}
40px		{{CDD|node_1|4|node_1|4|node}}
40px		[4,4,4]80px120px
align=center BGCOLOR="#e0f0e0"

!82

omnitruncated order-4 square (gipiddish)
{{CDD|node_1|4|node_1|4|node_1|4|node_1}}
t0,1,2,3{4,4,4}		{{CDD|node_1|4|node_1|4|node_1}}
40px		{{CDD|node_1|2|node_1|4|node_1}}
40px		{{CDD|node_1|4|node_1|2|node_1}}
40px		{{CDD|node_1|4|node_1|4|node_1}}
40px		4,4,480px120px
align=center BGCOLOR="#f0e0e0"

|[62]

square (squah)
{{CDD|node|4|node_1|4|node|4|node_h0}} ↔ {{CDD|node_1|4|node|4|node_g|3sg|node_g}}
t1{4,4,4} or r{4,4,4}		{{CDD|node_1|4|node|4|node}}
40px		--{{CDD|node|4|node_1|4|node}}
40px		[4,4,4]40px
Square tiling		120px
align=center BGCOLOR="#f0e0e0"

|[63]

rectified square (risquah)
{{CDD|node_1|4|node|4|node_1|4|node_h0}} ↔ {{CDD|node|4|node_1|4|node_g|3sg|node_g}}
t0,2{4,4,4} or rr{4,4,4}		{{CDD|node|4|node_1|4|node}}
40px		{{CDD|node_1|2|node_1|4|node}}
40px		-{{CDD|node_1|4|node|4|node_1}}
40px		[4,4,4]80px120px
align=center BGCOLOR="#f0e0e0"

|[66]

truncated order-4 square (tisquah)
{{CDD|node_1|4|node_1|4|node_1|4|node_h0}} ↔ {{CDD|node_1|4|node_1|4|node_g|3sg|node_g}}
t0,1,2{4,4,4} or tr{4,4,4}		{{CDD|node_1|4|node_1|4|node}}
40px		{{CDD|node_1|2|node_1|4|node}}
40px		-{{CDD|node_1|4|node_1|4|node_1}}
40px		[4,4,4]80px120px

class="wikitable"

|+ Alternated constructions

rowspan=2|#

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

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

!rowspan=2|Symmetry

!rowspan=2|Vertex figure

!rowspan=2|Picture

align=center

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

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

!2
{{CDD|node_n1|4|node_n2|2|node_n4}}

!3
{{CDD|node_n1|4|node_n2|4|node_n4}}

!Alt

align=center BGCOLOR="#e0f0f0"

| [62]

|Square (squah)
( {{CDD|node_h1|4|node|4|node|4|node_1}} ↔ {{CDD|nodes_10ru|split2-44|node|4|node_1}} ↔ {{CDD|nodes_11|split2-44|node|4|node}} ↔ {{CDD|node_h0|4|node_1|4|node|4|node}} ) = {{CDD|node_1|4|node|4|node|3|node}}

|40px
(4.4.4.4)

| -

| -

|

|40px
(4.4.4.4)

|[1+,4,4,4]
=[4,4,4]

|80px

|120px

align=center BGCOLOR="#e0f0f0"

|[63]

rectified square (risquah)
{{CDD|node_h|4|node_h|4|node_1|4|node}} = {{CDD|node_1|4|node|4|node_1|4|node}}
s2{4,4,4}		{{CDD|node|4|node_1|4|node}}
40px		{{CDD|node_1|2|node_1|4|node}}
40px		-{{CDD|node_1|4|node|4|node_1}}
40px		[4+,4,4]80px120px
align=center BGCOLOR="#e0f0f0"

|[77]

order-4 square (sisquah)
{{CDD|node_h1|4|node|4|node|4|node}} ↔ {{CDD|nodes_10ru|split2-44|node|4|node}} ↔ {{CDD|node_1|4|node|split1-44|nodes}} ↔ {{CDD|node_1|4|node|4|node|4|node_h0}}		---{{CDD|node_h1|4|node|4|node}}
40px		{{CDD|node_1|4|node|4|node}}
40px		[1+,4,4,4]
=[4,4,4]		{{CDD|node_1|4|node|4|node}}
40px
Cube		120px
align=center BGCOLOR="#e0f0f0"

| [78]

|truncated order-4 square (tissish)
{{CDD|node_h1|4|node|4|node_1|4|node}} ↔ {{CDD|nodes_10ru|split2-44|node_1|4|node}} ↔ {{CDD|node_1|4|node_1|split1-44|nodes}} ↔ {{CDD|node_1|4|node_1|4|node|4|node_h0}}

|40px
(4.8.8)

| -

|40px
(4.8.8)

| -

|40px
(4.4.4.4)

|[1+,4,4,4]
=[4,4,4]

|80px

| 120px

align=center BGCOLOR="#e0f0f0"

| [79]

|bitruncated order-4 square (dish)
{{CDD|node_h1|4|node|4|node_1|4|node_1}} ↔ {{CDD|nodes_10ru|split2-44|node_1|4|node_1}} ↔ {{CDD|nodes_11|split2-44|node_1|4|node}} ↔ {{CDD|node_h0|4|node_1|4|node_1|4|node}}

|40px
(4.8.8)

| -

| -

|40px
(4.8.8)

|40px
(4.8.8)

|[1+,4,4,4]
=[4,4,4]

|80px

|120px

align=center BGCOLOR="#e0f0f0"

|[81]

runcitruncated order-4 square tiling (prissish)
{{CDD|node_h|4|node_h|4|node_1|4|node_1}} = {{CDD|node_1|4|node|4|node_1|4|node_1}}
s2,3{4,4,4}		{{CDD|node_1|4|node|4|node_1}}
40px		{{CDD|node_1|2|node|4|node_1}}
40px		{{CDD|node_1|4|node_1|2|node_1}}
40px		{{CDD|node_1|4|node_1|4|node}}
40px		[4,4,4]80px120px
align=center BGCOLOR="#e0f0f0"

|[83]

alternated square
( {{CDD|node|4|node_h1|4|node|4|node}} ↔ {{CDD|node_1|ultra|node|4|node|4|node_1|ultra|node}} ) ↔ {{CDD|nodes_10ru|split2-44|node|3|node}}
hr{4,4,4}		{{CDD|node_h1|4|node|4|node}}
40px		--{{CDD|node|4|node_h1|4|node}}
40px		{{CDD|node_1|4|node|3|node}}40px[4,1+,4,4]40px
(4.3.4.3)
align=center BGCOLOR="#e0f0f0"

|[104]

quarter order-4 square
{{CDD|node_h1|4|node|4|node|4|node_h1}} ↔ {{CDD|label4|branch_11|4a4b|branch|label4}}
q{4,4,4}

|

1+,4,4,4,1+
=4[4]		80px

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

There are 5 forms, 1 unique, generated by ring permutations of the Coxeter group: {{CDD|label4|branch|4a4b|branch|label4}}. Repeat constructions are related as: {{CDD|node_c3|split1-44|nodeab_c1-2|split2-44|node_c3}} ↔ {{CDD|node_h0|4|node_c3|split1-44|nodeab_c1-2}}, {{CDD|node_c1|split1-44|nodeab_c2|split2-44|node_c1}} ↔ {{CDD|node_h0|4|node_c1|4|node_c2|4|node_h0}}, and {{CDD|label4|branch_c1|4-4|branch_c1|label4}} ↔ {{CDD|label4|branch_c1|4-4|nodes}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|node|4|node|4|node}}

!1
{{CDD|node|4|node|4|node}}

!2
{{CDD|node|4|node|4|node}}

!3
{{CDD|node|4|node|4|node}}

align=center

!104

|quarter order-4 square
{{CDD|label4|branch_10r|4a4b|branch_10l|label4}} ↔ {{CDD|node_h1|4|node|4|node|4|node_h1}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node}}

|40px
(4.4.4.4)
{{CDD|node|4|node|4|node_1}}

|40px
(4.4.4.4)
{{CDD|node_1|4|node|4|node}}

|40px
(4.8.8)
{{CDD|node|4|node_1|4|node_1}}

|80px

|

align=center

|[62]

|square (squah)
{{CDD|label4|branch_01r|4a4b|branch_10l|label4}} ↔ {{CDD|node_h0|4|node_1|4|node|4|node_h0}} ↔ {{CDD|node_1|4|node|4|node_g|3sg|node_g}}

|40px
(4.4.4.4)
{{CDD|node|4|node_1|4|node}}

|40px
(4.4.4.4)
{{CDD|node_1|4|node|4|node_1}}

|40px
(4.4.4.4)
{{CDD|node|4|node_1|4|node}}

|40px
(4.4.4.4)
{{CDD|node_1|4|node|4|node_1}}

|80px

|120px

align=center

|[77]

|order-4 square (sisquah)
({{CDD|label4|branch_10r|4a4b|branch|label4}} ↔ {{CDD|node_h0|4|node|split1-44|nodes_10lu}} ) = {{CDD|node_1|4|node|4|node|4|node}}

|40px
(4.4.4.4)
{{CDD|node_1|4|node|4|node}}

| -

|40px
(4.4.4.4)
{{CDD|node_1|4|node|4|node}}

|40px
(4.4.4.4)
{{CDD|node|4|node_1|4|node}}

| 40px
(4.4.4.4)

|120px

align=center

|[78]

|truncated order-4 square (tissish)
( {{CDD|label4|branch_11|4a4b|branch_10l|label4}} ↔ {{CDD|node_h0|4|node_1|split1-44|nodes_10lu}} ) = {{CDD|node_1|4|node_1|4|node|4|node}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node}}

|40px
(4.4.4.4)
{{CDD|node_1|4|node|4|node_1}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node_1}}

|80px

|120px

align=center

|[79]

|bitruncated order-4 square (dish)
{{CDD|label4|branch_11|4a4b|branch_11|label4}} ↔ {{CDD|node_h0|4|node_1|4|node_1|4|node_h0}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node_1}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node_1}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node_1}}

|40px
(4.8.8)
{{CDD|node_1|4|node_1|4|node_1}}

|80px

|120px

class="wikitable"

|+ Alternated forms

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

0
{{CDD|node|4|node|4|node}}

!1
{{CDD|node|4|node|4|node}}

!2
{{CDD|node|4|node|4|node}}

!3
{{CDD|node|4|node|4|node}}

!Alt

align=center

|[83]

|alternated square
({{CDD|node_h|split1-44|nodes|split2-44|node_h}} ↔ {{CDD|node_h0|4|node_h1|4|node|4|node_h0}} ↔ {{CDD|node_h1|4|node|4|node_g|3sg|node_g}}) = {{CDD|nodes_10ru|split2-44|node|3|node}}

|(6)
40px
(4.4.4.4)
{{CDD|node_h|4|node|4|node_h}}

|(6)
40px
(4.4.4.4)
{{CDD|node|4|node_h|4|node}}

|(6)
40px
(4.4.4.4)
{{CDD|node_h|4|node|4|node_h}}

|(6)
40px
(4.4.4.4)
{{CDD|node|4|node_h|4|node}}

|(8)
40px
(4.4.4)

|40px
(4.3.4.3)

align=center

|[77]

|alternated order-4 square (sisquah)
{{CDD|node_h1|split1-44|nodes|split2-44|node}} ↔ {{CDD|branchu_10|split2-44|node|split1-44|branchu_01}}

|
{{CDD|node_h|4|node|4|node}}

| -

|
{{CDD|node|4|node|4|node_h}}

|
{{CDD|node|4|node_h|4|node}}

|

|

align=center

|158

|cantic order-4 square
{{CDD|node_h1|split1-44|nodes|split2-44|node_1}} ↔ {{CDD|branchu_10|split2-44|node_1|split1-44|branchu_01}}

|
{{CDD|node_h|4|node|4|node_1}}

|
{{CDD|node|4|node_1|4|node}}

|
{{CDD|node_1|4|node|4|node_h}}

|
{{CDD|node|4|node_h|4|node}}

|

|

align=center

|Nonuniform

|cyclosnub square
{{CDD|label4|branch_hh|4a4b|branch|label4}}

|
{{CDD|node_h|4|node|4|node}}

|
{{CDD|node|4|node|4|node_h}}

|
{{CDD|node|4|node_h|4|node_h}}

|
{{CDD|node_h|4|node_h|4|node}}

|

|

align=center

|Nonuniform

|snub order-4 square
{{CDD|node_h|split1-44|nodes_hh|split2-44|node}}

|
{{CDD|node_h|4|node_h|4|node}}

|
{{CDD|node_h|4|node|4|node_h}}

|
{{CDD|node|4|node_h|4|node_h}}

|
{{CDD|node_h|4|node_h|4|node_h}}

|

|

align=center

|Nonuniform

|bisnub order-4 square
{{CDD|label4|branch_hh|4a4b|branch_hh|label4}} ↔ {{CDD|node_h0|4|node_h|4|node_h|4|node_h0}}

| 40px
(3.3.4.3.4)
{{CDD|node_h|4|node_h|4|node_h}}

| 40px
(3.3.4.3.4)
{{CDD|node_h|4|node_h|4|node_h}}

| 40px
(3.3.4.3.4)
{{CDD|node_h|4|node_h|4|node_h}}

| 40px
(3.3.4.3.4)
{{CDD|node_h|4|node_h|4|node_h}}

| 40px
+(3.3.3)

| 80px

= [(6,3,3,3)] family =

There are 9 forms, generated by ring permutations of the Coxeter group: {{CDD|label6|branch|3ab|branch}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

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

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

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

align=center

!105

|tetrahedral-hexagonal
{{CDD|label6|branch_10r|3ab|branch}}

| (4)
40px
(3.3.3)

| -

| (4)
40px
(6.6.6)

| (6)
40px
(3.6.3.6)

| 40px {{CDD|nodeb_1|3b|branch_10l}}
(3.4.3.4)

align=center

!106

|tetrahedral-triangular
{{CDD|label6|branch|3ab|branch_10l}}

|
40px
(3.3.3.3)

|
40px
(3.3.3)

| -

|
40px
(3.3.3.3.3.3)

|40px {{CDD|label6|branch_10r|3b|nodeb_1}}
(3.4.6.4)

align=center

!107

|cyclotruncated tetrahedral-hexagonal
{{CDD|label6|branch_10r|3ab|branch_10l}}

| (3)
40px
(3.6.6)

| (1)
40px
(3.3.3)

| (1)
40px
(6.6.6)

| (3)
40px
(6.6.6)

|80px

align=center

!108

|cyclotruncated hexagonal-tetrahedral
{{CDD|label6|branch_11|3ab|branch}}

| (1)
40px
(3.3.3)

| (1)
40px
(3.3.3)

| (4)
40px
(3.12.12)

| (4)
40px
(3.12.12)

|80px

align=center

!109

|cyclotruncated tetrahedral-triangular
{{CDD|label6|branch|3ab|branch_11}}

| (6)
40px
(3.6.6)

| (6)
40px
(3.6.6)

| (1)
40px
(3.3.3.3.3.3)

| (1)
40px
(3.3.3.3.3.3)

|80px

align=center

!110

|rectified tetrahedral-hexagonal
{{CDD|label6|branch_01r|3ab|branch_10l}}

| (1)
40px
(3.3.3.3)

| (2)
40px
(3.4.3.4)

| (1)
40px
(3.6.3.6)

| (2)
40px
(3.4.6.4)

|80px

align=center

!111

|truncated tetrahedral-hexagonal
{{CDD|label6|branch_11|3ab|branch_10l}}

| (1)
40px
(3.6.6)

| (1)
40px
(3.4.3.4)

| (1)
40px
(3.12.12)

| (2)
40px
(4.6.12)

|80px

align=center

!112

|truncated tetrahedral-triangular
{{CDD|label6|branch_10r|3ab|branch_11}}

| (2)
40px
(4.6.6)

| (1)
40px
(3.6.6)

| (1)
40px
(3.4.6.4)

| (1)
40px
(6.6.6)

|80px

align=center

!113

|omnitruncated tetrahedral-hexagonal
{{CDD|label6|branch_11|3ab|branch_11}}

| (1)
40px
(4.6.6)

| (1)
40px
(4.6.6)

| (1)
40px
(4.6.12)

| (1)
40px
(4.6.12)

|80px

class="wikitable"

|+ Alternated forms

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

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

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

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

!Alt

align=center

|Nonuniform

|omnisnub tetrahedral-hexagonal
{{CDD|label6|branch_hh|3ab|branch_hh}}

| 40px
(3.3.3.3.3)

| 40px
(3.3.3.3.3)

| 40px
(3.3.3.3.6)

| 40px
(3.3.3.3.6)

| 40px
+(3.3.3)

|80px

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

There are 9 forms, generated by ring permutations of the Coxeter group: {{CDD|label6|branch|3ab|branch|label4}}

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

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

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

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

align=center

!114

|octahedral-hexagonal
{{CDD|label6|branch_10r|3ab|branch|label4}}

| (6)
40px
(3.3.3.3)
{{CDD|nodea_1|3a|branch|label4}}

| -

| (8)
40px
(6.6.6)
{{CDD|label6|branch_01|3b|nodeb}}

| (12)
40px
(3.6.3.6)
{{CDD|label6|branch_10|3a|nodea}}

| 80px

align=center

!115

|cubic-triangular
{{CDD|label6|branch|3ab|branch_10l|label4}}

| (∞)
40px
(3.4.3.4)
{{CDD|nodea|3a|branch_10|label4}}

| (∞)
40px
(4.4.4)
{{CDD|nodeb|3b|branch_10l|label4}}

| -

| (∞)
40px
(3.3.3.3.3.3)
{{CDD|label6|branch|3a|nodea_1}}

|40px {{CDD|label6|branch_10r|3b|nodeb_1}}
(3.4.6.4)

align=center

!116

|cyclotruncated octahedral-hexagonal
{{CDD|label6|branch_10r|3ab|branch_10l|label4}}

| (3)
40px
(4.6.6)
{{CDD|nodea_1|3a|branch_10|label4}}

| (1)
40px
(4.4.4)
{{CDD|nodeb|3b|branch_10l|label4}}

| (1)
40px
(6.6.6)
{{CDD|label6|branch_10r|3b|nodeb}}

| (3)
40px
(6.6.6)
{{CDD|label6|branch_10|3a|nodea_1}}

|80px

align=center

!117

|cyclotruncated hexagonal-octahedral
{{CDD|label6|branch_11|3ab|branch|label4}}

| (1)
40px
(3.3.3.3)
{{CDD|nodea_1|3a|branch|label4}}

| (1)
40px
(3.3.3.3)
{{CDD|nodeb_1|3b|branch|label4}}

| (4)
40px
(3.12.12)
{{CDD|label6|branch_01|3b|nodeb_1}}

| (4)
40px
(3.12.12)
{{CDD|label6|branch_11|3a|nodea}}

|80px

align=center

!118

|cyclotruncated cubic-triangular
{{CDD|label6|branch|3ab|branch_11|label4}}

| (6)
40px
(3.8.8)
{{CDD|nodea|3a|branch_11|label4}}

| (6)
40px
(3.8.8)
{{CDD|nodeb|3b|branch_11|label4}}

| (1)
40px
(3.3.3.3.3.3)
{{CDD|label6|branch|3b|nodeb_1}}

| (1)
40px
(3.3.3.3.3.3)
{{CDD|label6|branch|3a|nodea_1}}

|80px

align=center

!119

|rectified octahedral-hexagonal
{{CDD|label6|branch_01r|3ab|branch_10l|label4}}

| (1)
40px
(3.4.3.4)
{{CDD|nodea|3a|branch_10|label4}}

| (2)
40px
(3.4.4.4)
{{CDD|nodeb_1|3b|branch_10l|label4}}

| (1)
40px
(3.6.3.6)
{{CDD|label6|branch_01|3b|nodeb}}

| (2)
40px
(3.4.6.4)
{{CDD|label6|branch_01r|3a|nodea_1}}

|80px

align=center

!120

|truncated octahedral-hexagonal
{{CDD|label6|branch_11|3ab|branch_10l|label4}}

| (1)
40px
(4.6.6)
{{CDD|nodea_1|3a|branch_10|label4}}

| (1)
40px
(3.4.4.4)
{{CDD|nodeb_1|3b|branch_10l|label4}}

| (1)
40px
(3.12.12)
{{CDD|label6|branch_11|3b|nodeb}}

| (2)
40px
(4.6.12)
{{CDD|label6|branch_11|3a|nodea_1}}

|80px

align=center

!121

|truncated cubic-triangular
{{CDD|label6|branch_10r|3ab|branch_11|label4}}

| (2)
40px
(4.6.8)
{{CDD|nodea_1|3a|branch_11|label4}}

| (1)
40px
(3.8.8)
{{CDD|nodeb|3b|branch_11|label4}}

| (1)
40px
(3.4.6.4)
{{CDD|label6|branch_10r|3b|nodeb_1}}

| (1)
40px
(6.6.6)
{{CDD|label6|branch_11|3a|nodea}}

|80px

align=center

!122

|omnitruncated octahedral-hexagonal
{{CDD|label6|branch_11|3ab|branch_11|label4}}

| (1)
40px
(4.6.8)
{{CDD|nodea_1|3a|branch_11|label4}}

| (1)
40px
(4.6.8)
{{CDD|nodeb_1|3b|branch_11|label4}}

| (1)
40px
(4.6.12)
{{CDD|label6|branch_11|3b|nodeb_1}}

| (1)
40px
(4.6.12)
{{CDD|label6|branch_11|3a|nodea_1}}

|80px

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

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

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

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

!Alt

align=center

|Nonuniform

|cyclosnub octahedral-hexagonal
{{CDD|label6|branch_h0r|3ab|branch_h0l|label4}}

| 40px
(3.3.3.3.3)
{{CDD|nodea_h|3a|branch_h0|label4}}

| 40px
(3.3.3)
{{CDD|nodeb|3b|branch_h0l|label4}}

| 40px
(3.3.3.3.3.3)
{{CDD|label6|branch_h0r|3b|nodeb}}

| 40px
(3.3.3.3.3.3)
{{CDD|label6|branch_h0|3a|nodea_h}}

| 40px
irr. {3,4}

| 80px

align=center

|Nonuniform

|omnisnub octahedral-hexagonal
{{CDD|label6|branch_hh|3ab|branch_hh|label4}}

| 40px
(3.3.3.3.4)
{{CDD|nodea_h|3a|branch_hh|label4}}

| 40px
(3.3.3.3.4)
{{CDD|nodeb_h|3b|branch_hh|label4}}

| 40px
(3.3.3.3.6)
{{CDD|label6|branch_hh|3b|nodeb_h}}

| 40px
(3.3.3.3.6)
{{CDD|label6|branch_hh|3a|nodea_h}}

| 40px
irr. {3,3}

| 80px

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

There are 9 forms, generated by ring permutations of the Coxeter group: {{CDD|label6|branch|3ab|branch|label5}}

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

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

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

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

align=center

!123

|icosahedral-hexagonal
{{CDD|label6|branch_10r|3ab|branch|label5}}

| (6)
50px
(3.3.3.3.3)

| -

| (8)
50px
(6.6.6)

| (12)
50px
(3.6.3.6)

|50px
3.4.5.4

|

align=center

!124

|dodecahedral-triangular
{{CDD|label6|branch|3ab|branch_10l|label5}}

| (30)
50px
(3.5.3.5)

| (20)
50px
(5.5.5)

| -

| (12)
50px
(3.3.3.3.3.3)

| 50px
(3.4.6.4)

|

align=center

!125

|cyclotruncated icosahedral-hexagonal
{{CDD|label6|branch_10r|3ab|branch_10l|label5}}

| (3)
50px
(5.6.6)

| (1)
50px
(5.5.5)

| (1)
50px
(6.6.6)

| (3)
50px
(6.6.6)

|80px

|

align=center

!126

|cyclotruncated hexagonal-icosahedral
{{CDD|label6|branch_11|3ab|branch|label5}}

| (1)
50px
(3.3.3.3.3)

| (1)
50px
(3.3.3.3.3)

| (5)
50px
(3.12.12)

| (5)
50px
(3.12.12)

|80px

|

align=center

!127

|cyclotruncated dodecahedral-triangular
{{CDD|label6|branch|3ab|branch_11|label5}}

| (6)
50px
(3.10.10)

| (6)
50px
(3.10.10)

| (1)
50px
(3.3.3.3.3.3)

| (1)
50px
(3.3.3.3.3.3)

|80px

|

align=center

!128

|rectified icosahedral-hexagonal
{{CDD|label6|branch_01r|3ab|branch_10l|label5}}

| (1)
50px
(3.5.3.5)

| (2)
50px
(3.4.5.4)

| (1)
50px
(3.6.3.6)

| (2)
50px
(3.4.6.4)

|80px

|

align=center

!129

|truncated icosahedral-hexagonal
{{CDD|label6|branch_11|3ab|branch_10l|label5}}

| (1)
50px
(5.6.6)

| (1)
50px
(3.5.5.5)

| (1)
50px
(3.12.12)

| (2)
50px
(4.6.12)

|80px

|

align=center

!130

|truncated dodecahedral-triangular
{{CDD|label6|branch_10r|3ab|branch_11|label5}}

| (2)
50px
(4.6.10)

| (1)
50px
(3.10.10)

| (1)
50px
(3.4.6.4)

| (1)
50px
(6.6.6)

|80px

|

align=center

!131

|omnitruncated icosahedral-hexagonal
{{CDD|label6|branch_11|3ab|branch_11|label5}}

| (1)
50px
(4.6.10)

| (1)
50px
(4.6.10)

| (1)
50px
(4.6.12)

| (1)
50px
(4.6.12)

|80px

|

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

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

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

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

!Alt

align=center

|Nonuniform

|omnisnub icosahedral-hexagonal
{{CDD|label6|branch_hh|3ab|branch_hh|label5}}

| 50px
(3.3.3.3.5)

| 50px
(3.3.3.3.5)

| 50px
(3.3.3.3.6)

| 50px
(3.3.3.3.6)

| 50px
+(3.3.3)

|80px

|

= [(6,3,6,3)] family =

There are 6 forms, generated by ring permutations of the Coxeter group: {{CDD|label6|branch|3ab|branch|label6}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|branch|label6}}

!1
{{CDD|nodeb|3b|branch|label6}}

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

align=center

!132

|hexagonal-triangular
{{CDD|label6|branch_10r|3ab|branch|label6}}

|40px
(3.3.3.3.3.3)

| -

|40px
(6.6.6)

|40px
(3.6.3.6)

|40px
(3.4.6.4)

|

align=center

!133

|cyclotruncated hexagonal-triangular
{{CDD|label6|branch_11|3ab|branch|label6}}

| (1)
40px
(3.3.3.3.3.3)

| (1)
40px
(3.3.3.3.3.3)

| (3)
40px
(3.12.12)

| (3)
40px
(3.12.12)

|80px

|

align=center

!134

|cyclotruncated triangular-hexagonal
{{CDD|label6|branch_01r|3ab|branch_10l|label6}}

| (1)
40px
(3.6.3.6)

| (2)
40px
(3.4.6.4)

| (1)
40px
(3.6.3.6)

| (2)
40px
(3.4.6.4)

|80px

|

align=center

!135

|rectified hexagonal-triangular
{{CDD|label6|branch_11|3ab|branch_10l|label6}}

| (1)
40px
(6.6.6)

| (1)
40px
(3.4.6.4)

| (1)
40px
(3.12.12)

| (2)
40px
(4.6.12)

|80px

|

align=center

!136

|truncated hexagonal-triangular
{{CDD|label6|branch_11|3ab|branch_11|label6}}

| (1)
40px
(4.6.12)

| (1)
40px
(4.6.12)

| (1)
40px
(4.6.12)

| (1)
40px
(4.6.12)

|80px

|

align=center

|[16]

|order-4 hexagonal tiling (shexah)
{{CDD|label6|branch_10r|3ab|branch_10l|label6}}
={{CDD|node_1|6|node|3|node|4|node}}

| (3)
40px
(6.6.6)

| (1)
40px
(6.6.6)

| (1)
40px
(6.6.6)

| (3)
40px
(6.6.6)

| 80px
(3.3.3.3)

|120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|branch|label6}}

!1
{{CDD|nodeb|3b|branch|label6}}

!2
{{CDD|label6|branch|3b|nodeb}}

!3
{{CDD|label6|branch|3a|nodea}}

!Alt

align=center

|[141]

|alternated order-4 hexagonal (ashexah)
{{CDD|label6|branch_h0r|3ab|branch_h0l|label6}} ↔ {{CDD|branch_10ru|split2|node|4|node}} ↔ {{CDD|node_h1|6|node|3|node|4|node}} ↔ {{CDD|node|split1|branch_10luru|split2|node}}

| 40px
(3.3.3.3.3.3)

| 40px
(3.3.3.3.3.3)

| 40px
(3.3.3.3.3.3)

| 40px
(3.3.3.3.3.3)

|40px
+(3.3.3.3)

| 40px
(4.6.6)

|

align=center

|Nonuniform

|cyclocantisnub hexagonal-triangular
{{CDD|branch_hh|6a6b|branch_10l}}

|

|

|

|

|

|

|

align=center

|Nonuniform

|cycloruncicantisnub hexagonal-triangular
{{CDD|branch_hh|6a6b|branch_11}}

|

|

|

|

|

|

|

align=center

|Nonuniform

|snub rectified hexagonal-triangular
{{CDD|label6|branch_hh|3ab|branch_hh|label6}}

| 40px
(3.3.3.3.6)

| 40px
(3.3.3.3.6)

| 40px
(3.3.3.3.6)

| 40px
(3.3.3.3.6)

|40px
+(3.3.3)

|80px

|

Loop-n-tail graphs

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

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [3,3[3]] or {{CDD|node|3|node|split1|branch}}. 7 are half symmetry forms of [3,3,6]: {{CDD|node_c1|3|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|3|node_c2|3|node_c3|6|node_h0}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

!rowspan=2|Picture

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

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

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

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

align=center

!137

|alternated hexagonal (ahexah)
({{CDD|branch_10ru|split2|node|3|node}} ↔ {{CDD|node_h1|6|node|3|node|3|node}}) = {{CDD|branch_hh|splitcross|branch_hh}}

| -

| -

|40px
(3.3.3)

|40px
(3.3.3.3.3.3)

|40px {{CDD|node_1|3|node_1|3|node}}
(3.6.6)

|

align=center

!138

|cantic hexagonal (tahexah)
{{CDD|branch_10ru|split2|node_1|3|node}} ↔ {{CDD|node_h1|6|node|3|node_1|3|node}}

|(1)
40px
(3.3.3.3)

| -

|(2)
40px
(3.6.6)

|(2)
40px
(3.6.3.6)

|80px

|

align=center

!139

|runcic hexagonal (birahexah)
{{CDD|branch_10ru|split2|node|3|node_1}} ↔ {{CDD|node_h1|6|node|3|node|3|node_1}}

|(1)
40px
(4.4.4)

|(1)
40px
(4.4.3)

|(3)
40px
(3.4.3.4)

| (1)
40px
(3.3.3.3.3.3)

|80px

|

align=center

!140

|runcicantic hexagonal (bitahexah)
{{CDD|branch_10ru|split2|node_1|3|node_1}} ↔ {{CDD|node_h1|6|node|3|node_1|3|node_1}}

|(1)
40px
(3.10.10)

|(1)
40px
(4.4.3)

|(2)
40px
(4.6.6)

| (1)
40px
(3.6.3.6)

|80px

|

align=center

|[2]

|rectified hexagonal (rihexah)
{{CDD|branch_11|split2|node|3|node}} ↔ {{CDD|node_h0|6|node_1|3|node|3|node}}

|(1)
40px
(3.3.3)

| -

|(1)
40px
(3.3.3)

| (6)
40px
(3.6.3.6)

|80px {{CDD|node_1|2|node_1|3|node}}
Triangular prism

|120px

align=center

|[3]

|rectified order-6 tetrahedral (rath)
{{CDD|branch|split2|node_1|3|node}} ↔ {{CDD|node_h0|6|node|3|node_1|3|node}}

|(2)
40px
(3.3.3.3)

| -

|(2)
40px
(3.3.3.3)

| (2)
40px
(3.3.3.3.3.3)

|80px {{CDD|node|6|node_1|2|node_1}}
Hexagonal prism

|120px

align=center

|[4]

|order-6 tetrahedral (thon)
{{CDD|branch|split2|node|3|node_1}} ↔ {{CDD|node_h0|6|node|3|node|3|node_1}}

|(4)
40px
(4.4.4)

| -

|(4)
40px
(4.4.4)

| -

|60px {{CDD|node|6|node|3|node_1}}

|120px

align=center

|[8]

|cantellated order-6 tetrahedral (srath)
{{CDD|branch_11|split2|node|3|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node|3|node_1}}

|(1)
40px
(3.3.3.3)

| (2)
40px
(4.4.6)

|(1)
40px
(3.3.3.3)

| (1)
40px
(3.6.3.6)

|80px

|120px

align=center

|[9]

|bitruncated order-6 tetrahedral (tehexah)
{{CDD|branch_11|split2|node_1|3|node}} ↔ {{CDD|node_h0|6|node_1|3|node_1|3|node}}

|(1)
40px
(3.6.6)

| -

|(1)
40px
(3.6.6)

| (2)
40px
(6.6.6)

|80px

|120px

align=center

|[10]

|truncated order-6 tetrahedral (tath)
{{CDD|branch|split2|node_1|3|node_1}} ↔ {{CDD|node_h0|6|node|3|node_1|3|node_1}}

|(2)
40px
(3.10.10)

| -

|(2)
40px
(3.10.10)

| (1)
40px
(3.6.3.6)

|80px

|120px

align=center

|[14]

|cantitruncated order-6 tetrahedral (grath)
{{CDD|branch_11|split2|node_1|3|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node_1|3|node_1}}

|(1)
40px
(4.6.6)

| (1)
40px
(4.4.6)

|(1)
40px
(4.6.6)

| (1)
40px
(6.6.6)

|80px

|120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

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

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

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

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

!Alt

align=center

|Nonuniform

|snub rectified order-6 tetrahedral
{{CDD|branch_hh|split2|node_h|3|node_h}} ↔ {{CDD|node_h0|6|node_h|3|node_h|3|node_h}}

| 40px
(3.3.3.3.3)

| 40px
(3.3.3.3)

| 40px
(3.3.3.3.3)

| 40px
(3.3.3.3.3.3)

| 40px
+(3.3.3)

| 80px

= [4,3<sup>[3]</sup>] family =

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [4,3[3]] or {{CDD|node|4|node|split1|branch}}. 7 are half symmetry forms of [4,3,6]: {{CDD|node_c1|4|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|4|node_c2|3|node_c3|6|node_h0}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|nodea|4a|nodea}}

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

!0'
{{CDD|nodea|3a|nodea|4a|nodea}}

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

align=center

!141

|alternated order-4 hexagonal (ashexah)
{{CDD|branch_10ru|split2|node|4|node}} ↔ {{CDD|node_h1|6|node|3|node|4|node}}

| -

| -

|40px
(3.3.3.3)

|40px
(3.3.3.3.3.3)

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

|

align=center

!142

|cantic order-4 hexagonal (tashexah)
{{CDD|branch_10ru|split2|node_1|4|node_h0}} ↔ {{CDD|node_h1|6|node|3|node_1|4|node_h0}} ↔ {{CDD|node_1|split1|branch_10luru|split2|node_1}}

|(1)
40px
(3.4.3.4)

| -

|(2)
40px
(4.6.6)

|(2)
40px
(3.6.3.6)

|80px

|

align=center

!143

|runcic order-4 hexagonal (birashexah)
{{CDD|branch_10ru|split2|node|4|node_1}} ↔ {{CDD|node_h1|6|node|3|node|4|node_1}}

|(1)
40px
(4.4.4)

|(1)
40px
(4.4.3)

|(3)
40px
(3.4.4.4)

| (1)
40px
(3.3.3.3.3.3)

|80px

|

align=center

!144

|runcicantic order-4 hexagonal (bitashexah)
{{CDD|branch_10ru|split2|node_1|4|node_1}} ↔ {{CDD|node_h1|6|node|3|node_1|4|node_1}}

|(1)
40px
(3.8.8)

|(1)
40px
(4.4.3)

|(2)
40px
(4.6.8)

| (1)
40px
(3.6.3.6)

|80px

|

align=center

|[16]

|order-4 hexagonal (shexah)
{{CDD|branch|split2|node|4|node_1}} ↔ {{CDD|node_h0|6|node|3|node|4|node_1}}

|(4)
40px
(4.4.4)

| -

|(4)
40px
(4.4.4)

| -

|80px

|120px

align=center

|[17]

|rectified order-4 hexagonal (rishexah)
{{CDD|branch_11|split2|node|4|node}} ↔ {{CDD|node_h0|6|node_1|3|node|4|node}}

|(1)
40px
(3.3.3.3)

| -

|(1)
40px
(3.3.3.3)

| (6)
40px
(3.6.3.6)

|80px

|120px

align=center

|[18]

|rectified order-6 cubic (rihach)
{{CDD|branch|split2|node_1|4|node}} ↔ {{CDD|node_h0|6|node|3|node_1|4|node}}

|(2)
40px
(3.4.3.4)

| -

|(2)
40px
(3.4.3.4)

| (2)
40px
(3.3.3.3.3.3)

|80px

|120px

align=center

| [21]

|bitruncated order-4 hexagonal (chexah)
{{CDD|branch_11|split2|node_1|4|node}} ↔ {{CDD|node_h0|6|node_1|3|node_1|4|node}}

|(1)
40px
(4.6.6)

| -

|(1)
40px
(4.6.6)

| (2)
40px
(6.6.6)

|80px

|120px

align=center

|[22]

|truncated order-6 cubic (thach)
{{CDD|branch|split2|node_1|4|node_1}} ↔ {{CDD|node_h0|6|node|3|node_1|4|node_1}}

|(2)
40px
(3.8.8)

| -

|(2)
40px
(3.8.8)

| (1)
40px
(3.6.3.6)

|80px

|120px

align=center

|[23]

|cantellated order-4 hexagonal (srishexah)
{{CDD|branch_11|split2|node|4|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node|4|node_1}}

|(1)
40px
(3.4.4.4)

| (2)
40px
(4.4.6)

|(1)
40px
(3.4.4.4)

| (1)
40px
(3.6.3.6)

|80px

|120px

align=center

|[26]

|cantitruncated order-4 hexagonal (grishexah)
{{CDD|branch_11|split2|node_1|4|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node_1|4|node_1}}

|(1)
40px
(4.6.8)

| (1)
40px
(4.4.6)

|(1)
40px
(4.6.8)

| (1)
40px
(6.6.6)

|80px

|120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

0
{{CDD|nodea|3a|nodea|4a|nodea}}

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

!0'
{{CDD|nodea|3a|nodea|4a|nodea}}

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

!Alt

align=center

|Nonuniform

|snub rectified order-4 hexagonal
{{CDD|branch_hh|split2|node_h|4|node_h}} ↔ {{CDD|node_h0|6|node_h|3|node_h|4|node_h}}

| 40px
(3.3.3.3.4)

| 40px
(3.3.3.3)

| 40px
(3.3.3.3.4)

| 40px
(3.3.3.3.3.3)

| 40px
+(3.3.3)

|

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

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [5,3[3]] or {{CDD|node|5|node|split1|branch}}. 7 are half symmetry forms of [5,3,6]: {{CDD|node_c1|5|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|5|node_c2|3|node_c3|6|node_h0}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

!rowspan=2|Picture

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

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

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

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

align=center

!145

|alternated order-5 hexagonal (aphexah)
{{CDD|branch_10ru|split2|node|5|node}} ↔ {{CDD|node_h1|6|node|3|node|5|node}}

| -

| -

|40px
(3.3.3.3.3)

|40px
(3.3.3.3.3.3)

|40px {{CDD|node|5|node_1|3|node}}
(3.6.3.6)

|

align=center

!146

|cantic order-5 hexagonal (taphexah)
{{CDD|branch_10ru|split2|node_1|5|node}} ↔ {{CDD|node_h1|6|node|3|node_1|5|node}}

|(1)
40px
(3.5.3.5)

| -

|(2)
40px
(5.6.6)

|(2)
40px
(3.6.3.6)

|80px

|

align=center

!147

|runcic order-5 hexagonal (biraphexah)
{{CDD|branch_10ru|split2|node|5|node_1}} ↔ {{CDD|node_h1|6|node|3|node|5|node_1}}

|(1)
40px
(5.5.5)

|(1)
40px
(4.4.3)

|(3)
40px
(3.4.5.4)

| (1)
40px
(3.3.3.3.3.3)

|80px

|

align=center

!148

|runcicantic order-5 hexagonal (bitaphexah)
{{CDD|branch_10ru|split2|node_1|5|node_1}} ↔ {{CDD|node_h1|6|node|3|node_1|5|node_1}}

|(1)
40px
(3.10.10)

|(1)
40px
(4.4.3)

|(2)
40px
(4.6.10)

| (1)
40px
(3.6.3.6)

|80px

|

align=center

|[32]

|rectified order-5 hexagonal (riphexah)
{{CDD|branch_11|split2|node|5|node}} ↔ {{CDD|node_h0|6|node_1|3|node|5|node}}

|(1)
40px
(3.3.3.3.3)

| -

|(1)
40px
(3.3.3.3.3)

| (6)
40px
(3.6.3.6)

|80px

|120px

align=center

|[33]

|rectified order-6 dodecahedral (rihed)
{{CDD|branch|split2|node_1|5|node}} ↔ {{CDD|node_h0|6|node|3|node_1|5|node}}

|(2)
40px
(3.5.3.5)

| -

|(2)
40px
(3.5.3.5)

| (2)
40px
(3.3.3.3.3.3)

|80px

|120px

align=center

| [34]

|Order-5 hexagonal (hedhon)
{{CDD|branch|split2|node|5|node_1}} ↔ {{CDD|node_h0|6|node|3|node|5|node_1}}

|(4)
40px
(5.5.5)

| -

|(4)
40px
(5.5.5)

| -

|80px

|120px

align=center

| [40]

|truncated order-6 dodecahedral (thed)
{{CDD|branch|split2|node_1|5|node_1}} ↔ {{CDD|node_h0|6|node|3|node_1|5|node_1}}

|(2)
40px
(3.10.10)

| -

|(2)
40px
(3.10.10)

| (1)
40px
(3.6.3.6)

|80px

|120px

align=center

| [36]

|cantellated order-5 hexagonal (sriphexah)
{{CDD|branch_11|split2|node|5|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node|5|node_1}}

|(1)
40px
(3.4.5.4)

| (2)
40px
(6.4.4)

|(1)
40px
(3.4.5.4)

| (1)
40px
(3.6.3.6)

|80px

|120px

align=center

|[39]

|bitruncated order-5 hexagonal (dohexah)
{{CDD|branch_11|split2|node_1|5|node}} ↔ {{CDD|node_h0|6|node_1|3|node_1|5|node}}

|(1)
40px
(5.6.6)

| -

|(1)
40px
(5.6.6)

|(2)
40px
(6.6.6)

|80px

|120px

align=center

|[41]

|cantitruncated order-5 hexagonal (griphexah)
{{CDD|branch_11|split2|node_1|5|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node_1|5|node_1}}

|(1)
40px
(4.6.10)

| (1)
40px
(6.4.4)

|(1)
40px
(4.6.10)

| (1)
40px
(6.6.6)

|80px

|120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

!rowspan=2|Picture

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

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

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

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

!Alt

align=center

|Nonuniform

|snub rectified order-5 hexagonal
{{CDD|branch_hh|split2|node_h|5|node_h}} ↔ {{CDD|node_h0|6|node_h|3|node_h|5|node_h}}

| 40px
(3.3.3.3.5)

| 40px
(3.3.3)

| 40px
(3.3.3.3.5)

| 40px
(3.3.3.3.3.3)

| 40px
+(3.3.3)

|

|

= [6,3<sup>[3]</sup>] family =

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [6,3[3]] or {{CDD|branch|split2|node|6|node}}. 7 are half symmetry forms of [6,3,6]: {{CDD|node_c1|6|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|6|node_c2|3|node_c3|6|node_h0}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|nodea|6a|nodea}}

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

!0'
{{CDD|nodea|3a|nodea|6a|nodea}}

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

align=center

!149

|runcic order-6 hexagonal
{{CDD|branch_10ru|split2|node|6|node_1}} ↔ {{CDD|node_h1|6|node|3|node|6|node_1}}

|(1)
40px
(6.6.6)

|(1)
40px
(4.4.3)

|(3)
40px
(3.4.6.4)

|(1)
40px
(3.3.3.3.3.3)

|80px

|

align=center

!150

|runcicantic order-6 hexagonal
{{CDD|branch_10ru|split2|node_1|6|node_1}} ↔ {{CDD|node_h1|6|node|3|node_1|6|node_1}}

|(1)
40px
(3.12.12)

|(1)
40px
(4.4.3)

|(2)
40px
(4.6.12)

|(1)
40px
(3.6.3.6)

|80px

|

align=center

| [1]

|hexagonal (hexah)
{{CDD|branch_11|split2|node_1|6|node_h0}} ↔ {{CDD|node_h0|6|node_1|3|node_1|6|node_h0}} ↔ {{CDD|branch_11|splitcross|branch_11}} ↔ {{CDD|node_1|6|node_g|3sg|node_g|3g|node_g}}

| (1)
40px
(6.6.6)

| -

| (1)
40px
(6.6.6)

| (2)
40px
(6.6.6)

| 80px

|120px

align=center

| [46]

|order-6 hexagonal (hihexah)
{{CDD|branch|split2|node|6|node_1}} ↔ {{CDD|node_h0|6|node|3|node|6|node_1}}

| (4)
40px
(6.6.6)

| -

| (4)
40px
(6.6.6)

| -

|40px

|120px

align=center

| [47]

|rectified order-6 hexagonal (rihihexah)
{{CDD|branch|split2|node_1|6|node}} ↔ {{CDD|node_h0|6|node|3|node_1|6|node}}

| (2)
40px
(3.6.3.6)

| -

| (2)
40px
(3.6.3.6)

| (2)
40px
(3.3.3.3.3.3)

| 80px

|120px

align=center

| [47]

|rectified order-6 hexagonal (rihihexah)
{{CDD|branch_11|split2|node|6|node}} ↔ {{CDD|node_h0|6|node_1|3|node|6|node}}

| (1)
40px
(3.3.3.3.3.3)

| -

| (1)
40px
(3.3.3.3.3.3)

| (6)
40px
(3.6.3.6)

| 80px

|120px

align=center

| [48]

|truncated order-6 hexagonal (thihexah)
{{CDD|branch|split2|node_1|6|node_1}} ↔ {{CDD|node_h0|6|node|3|node_1|6|node_1}}

| (2)
40px
(3.12.12)

| -

| (2)
40px
(3.12.12)

| (1)
40px
(3.3.3.3.3.3)

| 80px

|120px

align=center

| [49]

|cantellated order-6 hexagonal (srihihexah)
{{CDD|branch_11|split2|node|6|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node|6|node_1}}

| (1)
40px
(3.4.6.4)

| (2)
40px
(6.4.4)

| (1)
40px
(3.4.6.4)

| (1)
40px
(3.6.3.6)

| 80px

|120px

align=center

| [51]

|cantitruncated order-6 hexagonal (grihihexah)
{{CDD|branch_11|split2|node_1|6|node_1}} ↔ {{CDD|node_h0|6|node_1|3|node_1|6|node_1}}

| (1)
40px
(4.6.12)

| (1)
40px
(6.4.4)

| (1)
40px
(4.6.12)

| (1)
40px
(6.6.6)

| 80px

|120px

align=center

|[54]

|triangular tiling honeycomb (trah)
( {{CDD|branch_10ru|split2|node|6|node}} ↔ {{CDD|node_h1|6|node|3|node|6|node}} ) = {{CDD|node_1|3|node|6|node|3|node}}

| -

| -

|40px
(3.3.3.3.3.3)

|40px
(3.3.3.3.3.3)

|40px {{CDD|node|6|node_1|3|node_1}}
(6.6.6)

|120px

align=center

|[55]

|cantic order-6 hexagonal (ritrah)
( {{CDD|branch_10ru|split2|node_1|6|node}} ↔ {{CDD|node_h1|6|node|3|node_1|6|node}} ) = {{CDD|node|3|node_1|6|node|3|node}}

|(1)
40px
(3.6.3.6)

| -

|(2)
40px
(6.6.6)

|(2)
40px
(3.6.3.6)

|80px

|120px

class="wikitable"

|+ Alternated forms

rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|vertex figure

!rowspan=2|Picture

0
{{CDD|nodea|3a|nodea|6a|nodea}}

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

!0'
{{CDD|nodea|3a|nodea|6a|nodea}}

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

!Alt

align=center

| [54]

|triangular tiling honeycomb (trah)
( {{CDD|branch|split2|node|6|node_h1}} ↔ {{CDD|node_h0|6|node|3|node|6|node_h1}} ↔ {{CDD|node_h0|6|node|split1|branch_10lu}} ) = {{CDD|node_1|3|node|6|node|3|node}}

|40px
{{CDD|node|3|node|6|node_h1}}

| -

|40px
{{CDD|node|3|node|6|node_h1}}

| -

|40px

|40px {{CDD|node|6|node_1|3|node_1}}
(6.6.6)

|120px

align=center

|[137]

|alternated hexagonal (ahexah)
( {{CDD|branch_hh|split2|node_h|6|node}} ↔ {{CDD|node_h0|6|node_h|3|node_h|6|node}} ) = ( {{CDD|node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}} )

|40px
{{CDD|node_h|3|node_h|6|node}}

| -

|40px
{{CDD|node_h|3|node_h|6|node}}

|40px
{{CDD|branch_hh|split2|node_h}}

| 40px
+(3.6.6)

|40px {{CDD|node_1|3|node_1|3|node}}
(3.6.6)

|

align=center

|[47]

|rectified order-6 hexagonal (rihihexah)
{{CDD|branch_10ru|split2|node|6|node_h1}} ↔ {{CDD|node_h1|6|node|3|node|6|node_h1}} ↔ {{CDD|node|splitsplit1|branch4_11|splitsplit2|node}} ↔ {{CDD|node|6|node_1|3|node|6|node}}

|40px
(3.6.3.6)

| -

|40px
(3.6.3.6)

|40px
(3.3.3.3.3.3)

|

| 80px

|120px

align=center

|[55]

|cantic order-6 hexagonal (ritrah)
( {{CDD|branch_11|split2|node|6|node_h1}} ↔ {{CDD|node_h0|6|node_1|3|node|6|node_h1}} ) = ( {{CDD|node_h0|6|node_1|split1|branch_10lu}} ↔ {{CDD|node_1|splitsplit1|branch4_11|splitsplit2|node}} ) = {{CDD|node|3|node_1|6|node|3|node}}

|(1)
40px
(3.6.3.6)

| -

|(2)
40px
(6.6.6)

|(2)
40px
(3.6.3.6)

|

|80px

|120px

align=center

|Nonuniform

|snub rectified order-6 hexagonal
{{CDD|branch_hh|split2|node_h|6|node_h}} ↔ {{CDD|node_h0|6|node_h|3|node_h|6|node_h}}

|{{CDD|node_h|3|node_h|6|node_h}}
40px
(3.3.3.3.6)

|{{CDD|branch_hh|2x|node_h}}
40px
(3.3.3.3)

|{{CDD|node_h|3|node_h|6|node_h}}
40px
(3.3.3.3.6)

|{{CDD|branch_hh|split2|node_h}}
40px
(3.3.3.3.3.3)

| 40px
+(3.3.3)

|

|

Multicyclic graphs

= [3<sup>[ ]×[ ]</sup>] family =

There are 8 forms, 1 unique, generated by ring permutations of the Coxeter group: {{CDD|node|split1|branch|split2|node}}. Two are duplicated as {{CDD|node_c1|split1-44|branch_c3|split2|node_c2}} ↔ {{CDD|node_h0|6|node_c3|split1|nodeab_c1-2}}, two as {{CDD|node_c3|split1-44|branch_c1-2|split2|node_c3}} ↔ {{CDD|node_h0|4|node_c3|split1|branch_c1-2}}, and three as {{CDD|node_c2|split1|branch_c1|split2|node_c2}} ↔ {{CDD|node_h0|6|node_c1|3|node_c2|4|node_h0}}.

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|branch|split2|node}}

!1
{{CDD|node|3|node|3|node}}

!2
{{CDD|node|3|node|3|node}}

!3
{{CDD|node|split1|branch}}

align=center

!151

|Quarter order-4 hexagonal (quishexah)
{{CDD|node_1|split1|branch_10luru|split2|node}} ↔ {{CDD|node_h1|6|node|3|node|4|node_h1}}

|{{CDD|branch_10ru|split2|node}}
40px

|{{CDD|node_1|3|node|3|node}}
40px

|{{CDD|node_1|3|node_1|3|node}}
40px

|{{CDD|node_1|split1|branch_10lu}}
40px

|80px

|

align=center

|[17]

|rectified order-4 hexagonal (rishexah)
{{CDD|node|split1|branch_11|split2|node}} ↔ {{CDD|node_h0|6|node_1|split1|nodes}} ↔ {{CDD|node_h0|4|node|split1|branch_11}} ↔ {{CDD|node_h0|6|node_1|3|node|4|node_h0}}

|{{CDD|branch_11|split2|node}}
40px

|{{CDD|node|3|node_1|3|node}}
40px

|{{CDD|node|3|node_1|3|node}}
40px

|{{CDD|node|split1|branch_11}}
40px

|60px {{CDD|node_1|2|node_1|4|node}}
(4.4.4)

|120px

align=center

|[18]

|rectified order-6 cubic (rihach)
{{CDD|node_1|split1|branch|split2|node_1}} ↔ {{CDD|node_h0|6|node|split1|nodes_11}} ↔ {{CDD|node_h0|4|node_1|split1|branch}} ↔ {{CDD|node_h0|6|node|3|node_1|4|node_h0}}

|{{CDD|branch|split2|node_1}}
40px

|{{CDD|node_1|3|node|3|node_1}}
40px

|{{CDD|node_1|3|node|3|node_1}}
40px

|{{CDD|node_1|split1|branch}}
40px

|60px {{CDD|node|6|node_1|2|node_1}}
(6.4.4)

|120px

align=center

|[21]

|bitruncated order-6 cubic (chexah)
{{CDD|node_1|split1|branch_11|split2|node_1}} ↔ {{CDD|node_h0|6|node_1|split1|nodes_11}} ↔ {{CDD|node_h0|4|node_1|split1|branch_11}} ↔ {{CDD|node_h0|6|node_1|3|node_1|4|node_h0}}

|{{CDD|branch_11|split2|node_1}}
40px

|{{CDD|node_1|3|node_1|3|node_1}}
40px

|{{CDD|node_1|3|node_1|3|node_1}}
40px

|{{CDD|node_1|split1|branch_11}}
40px

|60px

|120px

align=center

|[87]

|alternated order-6 cubic (ahach)
{{CDD|node_1|split1|branch|split2|node}} ↔ {{CDD|node_h0|6|node|split1|nodes_10lu}} ↔ {{CDD|node_h0|6|node|3|node|4|node_h1}}

| -

|{{CDD|node_1|3|node|3|node}}
40px

|{{CDD|node_1|3|node|3|node}}
40px

|{{CDD|node_1|split1|branch}}
40px

|40px {{CDD|branch_11|split2|node}}
(3.6.3.6)

|

align=center

|[88]

|cantic order-6 cubic (tachach)
{{CDD|node_1|split1|branch_11|split2|node}} ↔ {{CDD|node_h0|6|node_1|split1|nodes_10lu}} ↔ {{CDD|node_h0|6|node_1|3|node|4|node_h1}}

|{{CDD|branch_11|split2|node}}
40px

|{{CDD|node_1|3|node_1|3|node}}
40px

|{{CDD|node_1|3|node_1|3|node}}
40px

|{{CDD|node_1|split1|branch_11}}
40px

|60px

|

align=center

|[141]

|alternated order-4 hexagonal (ashexah)
{{CDD|node|split1|branch_10luru|split2|node}} ↔ {{CDD|node_h0|4|node|split1|branch_10lu}} ↔ {{CDD|node_h0|4|node|3|node|6|node_h1}}

|{{CDD|branch_10ru|split2|node}}
40px

| -

|{{CDD|node|3|node_1|3|node}}
40px

|{{CDD|node|split1|branch_10lu}}
40px

|40px {{CDD|node_1|3|node_1|3|node_1}}
(4.6.6)

|

align=center

|[142]

|cantic order-4 hexagonal (tashexah)
{{CDD|node_1|split1|branch_10luru|split2|node_1}} ↔ {{CDD|node_h0|4|node_1|split1|branch_10lu}} ↔ {{CDD|node_h0|4|node_1|3|node|6|node_h1}}

|{{CDD|branch_10ru|split2|node_1}}
40px

|{{CDD|node_1|3|node|3|node_1}}
40px

|{{CDD|node_1|3|node_1|3|node_1}}
40px

|{{CDD|node_1|split1|branch_10lu}}
40px

|80px

|

class="wikitable"

!rowspan=2|#

!rowspan=2|Honeycomb name
Coxeter diagram

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

!rowspan=2|Vertex figure

!rowspan=2|Picture

0
{{CDD|branch|split2|node}}

!1
{{CDD|node|3|node|3|node}}

!2
{{CDD|node|3|node|3|node}}

!3
{{CDD|node|split1|branch}}

!Alt

align=center
align=center

|Nonuniform

|bisnub order-6 cubic
{{CDD|node_h|split1|branch_hh|split2|node_h}} ↔ {{CDD|node_h0|6|node_h|3|node_h|4|node_h0}}

|40px
{{CDD|branch_hh|split2|node_h}}

|40px
{{CDD|node_h|3|node_h|3|node_h}}

|40px
{{CDD|node_h|3|node_h|3|node_h}}

|40px
{{CDD|node_h|split1|branch_hh}}

|40px
irr. {3,3}

|80px

|

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

There are 4 forms, 0 unique, generated by ring permutations of the Coxeter group: {{CDD|branch|splitcross|branch}}. They are repeated in four families: {{CDD|node_c3|splitsplit1|branch4_c1-2|splitsplit2|node_c3}} ↔ {{CDD|node_h0|6|node_c3|split1|branch_c1-2}} (index 2 subgroup),

{{CDD|node_c2|splitsplit1|branch4_c1|splitsplit2|node_c2}} ↔ {{CDD|node_h0|6|node_c1|3|node_c2|6|node_h0}} (index 4 subgroup),

{{CDD|node_c2|splitsplit1|branch4_c1|splitsplit2|node_c1}} ↔ {{CDD|node_c2|3|node_c1|6|node_g|3sg|node_g}} (index 6 subgroup), and

{{CDD|branch_c1|splitcross|branch_c1}} ↔ {{CDD|node_c1|6|node_g|3sg|node_g|3g|node_g}} (index 24 subgroup).

class=wikitable

!#

!Name
Coxeter diagram

!0

!1

!2

!3

!vertex figure

!Picture

align=center

|[1]

|hexagonal (hexah)
{{CDD|branch_11|splitcross|branch_11}} ↔ {{CDD|node_1|6|node_g|3sg|node_g|3g|node_g}}

|40px
{{CDD|node_1|split1|branch_11}}

|40px
{{CDD|node_1|split1|branch_11}}

|40px
{{CDD|node_1|split1|branch_11}}

|40px
{{CDD|node_1|split1|branch_11}}

|60px {{CDD|node_1|3|node|3|node}}
{3,3}

|120px

align=center

|[47]

|rectified order-6 hexagonal (rihihexah)
{{CDD|node|splitsplit1|branch4_11|splitsplit2|node}} ↔ {{CDD|node_h0|6|node_1|3|node|6|node_h0}}

|40px
{{CDD|node_1|split1|branch}}

|40px
{{CDD|node|split1|branch_11}}

|40px
{{CDD|node_1|split1|branch}}

|40px
{{CDD|node|split1|branch_11}}

|60px {{CDD|node_1|2|node_1|6|node}}
t{2,3}

|120px

align=center

|[54]

|triangular tiling honeycomb (trah)
( {{CDD|branch|splitcross|branch_10l}} ↔ {{CDD|node_h0|6|node|split1|branch_10lu}} ) = {{CDD|node_1|3|node|6|node|3|node}}

|40px
{{CDD|node_1|split1|branch}}

| -

|40px
{{CDD|node|split1|branch_10lu}}

|40px
{{CDD|node|split1|branch_01ld}}

|40px {{CDD|node_1|split1|branch_11}}
t{3[3]}

|120px

align=center

|[55]

|rectified triangular (ritrah)
{{CDD|node|splitsplit1|branch4_11|splitsplit2|node_1}} ↔ {{CDD|node|3|node_1|6|node_g|3sg|node_g}}

|40px
{{CDD|node_1|split1|branch}}

|40px
{{CDD|node|split1|branch_11}}

|40px
{{CDD|node|split1|branch_11}}

|40px
{{CDD|node_1|split1|branch_11}}

|60px {{CDD|node_1|2|node_1|3|node}}
t{2,3}

|120px

class=wikitable

!#

!Name
Coxeter diagram

!0

!1

!2

!3

!Alt

!vertex figure

!Picture

align=center
align=center

|[137]

|alternated hexagonal (ahexah)
( {{CDD|branch_hh|splitcross|branch_hh}} ↔ {{CDD|node_h1|6|node_g|3sg|node_g|3g|node_g}} ) = {{CDD|branch_10ru|split2|node|3|node}}

|40px
{{CDD|node_h|split1|branch_hh}}
s{3[3]}

|40px
{{CDD|node_h|split1|branch_hh}}
s{3[3]}

|40px
{{CDD|node_h|split1|branch_hh}}
s{3[3]}

|40px
{{CDD|node_h|split1|branch_hh}}
s{3[3]}

|40px
{{CDD|node_1|3|node|3|node}}
{3,3}

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

Summary enumerations by family

= Linear graphs =

class=wikitable

|+ Paracompact hyperbolic enumeration

!Group

!Extended
symmetry

!colspan=2|Honeycombs

!Chiral
extended
symmetry

!colspan=2|Alternation honeycombs

align=center

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

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

rowspan=2|15

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

|[1+,4,1+,4,3+]

(6)

|{{CDD|node_h1|4|node|4|node|3|node}} (↔ {{CDD|nodes_10ru|split2-44|node|3|node}})
{{CDD|node|4|node_h1|4|node|3|node}} (↔ {{CDD|node_1|ultra|node|3|node|3|node_1|ultra|node}})
{{CDD|node_h|4|node_h|4|node|3|node}} | {{CDD|node|4|node|4|node_h|3|node_h}}
{{CDD|node_h|4|node|4|node_h|3|node_h}} | {{CDD|node|4|node_h|4|node_h|3|node_h}}

align=center

|[4,4,3]+

(1)

|{{CDD|node_h|4|node_h|4|node_h|3|node_h}}

align=center

!rowspan=4|{\bar{N}}_3
[4,4,4]
{{CDD|node|4|node|4|node|4|node}}

|[4,4,4]
{{CDD|node_c1|4|node_c2|4|node_c3|4|node_c4}}

3

| {{CDD|node_1|4|node|4|node|4|node}} | {{CDD|node_1|4|node_1|4|node|4|node}} | {{CDD|node_1|4|node_1|4|node|4|node_1}}

|[1+,4,1+,4,1+,4,1+]

(3)

| {{CDD|node_h1|4|node|4|node|4|node}} (↔ {{CDD|nodes_10ru|split2-44|node|4|node}} = {{CDD|node_1|4|node|4|node|4|node}})
{{CDD|node_h|4|node_h|4|node|4|node}} | {{CDD|node_h|4|node_h|4|node|4|node_h}}

align=center BGCOLOR="#e0f0e0"

|[4,4,4]
{{CDD|node_c1|4|node_c2|4|node_c1|4|node_h0}} ↔ {{CDD|node_c2|4|node_c1|4|node_g|3sg|node_g}}

|(3)

| {{CDD|node|4|node_1|4|node|4|node}} | {{CDD|node_1|4|node|4|node_1|4|node}} | {{CDD|node_1|4|node_1|4|node_1|4|node}}

|[1+,4,1+,4,1+,4,1+]

(3)

| {{CDD|node|4|node_h1|4|node|4|node}} (↔ {{CDD|node_1|ultra|node|4|node|4|node_1|ultra|node}})
{{CDD|node_h|4|node|4|node_h|4|node}} | {{CDD|node_h|4|node_h|4|node_h|4|node}}

align=center

|rowspan=2|[2+[4,4,4]]
{{CDD|node_c1|4|node_c2|4|node_c2|4|node_c1}}

rowspan=2|3

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

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

|{{CDD|node_h|4|node|4|node|4|node_h}} | {{CDD|node|4|node_h|4|node_h|4|node}}

align=center[2+[4,4,4]]+(1)

|{{CDD|node_h|4|node_h|4|node_h|4|node_h}}

align=center

!rowspan=2|{\bar{V}}_3
[6,3,3]
{{CDD|node|6|node|3|node|3|node}}

|rowspan=2|[6,3,3]
{{CDD|node_c1|6|node_c2|3|node_c3|3|node_c4}}

rowspan=2|15

|rowspan=2|{{CDD|node_1|6|node|3|node|3|node}} | {{CDD|node|6|node_1|3|node|3|node}} | {{CDD|node|6|node|3|node_1|3|node}} | {{CDD|node|6|node|3|node|3|node_1}} | {{CDD|node_1|6|node_1|3|node|3|node}}
{{CDD|node_1|6|node|3|node_1|3|node}} | {{CDD|node_1|6|node|3|node|3|node_1}} | {{CDD|node|6|node_1|3|node|3|node_1}} | {{CDD|node|6|node_1|3|node_1|3|node}} | {{CDD|node|6|node|3|node_1|3|node_1}}
{{CDD|node_1|6|node_1|3|node_1|3|node}} | {{CDD|node_1|6|node_1|3|node|3|node_1}} | {{CDD|node_1|6|node|3|node_1|3|node_1}} | {{CDD|node|6|node_1|3|node_1|3|node_1}} | {{CDD|node_1|6|node_1|3|node_1|3|node_1}}

|[1+,6,(3,3)+]

(2)

| {{CDD|node_h1|6|node|3|node|3|node}} (↔ {{CDD|branch_10ru|split2|node|3|node}})
{{CDD|node|6|node_h|3|node_h|3|node_h}}

align=center

|[6,3,3]+

(1)

| {{CDD|node_h|6|node_h|3|node_h|3|node_h}}

align=center

!rowspan=2|{\bar{BV}}_3
[6,3,4]
{{CDD|node|6|node|3|node|4|node}}

|rowspan=2|[6,3,4]
{{CDD|node_c1|6|node_c2|3|node_c3|4|node_c4}}

rowspan=2|15

|rowspan=2|{{CDD|node_1|6|node|3|node|4|node}} | {{CDD|node|6|node_1|3|node|4|node}} | {{CDD|node|6|node|3|node_1|4|node}} | {{CDD|node|6|node|3|node|4|node_1}} | {{CDD|node_1|6|node_1|3|node|4|node}}
{{CDD|node_1|6|node|3|node_1|4|node}} | {{CDD|node_1|6|node|3|node|4|node_1}} | {{CDD|node|6|node_1|3|node|4|node_1}} | {{CDD|node|6|node_1|3|node_1|4|node}} | {{CDD|node|6|node|3|node_1|4|node_1}}
{{CDD|node_1|6|node_1|3|node_1|4|node}} | {{CDD|node_1|6|node_1|3|node|4|node_1}} | {{CDD|node_1|6|node|3|node_1|4|node_1}} | {{CDD|node|6|node_1|3|node_1|4|node_1}} | {{CDD|node_1|6|node_1|3|node_1|4|node_1}}

|[1+,6,3+,4,1+]

(6)

| {{CDD|node_h1|6|node|3|node|4|node}} (↔ {{CDD|branch_10ru|split2|node|4|node}})
{{CDD|node|6|node|3|node|4|node_h1}} (↔ {{CDD|node|6|node|split1|nodes_10lu}})
{{CDD|node|6|node_h|3|node_h|4|node}} | {{CDD|node_h|6|node_h|3|node_h|4|node}}
{{CDD|node_h|6|node|3|node|4|node_h}} | {{CDD|node|6|node_h|3|node_h|4|node_h}}

align=center

|[6,3,4]+

(1)

|{{CDD|node_h|6|node_h|3|node_h|4|node_h}}

align=center

!rowspan=2|{\bar{HV}}_3
[6,3,5]
{{CDD|node|6|node|3|node|5|node}}

|rowspan=2|[6,3,5]
{{CDD|node_c1|6|node_c2|3|node_c3|5|node_c4}}

rowspan=2|15

|rowspan=2|{{CDD|node_1|6|node|3|node|5|node}} | {{CDD|node|6|node_1|3|node|5|node}} | {{CDD|node|6|node|3|node_1|5|node}} | {{CDD|node|6|node|3|node|5|node_1}} | {{CDD|node_1|6|node_1|3|node|5|node}}
{{CDD|node_1|6|node|3|node_1|5|node}} | {{CDD|node_1|6|node|3|node|5|node_1}} | {{CDD|node|6|node_1|3|node|5|node_1}} | {{CDD|node|6|node_1|3|node_1|5|node}} | {{CDD|node|6|node|3|node_1|5|node_1}}
{{CDD|node_1|6|node_1|3|node_1|5|node}} | {{CDD|node_1|6|node_1|3|node|5|node_1}} | {{CDD|node_1|6|node|3|node_1|5|node_1}} | {{CDD|node|6|node_1|3|node_1|5|node_1}} | {{CDD|node_1|6|node_1|3|node_1|5|node_1}}

|[1+,6,(3,5)+]

(2)

|{{CDD|node_h1|6|node|3|node|5|node}} (↔ {{CDD|branch_10ru|split2|node|5|node}})
{{CDD|node|6|node_h|3|node_h|5|node_h}}

align=center

|[6,3,5]+

(1)

|{{CDD|node_h|6|node_h|3|node_h|5|node_h}}

align=center

!rowspan=3|{\bar{Y}}_3
[3,6,3]
{{CDD|node|3|node|6|node|3|node}}

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

5

| {{CDD|node_1|3|node|6|node|3|node}} | {{CDD|node|3|node_1|6|node|3|node}} | {{CDD|node_1|3|node|6|node_1|3|node}} | {{CDD|node_1|3|node_1|6|node_1|3|node}} | {{CDD|node_1|3|node_1|6|node|3|node_1}}

align=center BGCOLOR="#e0f0e0"

|[3,6,3]
{{CDD|node_c1|3|node_c1|6|node_g|3sg|node_g}} ↔ {{CDD|node_c1|6|node_g|3sg|node_g|3g|node_g}}

(1)

| {{CDD|node_1|3|node_1|6|node|3|node}}

[2+[3+,6,3+]](1)

| {{CDD|node_h|3|node_h|6|node|3|node}}

align=center[2+[3,6,3]]
{{CDD|node_c1|3|node_c2|6|node_c2|3|node_c1}}		3

|{{CDD|node_1|3|node|6|node|3|node_1}} | {{CDD|node|3|node_1|6|node_1|3|node}} | {{CDD|node_1|3|node_1|6|node_1|3|node_1}}

[2+[3,6,3]]+(1)

|{{CDD|node_h|3|node_h|6|node_h|3|node_h}}

align=center

!rowspan=4|{\bar{Z}}_3
[6,3,6]
{{CDD|node|6|node|3|node|6|node}}

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

6

| {{CDD|node_1|6|node|3|node|6|node}} | {{CDD|node|6|node_1|3|node|6|node}} | {{CDD|node_1|6|node_1|3|node|6|node}}
{{CDD|node_1|6|node|3|node_1|6|node}} | {{CDD|node_1|6|node_1|3|node_1|6|node}} | {{CDD|node_1|6|node_1|3|node|6|node_1}}

|[1+,6,3+,6,1+]

(2)

|{{CDD|node_h1|6|node|3|node|6|node}} (↔ {{CDD|branch_10ru|split2|node|6|node}})
{{CDD|node_h|6|node_h|3|node_h|6|node}}

align=center BGCOLOR="#e0f0e0"

|[2+[6,3,6]]
{{CDD|node_h0|6|node_c1|3|node_c1|6|node_h0}} ↔ {{CDD|node_c1|6|node_g|3sg|node_g|3g|node_g}}

(1)

|{{CDD|node|6|node_1|3|node_1|6|node}}

|rowspan=2|[2+[(6,3+,6,2+)]]

rowspan=2|(2)

|{{CDD|node|6|node_h|3|node_h|6|node}}

align=center

|rowspan=2|[2+[6,3,6]]
{{CDD|node_c1|6|node_c2|3|node_c2|6|node_c1}}

rowspan=2|2

|rowspan=2|{{CDD|node_1|6|node|3|node|6|node_1}} | {{CDD|node_1|6|node_1|3|node_1|6|node_1}}

|{{CDD|node_h|6|node|3|node|6|node_h}}

align=center

|[2+[6,3,6]]+

(1)

|{{CDD|node_h|6|node_h|3|node_h|6|node_h}}

= Tridental graphs =

class=wikitable

|+ Paracompact hyperbolic enumeration

!Group

!Extended
symmetry

!colspan=2|Honeycombs

!Chiral
extended
symmetry

!colspan=2|Alternation honeycombs

align=center

|rowspan=3|{\bar{DV}}_3
[6,31,1]
{{CDD|node|6|node|split1|nodes}}

[6,31,1]4

| {{CDD|node|6|node|split1|nodes_10lu}} | {{CDD|node_1|6|node|split1|nodes_10lu}} | {{CDD|node|6|node_1|split1|nodes_10lu}} | {{CDD|node_1|6|node_1|split1|nodes_10lu}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center

|rowspan=2|[1[6,31,1]]=[6,3,4]
{{CDD|node_c1|6|node_c2|split1|nodeab_c3}} ↔ {{CDD|node_c1|6|node_c2|3|node_c3|4|node_h0}}

rowspan=2|(7)

|rowspan=2|{{CDD|node_1|6|node|split1|nodes}} | {{CDD|node|6|node_1|split1|nodes}} | {{CDD|node_1|6|node_1|split1|nodes}} | {{CDD|node|6|node|split1|nodes_11}} | {{CDD|node_1|6|node|split1|nodes_11}} | {{CDD|node|6|node_1|split1|nodes_11}} | {{CDD|node_1|6|node_1|split1|nodes_11}}

[1[1+,6,31,1]]+(2)

|{{CDD|node_h1|6|node|split1|nodes}} (↔ {{CDD|node|split1|branch_10luru|split2|node}})
{{CDD|node|6|node_h|split1|nodes_hh}}

BGCOLOR="#e0f0e0" align=center[1[6,31,1]]+=[6,3,4]+(1)

|{{CDD|node_h|6|node_h|split1|nodes_hh}}

align=center

|rowspan=3|{\bar{O}}_3
[3,41,1]
{{CDD|node|3|node|split1-44|nodes}}

[3,41,1]4

| {{CDD|node|3|node|split1-44|nodes_10lu}} | {{CDD|node_1|3|node|split1-44|nodes_10lu}} | {{CDD|node|3|node_1|split1-44|nodes_10lu}} | {{CDD|node_1|3|node_1|split1-44|nodes_10lu}}

|[3+,41,1]+

(2)

|{{CDD|node|3|node|split1-44|nodes_h0l}} ↔ {{CDD|node|split1-44|nodes_10luru|split2|node}}
{{CDD|node_h|3|node_h|split1-44|nodes_h0l}}

BGCOLOR="#e0f0e0" align=center

|rowspan=2|[1[3,41,1]]=[3,4,4]
{{CDD|node_c1|3|node_c2|split1-44|nodeab_c3}} ↔ {{CDD|node_c1|3|node_c2|4|node_c3|4|node_h0}}

rowspan=2|(7)

|rowspan=2|{{CDD|node_1|3|node|split1-44|nodes}} | {{CDD|node|3|node_1|split1-44|nodes}} | {{CDD|node_1|3|node_1|split1-44|nodes}} | {{CDD|node|3|node|split1-44|nodes_11}} | {{CDD|node_1|3|node|split1-44|nodes_11}} | {{CDD|node|3|node_1|split1-44|nodes_11}} | {{CDD|node_1|3|node_1|split1-44|nodes_11}}

|[1[3+,41,1]]+

(2)

|{{CDD|node_h|3|node_h|split1-44|nodes}} | {{CDD|node|3|node|split1-44|nodes_hh}}

BGCOLOR="#e0f0e0" align=center

|[1[3,41,1]]+

(1)

|{{CDD|node_h|3|node_h|split1-44|nodes_hh}}

align=center

|rowspan=4|{\bar{M}}_3
[41,1,1]
{{CDD|node|4|node|split1-44|nodes}}

[41,1,1]0

| (none)

BGCOLOR="#e0f0e0" align=center[1[41,1,1]]=[4,4,4]
{{CDD|node_c1|4|node_c2|split1-44|nodeab_c3}} ↔ {{CDD|node_c1|4|node_c2|4|node_c3|4|node_h0}}		(4)

| {{CDD|node_1|4|node|split1-44|nodes}} | {{CDD|node|4|node|split1-44|nodes_11}} | {{CDD|node_1|4|node_1|split1-44|nodes}} | {{CDD|node|4|node_1|split1-44|nodes_11}}

[1[1+,4,1+,41,1]]+=[(4,1+,4,1+,4,2+)](4)

| {{CDD|node_h1|4|node|split1-44|nodes}} (↔ {{CDD|node_1|split1-44|nodes|split2-44|node}})
{{CDD|node|4|node|split1-44|nodes_hh}} | {{CDD|node_h|4|node_h|split1-44|nodes}} | {{CDD|node|4|node_h|split1-44|nodes_hh}}

BGCOLOR="#e0f0e0" align=center

|rowspan=2|[3[41,1,1]]=[4,4,3]
{{CDD|node_c1|4|node_c2|split1-44|nodeab_c1}} ↔ {{CDD|node_c2|4|node_c1|4|node_g|3sg|node_g}}

rowspan=2|(3)

|rowspan=2|{{CDD|node|4|node_1|split1-44|nodes}} | {{CDD|node_1|4|node|split1-44|nodes_11}} | {{CDD|node_1|4|node_1|split1-44|nodes_11}}

[3[1+,41,1,1]]+=[1+,4,1+,4,3+](2)

| {{CDD|node|4|node_h1|split1-44|nodes}} (↔ {{CDD|node_1|split1-uu|nodes|2a2b-cross|nodes_11|split2-uu|node}})
{{CDD|node_h|4|node|split1-44|nodes_hh}}

BGCOLOR="#e0f0e0" align=center[3[41,1,1]]+=[4,4,3]+(1)

| {{CDD|node_h|4|node_h|split1-44|nodes_hh}}

= Cyclic graphs =

class=wikitable

|+ Paracompact hyperbolic enumeration

!Group

!Extended
symmetry

!colspan=2|Honeycombs

!Chiral
extended
symmetry

!colspan=2|Alternation honeycombs

align=center

|rowspan=3|{\widehat{CR}}_3
[(4,4,4,3)]
{{CDD|label4|branch|4-4|branch}}

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

|{{CDD|label4|branch_10r|4-4|branch}} | {{CDD|label4|branch|4-4|branch_10l}} | {{CDD|label4|branch_01r|4-4|branch_10l}} | {{CDD|label4|branch_10r|4-4|branch_10l}} | {{CDD|label4|branch_11|4-4|branch_10l}} | {{CDD|label4|branch_10r|4-4|branch_11}}

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

|(2)

{{CDD|label4|branch_h0r|4-4|branch}} ↔ {{CDD|branchu_10|split2-43|node|split1-43|branchu_01}}
{{CDD|label4|branch_h0r|4-4|branch_hh}}
align=center

|rowspan=2|[2+[(4,4,4,3)]]
{{CDD|label4|branch_c1|4-4|branch_c2}}

rowspan=2|3

| rowspan=2|{{CDD|label4|branch_11|4-4|branch}} | {{CDD|label4|branch|4-4|branch_11}} | {{CDD|label4|branch_11|4-4|branch_11}}

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

|(2)

{{CDD|label4|branch_hh|4-4|branch}} | {{CDD|label4|branch|4-4|branch_hh}}
align=center[2+[(4,4,4,3)]]+

|(1)

{{CDD|label4|branch_hh|4-4|branch_hh}}
align=center

|rowspan=5|{\widehat{RR}}_3
[4[4]]
{{CDD|label4|branch|4-4|branch|label4}}

[4[4]]

|colspan=5|(none)

align=center[2+[4[4]]]
{{CDD|label4|branch_c1|4-4|branch_c2|label4}}		1

| {{CDD|label4|branch_11|4-4|branch|label4}}

|[2+[(4+,4)[2]]]

|(1)

{{CDD|label4|branch_hh|4-4|branch|label4}}
BGCOLOR="#e0f0e0" align=center[1[4[4]]]=[4,41,1]
{{CDD|node_c3|split1-44|nodeab_c1-2|split2-44|node_c3}} ↔ {{CDD|node_h0|4|node_c3|split1-44|nodeab_c1-2}}		(2)

|{{CDD|node_1|split1-44|nodes|split2-44|node}} {{CDD|node_1|split1-44|nodes_11|split2-44|node}}

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

|(2)

{{CDD|node_h1|split1-44|nodes|split2-44|node}} ↔ {{CDD|branchu_10|split2-44|node|split1-44|branchu_01}}
{{CDD|node_h|split1-44|nodes_hh|split2-44|node}}
align=center BGCOLOR="#e0f0e0"[2[4[4]]]=[4,4,4]
{{CDD|node_c1|split1-44|nodeab_c2|split2-44|node_c1}} ↔ {{CDD|node_h0|4|node_c1|4|node_c2|4|node_h0}}		(1)

| {{CDD|node_1|split1-44|nodes|split2-44|node_1}}

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

|(1)

{{CDD|node_h|split1-44|nodes|split2-44|node_h}}
BGCOLOR="#e0f0e0" align=center

|[(2+,4)[4[4]]]=[2+[4,4,4]]
{{CDD|label4|branch_c1|4-4|branch_c1|label4}} = {{CDD|label4|branch_c1|4-4|nodes}}

(1)

| {{CDD|label4|branch_11|4-4|branch_11|label4}}

|[(2+,4)[4[4]]]+
= [2+[4,4,4]]+

|(1)

{{CDD|label4|branch_hh|4-4|branch_hh|label4}}
align=center

|rowspan=2|{\widehat{AV}}_3
[(6,3,3,3)]
{{CDD|label6|branch|3ab|branch}}

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

|{{CDD|label6|branch_10r|3ab|branch}} | {{CDD|label6|branch|3ab|branch_10l}} | {{CDD|label6|branch_01r|3ab|branch_10l}} | {{CDD|label6|branch_10r|3ab|branch_10l}} | {{CDD|label6|branch_11|3ab|branch_10l}} | {{CDD|label6|branch_10r|3ab|branch_11}}

|colspan=3|

align=center[2+[(6,3,3,3)]]
{{CDD|label6|branch_c1|3ab|branch_c2}}		3

| {{CDD|label6|branch_11|3ab|branch}} | {{CDD|label6|branch|3ab|branch_11}} | {{CDD|label6|branch_11|3ab|branch_11}}

[2+[(6,3,3,3)]]+(1)

| {{CDD|label6|branch_hh|3ab|branch_hh}}

align=center

|rowspan=2|{\widehat{BV}}_3
[(3,4,3,6)]
{{CDD|label6|branch|3ab|branch|label4}}

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

|{{CDD|label6|branch_10r|3ab|branch|label4}} | {{CDD|label6|branch|3ab|branch_10l|label4}} | {{CDD|label6|branch_01r|3ab|branch_10l|label4}} | {{CDD|label6|branch_10r|3ab|branch_10l|label4}} | {{CDD|label6|branch_11|3ab|branch_10l|label4}} | {{CDD|label6|branch_10r|3ab|branch_11|label4}}

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

(1)

|{{CDD|label6|branch_h0r|3ab|branch_h0l|label4}}

align=center[2+[(3,4,3,6)]]
{{CDD|label6|branch_c1|3ab|branch_c2|label4}}		3

| {{CDD|label6|branch_11|3ab|branch|label4}} | {{CDD|label6|branch|3ab|branch_11|label4}} | {{CDD|label6|branch_11|3ab|branch_11|label4}}

[2+[(3,4,3,6)]]+(1)

| {{CDD|label6|branch_hh|3ab|branch_hh|label4}}

align=center

|rowspan=2|{\widehat{HV}}_3
[(3,5,3,6)]
{{CDD|label6|branch|3ab|branch|label5}}

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

|{{CDD|label6|branch_10r|3ab|branch|label5}} | {{CDD|label6|branch|3ab|branch_10l|label5}} | {{CDD|label6|branch_01r|3ab|branch_10l|label5}} | {{CDD|label6|branch_10r|3ab|branch_10l|label5}} | {{CDD|label6|branch_11|3ab|branch_10l|label5}} | {{CDD|label6|branch_10r|3ab|branch_11|label5}}

|colspan=3|

align=center[2+[(3,5,3,6)]]
{{CDD|label6|branch_c1|3ab|branch_c2|label5}}		3

|{{CDD|label6|branch_11|3ab|branch|label5}} | {{CDD|label6|branch|3ab|branch_11|label5}} | {{CDD|label6|branch_11|3ab|branch_11|label5}}

[2+[(3,5,3,6)]]+(1)

|{{CDD|label6|branch_hh|3ab|branch_hh|label5}}

align=center

|rowspan=5|{\widehat{VV}}_3
[(3,6)[2]]
{{CDD|label6|branch|3ab|branch|label6}}

[(3,6)[2]]2

|{{CDD|label6|branch_10r|3ab|branch|label6}} | {{CDD|label6|branch_11|3ab|branch_10l|label6}}

|colspan=3|

align=center[2+[(3,6)[2]]]
{{CDD|label6|branch_c1-2|3ab|branch_c2-1|label6}}		1

|{{CDD|label6|branch_01r|3ab|branch_10l|label6}}

|colspan=3|

align=center[2+[(3,6)[2]]]
{{CDD|label6|branch_c1|3ab|branch_c2|label6}}		1

|{{CDD|label6|branch_11|3ab|branch|label6}}

|colspan=3|

align=center BGCOLOR="#e0f0e0"[2+[(3,6)[2]]]
{{CDD|label6|branch_c1-0|3ab|branch_c1-0|label6}} = {{CDD|node_c1|6|node|3|node|4|node}}		(1)

|{{CDD|label6|branch_10r|3ab|branch_10l|label6}}

[2+[(3+,6)[2]]](1)

|{{CDD|label6|branch_h0r|3ab|branch_h0l|label6}}

align=center[(2,2)+[(3,6)[2]]]
{{CDD|label6|branch_c1|3ab|branch_c1|label6}}		1

| {{CDD|label6|branch_11|3ab|branch_11|label6}}

[(2,2)+[(3,6)[2]]]+(1)

| {{CDD|label6|branch_hh|3ab|branch_hh|label6}}

class=wikitable

|+ Paracompact hyperbolic enumeration

!Group

!Extended
symmetry

!colspan=2|Honeycombs

!Chiral
extended
symmetry

!colspan=2|Alternation honeycombs

align=center

!rowspan=3|{\widehat{BR}}_3
[(3,3,4,4)]
{{CDD|node|split1-44|nodes|split2|node}}

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

|{{CDD|node|split1-44|nodes_10luru|split2|node}} | {{CDD|node_1|split1-44|nodes_10luru|split2|node}} | {{CDD|node|split1-44|nodes_10luru|split2|node_1}} | {{CDD|node_1|split1-44|nodes_10luru|split2|node_1}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center

|rowspan=2|[1[(4,4,3,3)]]=[3,41,1]
{{CDD|node_c1|split1-44|nodeab_c3|split2|node_c2}} ↔ {{CDD|node_h0|4|node_c3|split1-43|nodeab_c1-2}}

rowspan=2|(7)

|rowspan=2|{{CDD|node_1|split1-44|nodes|split2|node}} | {{CDD|node|split1-44|nodes|split2|node_1}} | {{CDD|node_1|split1-44|nodes|split2|node_1}} | {{CDD|node|split1-44|nodes_11|split2|node}} | {{CDD|node_1|split1-44|nodes_11|split2|node}} | {{CDD|node|split1-44|nodes_11|split2|node_1}} | {{CDD|node_1|split1-44|nodes_11|split2|node_1}}

|[1[(3,3,4,1+,4)]]+
= [3+,41,1]+

(2)

|{{CDD|node_h1|split1-44|nodes|split2|node}} (= {{CDD|branchu_10|split2|node|split1|branchu_01}})
{{CDD|node|split1-44|nodes_hh|split2|node_h}}

BGCOLOR="#e0f0e0" align=center[1[(3,3,4,4)]]+
= [3,41,1]+		(1)

|{{CDD|node_h|split1-44|nodes_hh|split2|node_h}}

align=center

!rowspan=4|{\bar{DP}}_3
[3[ ]x[ ]]
{{CDD|node|split1|branch|split2|node}}

[3[ ]x[ ]]1

|{{CDD|node_1|split1|branch_10luru|split2|node}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[1[3[ ]x[ ]]]=[6,31,1]
{{CDD|node_c1|split1-44|branch_c3|split2|node_c2}} ↔ {{CDD|node_h0|6|node_c3|split1|nodeab_c1-2}}		(2)

|{{CDD|node_1|split1|branch|split2|node}} | {{CDD|node_1|split1|branch_11|split2|node}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[1[3[ ]x[ ]]]=[4,3[3]]
{{CDD|node_c3|split1-44|branch_c1-2|split2|node_c3}} ↔ {{CDD|node_h0|4|node_c3|split1|branch_c1-2}}		(2)

| {{CDD|node|split1|branch_10luru|split2|node}} | {{CDD|node_1|split1|branch_10l|split2|node_1}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[2[3[ ]x[ ]]]=[6,3,4]
{{CDD|node_c2|split1|branch_c1|split2|node_c2}} ↔ {{CDD|node_h0|6|node_c1|3|node_c2|4|node_h0}}		(3)

| {{CDD|node|split1|branch_11|split2|node}} | {{CDD|node_1|split1|branch|split2|node_1}} | {{CDD|node_1|split1|branch_11|split2|node_1}}

[2[3[ ]x[ ]]]+
=[6,3,4]+		(1)

| {{CDD|node_h|split1|branch_hh|split2|node_h}}

BGCOLOR="#e0f0e0" align=center

!rowspan=5|{\bar{PP}}_3
[3[3,3]]
{{CDD|branch|splitcross|branch}}
{{CDD|node|splitsplit1|branch4|splitsplit2|node}}

[3[3,3]]0

|colspan=4|(none)

BGCOLOR="#e0f0e0" align=center[1[3[3,3]]]=[6,3[3]]
{{CDD|node_c3|splitsplit1|branch4_c1-2|splitsplit2|node_c3}} ↔ {{CDD|node_h0|6|node_c3|split1|branch_c1-2}}		0

|colspan=4|(none)

BGCOLOR="#e0f0e0" align=center[3[3[3,3]]]=[3,6,3]
{{CDD|node_c2|splitsplit1|branch4_c1|splitsplit2|node_c1}} ↔ {{CDD|node_c2|3|node_c1|6|node_g|3sg|node_g}}		(2)

| {{CDD|node_1|splitsplit1|branch4|splitsplit2|node}} | {{CDD|node_1|splitsplit1|branch4_11|splitsplit2|node}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[2[3[3,3]]]=[6,3,6]
{{CDD|node_c2|splitsplit1|branch4_c1|splitsplit2|node_c2}} ↔ {{CDD|node_h0|6|node_c1|3|node_c2|6|node_h0}}		(1){{CDD|node_1|splitsplit1|branch4|splitsplit2|node_1}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[(3,3)[3[3,3]]]=[6,3,3]
{{CDD|branch_c1|splitcross|branch_c1}} = {{CDD|node_c1|6|node_g|3sg|node_g|3g|node_g}}		(1)

| {{CDD|branch_11|splitcross|branch_11}}

[(3,3)[3[3,3]]]+
= [6,3,3]+		(1)

| {{CDD|branch_hh|splitcross|branch_hh}}

= Loop-n-tail graphs =

Symmetry in these graphs can be doubled by adding a mirror: [1[n,3[3]]] = [n,3,6]. Therefore ring-symmetry graphs are repeated in the linear graph families.

class=wikitable

|+ Paracompact hyperbolic enumeration

!Group

!Extended
symmetry

!colspan=2|Honeycombs

!Chiral
extended
symmetry

!colspan=2|Alternation honeycombs

align=center

|rowspan=2|{\bar{P}}_3
[3,3[3]]
{{CDD|node|3|node|split1|branch}}

[3,3[3]]4

| {{CDD|node|3|node|split1|branch_10lu}} | {{CDD|node_1|3|node|split1|branch_10lu}} | {{CDD|node|3|node_1|split1|branch_10lu}} | {{CDD|node_1|3|node_1|split1|branch_10lu}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[1[3,3[3]]]=[3,3,6]
{{CDD|node_c1|3|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|3|node_c2|3|node_c3|6|node_h0}}		(7)

|{{CDD|node_1|3|node|split1|branch}} | {{CDD|node|3|node_1|split1|branch}} | {{CDD|node_1|3|node_1|split1|branch}} | {{CDD|node|3|node|split1|branch_11}} | {{CDD|node_1|3|node|split1|branch_11}} | {{CDD|node|3|node_1|split1|branch_11}} | {{CDD|node_1|3|node_1|split1|branch_11}}

[1[3,3[3]]]+
= [3,3,6]+		(1)

|{{CDD|node_h|3|node_h|split1|branch_hh}}

align=center

|rowspan=3|{\bar{BP}}_3
[4,3[3]]
{{CDD|node|4|node|split1|branch}}

[4,3[3]]4

| {{CDD|node|4|node|split1|branch_10lu}} | {{CDD|node_1|4|node|split1|branch_10lu}} | {{CDD|node|4|node_1|split1|branch_10lu}} | {{CDD|node_1|4|node_1|split1|branch_10lu}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center

|rowspan=2|[1[4,3[3]]]=[4,3,6]
{{CDD|node_c1|4|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|4|node_c2|3|node_c3|6|node_h0}}

rowspan=2|(7)

|rowspan=2|{{CDD|node_1|4|node|split1|branch}} | {{CDD|node|4|node_1|split1|branch}} | {{CDD|node_1|4|node_1|split1|branch}} | {{CDD|node|4|node|split1|branch_11}} | {{CDD|node_1|4|node|split1|branch_11}} | {{CDD|node|4|node_1|split1|branch_11}} | {{CDD|node_1|4|node_1|split1|branch_11}}

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

(2)

|{{CDD|node_h1|4|node|split1|branch}} ↔ {{CDD|node_1|split1|branch|split2|node}}
{{CDD|node|4|node_h|split1|branch_hh}}

BGCOLOR="#e0f0e0" align=center

|[4,3[3]]+

(1)

|{{CDD|node_h|4|node_h|split1|branch_hh}}

align=center

|rowspan=2|{\bar{HP}}_3
[5,3[3]]
{{CDD|node|5|node|split1|branch}}

[5,3[3]]4

| {{CDD|node|5|node|split1|branch_10lu}} | {{CDD|node_1|5|node|split1|branch_10lu}} | {{CDD|node|5|node_1|split1|branch_10lu}} | {{CDD|node_1|5|node_1|split1|branch_10lu}}

|colspan=3|

BGCOLOR="#e0f0e0" align=center[1[5,3[3]]]=[5,3,6]
{{CDD|node_c1|5|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|5|node_c2|3|node_c3|6|node_h0}}		(7)

|{{CDD|node_1|5|node|split1|branch}} | {{CDD|node|5|node_1|split1|branch}} | {{CDD|node_1|5|node_1|split1|branch}} | {{CDD|node|5|node|split1|branch_11}} | {{CDD|node_1|5|node|split1|branch_11}} | {{CDD|node|5|node_1|split1|branch_11}} | {{CDD|node_1|5|node_1|split1|branch_11}}

[1[5,3[3]]]+
= [5,3,6]+		(1)

|{{CDD|node_h|5|node_h|split1|branch_hh}}

align=center

|rowspan=6|{\bar{VP}}_3
[6,3[3]]
{{CDD|node|6|node|split1|branch}}

[6,3[3]]2

| {{CDD|node_1|6|node|split1|branch_10lu}} | {{CDD|node_1|6|node_1|split1|branch_10lu}}

|colspan=3|

align=center BGCOLOR="#e0f0e0"

|[6,3[3]] =

(2)

| ({{CDD|node|6|node|split1|branch_10lu}} ↔ {{CDD|node_1|3|node|6|node|3|node}}) | ({{CDD|node|6|node_1|split1|branch_10lu}} = {{CDD|node|3|node_1|6|node|3|node}})

|colspan=3|

BGCOLOR="#e0f0e0" align=center

|[(3,3)[1+,6,3[3]]]=[6,3,3]
{{CDD|node_h0|6|node_c1|split1|branch_c1}} ↔ {{CDD|node_c1|6|node_g|3sg|node_g|3g|node_g}} ↔ {{CDD|branch_c1|splitcross|branch_c1}}

(1)

| {{CDD|node|6|node_1|split1|branch_11}}

[(3,3)[1+,6,3[3]]]+(1)

| {{CDD|node|6|node_h|split1|branch_hh}}

BGCOLOR="#e0f0e0" align=center

|rowspan=2|[1[6,3[3]]]=[6,3,6]
{{CDD|node_c1|6|node_c2|split1|branch_c3}} ↔ {{CDD|node_c1|6|node_c2|3|node_c3|6|node_h0}}

rowspan=3|(6)

|rowspan=3|{{CDD|node_1|6|node|split1|branch}} | {{CDD|node|6|node_1|split1|branch}} | {{CDD|node_1|6|node_1|split1|branch}} | {{CDD|node|6|node|split1|branch_11}} | {{CDD|node_1|6|node|split1|branch_11}} | {{CDD|node_1|6|node_1|split1|branch_11}}

[3[1+,6,3[3]]]+
= [3,6,3]+		(1)

|{{CDD|node_h1|6|node|split1|branch}} ↔ {{CDD|node_1|splitsplit1|branch4|splitsplit2|node}} (= {{CDD|node_1|3|node|6|node|3|node}} )

BGCOLOR="#e0f0e0" align=center[1[6,3[3]]]+
= [6,3,6]+		(1)

|{{CDD|node_h|6|node_h|split1|branch_hh}}

See also

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] {{Webarchive|url=https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |date=2016-06-10 }})
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Jeffrey R. Weeks The Shape of Space, 2nd edition {{isbn|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I, II)
  • [https://arxiv.org/abs/math/0212010 Coxeter Decompositions of Hyperbolic Tetrahedra], 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 }}
  • 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] [https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf]
  • N.W. Johnson, R. Kellerhals, J.G. Ratcliffe, S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [http://www.sciencedirect.com/science/article/pii/S0024379501004773]
  • {{KlitzingPolytopes|hyperbolic.htm#3D-non-compact|Hyperbolic honeycombs|H3 paracompact}}

Category:Honeycombs (geometry)