Paracompact uniform honeycombs

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Example paracompact regular honeycombs
H3 336 CC center.png
{3,3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H3 633 FC boundary.png
{6,3,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
H3 436 CC center.png
{4,3,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H3 634 FC boundary.png
{6,3,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
H3 536 CC center.png
{5,3,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H3 635 FC boundary.png
{6,3,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H3 636 FC boundary.png
{6,3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H3 363 FC boundary.png
{3,6,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H3 443 FC boundary.png
{4,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H3 344 CC center.png
{3,4,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
H3 444 FC boundary.png
{4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png

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[edit]

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.

Name Schläfli
Symbol
{p,q,r}
Coxeter
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure

{q,r}
Dual Coxeter
group
Order-6 tetrahedral honeycomb {3,3,6} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {3,3} {3} {6} {3,6} {6,3,3} [6,3,3]
Hexagonal tiling honeycomb {6,3,3} CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png {6,3} {6} {3} {3,3} {3,3,6}
Order-4 octahedral honeycomb {3,4,4} CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png {3,4} {3} {4} {4,4} {4,4,3} [4,4,3]
Square tiling honeycomb {4,4,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png {4,4} {4} {3} {4,3} {3,4,4}
Triangular tiling honeycomb {3,6,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png {3,6} {3} {3} {6,3} Self-dual [3,6,3]
Order-6 cubic honeycomb {4,3,6} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {4,3} {4} {4} {3,4} {6,3,4} [6,3,4]
Order-4 hexagonal tiling honeycomb {6,3,4} CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {6,3} {6} {4} {3,4} {4,3,6}
Order-4 square tiling honeycomb {4,4,4} CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png {4,4} {4} {4} {4,4} Self-dual [4,4,4]
Order-6 dodecahedral honeycomb {5,3,6} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {5,3} {5} {5} {3,6} {6,3,5} [6,3,5]
Order-5 hexagonal tiling honeycomb {6,3,5} CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png {6,3} {6} {5} {3,5} {5,3,6}
Order-6 hexagonal tiling honeycomb {6,3,6} CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {6,3} {6} {6} {3,6} Self-dual [6,3,6]

Coxeter groups of paracompact uniform honeycombs[edit]

Hyperbolic subgroup tree 36.png Hyperbolic subgroup tree 344.png
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.

Tetrahedral hyperbolic paracompact group summary
Coxeter group Simplex
volume
Commutator subgroup Unique honeycomb count
[6,3,3] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 0.0422892336 [1+,6,(3,3)+] = [3,3[3]]+ 15
[4,4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 0.0763304662 [1+,4,1+,4,3+] 15
[3,3[3]] CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png 0.0845784672 [3,3[3]]+ 4
[6,3,4] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 0.1057230840 [1+,6,3+,4,1+] = [3[]x[]]+ 15
[3,41,1] CDel node.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel nodes.png 0.1526609324 [3+,41+,1+] 4
[3,6,3] CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png 0.1691569344 [3+,6,3+] 8
[6,3,5] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 0.1715016613 [1+,6,(3,5)+] = [5,3[3]]+ 15
[6,31,1] CDel node.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png 0.2114461680 [1+,6,(31,1)+] = [3[]x[]]+ 4
[4,3[3]] CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png 0.2114461680 [1+,4,3[3]]+ = [3[]x[]]+ 4
[4,4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png 0.2289913985 [4+,4+,4+]+ 6
[6,3,6] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png 0.2537354016 [1+,6,3+,6,1+] = [3[3,3]]+ 8
[(4,4,3,3)] CDel node.pngCDel split1-44.pngCDel nodes.pngCDel split2.pngCDel node.png 0.3053218647 [(4,1+,4,(3,3)+)] 4
[5,3[3]] CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png 0.3430033226 [5,3[3]]+ 4
[(6,3,3,3)] CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.png 0.3641071004 [(6,3,3,3)]+ 9
[3[]x[]] CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png 0.4228923360 [3[]x[]]+ 1
[41,1,1] CDel node.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.png 0.4579827971 [1+,41+,1+,1+] 0
[6,3[3]] CDel node.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png 0.5074708032 [1+,6,3[3]] = [3[3,3]]+ 2
[(6,3,4,3)] CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png 0.5258402692 [(6,3+,4,3+)] 9
[(4,4,4,3)] CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.png 0.5562821156 [(4,1+,4,1+,4,3+)] 9
[(6,3,5,3)] CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png 0.6729858045 [(6,3,5,3)]+ 9
[(6,3,6,3)] CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png 0.8457846720 [(6,3+,6,3+)] 5
[(4,4,4,4)] CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.pngCDel label4.png 0.9159655942 [(4+,4+,4+,4+)] 1
[3[3,3]] CDel branch.pngCDel splitcross.pngCDel branch.png 1.014916064 [3[3,3]]+ 0

The complete list of nonsimplectic (non-tetrahedral) paracompact Coxeter groups was published by P. Tumarkin in 2003.[1] The smallest paracompact form in H3 can be represented by CDel node.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png or CDel node.pngCDel split1.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, or [∞,3,3,∞] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1+,4] : CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c3.png = CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 2a2b-cross.pngCDel nodeab c3.png. The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramid is CDel node.pngCDel ultra.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png or CDel node.pngCDel split1-44.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, constructed as [4,4,1+,4] = [∞,4,4,∞] : CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c3.png = CDel node c1.pngCDel split1-44.pngCDel nodeab c2.pngCDel 2a2b-cross.pngCDel nodeab c3.png.

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 CDel branchu.pngCDel split2.pngCDel node.pngCDel split1.pngCDel branchu.png, [(3,4,4,1+,4)] = [((4,∞,3)),((3,∞,4))] or CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1-43.pngCDel branchu.png, [(4,4,4,1+,4)] = [((4,∞,4)),((4,∞,4))] or CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-44.pngCDel branchu.png. CDel labelh.pngCDel node.pngCDel split1-44.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png = CDel labelinfin.pngCDel branch c1-2.pngCDel split2.pngCDel node c3.pngCDel split1.pngCDel branch c1-2.pngCDel labelinfin.png, CDel labelh.pngCDel node.pngCDel split1-44.pngCDel nodeab c1-2.pngCDel split2-43.pngCDel node c3.png = CDel labelinfin.pngCDel branch c1-2.pngCDel split2-43.pngCDel node c3.pngCDel split1-43.pngCDel branch c1-2.pngCDel labelinfin.png, CDel labelh.pngCDel node.pngCDel split1-44.pngCDel nodeab c1-2.pngCDel split2-44.pngCDel node c3.png = CDel labelinfin.pngCDel branch c1-2.pngCDel split2-44.pngCDel node c3.pngCDel split1-44.pngCDel branch c1-2.pngCDel labelinfin.png.

Another nonsimplectic half groups is CDel nodeab c1-2.pngCDel split2-44.pngCDel node h0.pngCDel 4.pngCDel node c3.pngCDel node c3.pngCDel split1-uu.pngCDel nodeab c1-2.pngCDel 2a2b-cross.pngCDel nodeab c1-2.pngCDel split2-uu.pngCDel node c3.png.

A radial nonsimplectic subgroup is CDel label4.pngCDel branch c1-2.pngCDel 4a4b.pngCDel branch.pngCDel labels.pngCDel node c1.pngCDel splitplit1u-44.pngCDel branch3u c2.pngCDel 4a4buc-cross.pngCDel branch3u c1.pngCDel splitplit2u-44.pngCDel node c2.png, which can be doubled into a triangular prism domain as CDel node c1.pngCDel splitplit1u-44.pngCDel branch3u c2.pngCDel 4a4buc-cross.pngCDel branch3u c3.pngCDel splitplit2u-44.pngCDel node c4.pngCDel branchu c1-4.pngCDel 4a4b.pngCDel branch c2-3.pngCDel split2-44.pngCDel node.pngCDel labelh.png.

Pyramidal hyperbolic paracompact group summary
Dimension Rank Graphs
H3 5

CDel node.pngCDel split1.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png | CDel node.pngCDel split1-43.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png | CDel node.pngCDel split1-44.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png | CDel node.pngCDel split1-53.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png | CDel node.pngCDel split1-63.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png
CDel branchu.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-43.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2-53.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-54.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-55.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-63.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-64.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-65.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png | CDel branchu.pngCDel split2-66.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-53.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1-43.pngCDel branchu.png | CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-43.pngCDel branchu.png | CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-44.pngCDel branchu.png | CDel branchu.pngCDel split2-54.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-55.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-63.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-64.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-65.pngCDel node.pngCDel split1.pngCDel branchu.png | CDel branchu.pngCDel split2-66.pngCDel node.pngCDel split1.pngCDel branchu.png

Linear graphs[edit]

[6,3,3] family[edit]

# Honeycomb name
Coxeter diagram: CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.pngCDel 3.pngCDel node n4.png
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
1
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 2.pngCDel node n4.png
4
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
1 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{6,3,3}
- - - (4)
Uniform tiling 63-t0.png
(6.6.6)
Order-3 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Tetrahedron
H3 633 FC boundary.png
2 rectified hexagonal
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t1{6,3,3} or r{6,3,3}
(2)
Uniform polyhedron-33-t0.png
(3.3.3)
- - (3)
Uniform tiling 63-t1.png
(3.6.3.6)
Rectified order-3 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Triangular prism
H3 633 boundary 0100.png
3 rectified order-6 tetrahedral
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1{3,3,6} or r{3,3,6}
(6)
Uniform polyhedron-33-t1.png
(3.3.3.3)
- - (2)
Uniform tiling 63-t2.png
(3.3.3.3.3.3)
Rectified order-6 tetrahedral honeycomb verf.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Hexagonal prism
H3 336 CC center 0100.png
4 order-6 tetrahedral
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,3,6}
(∞)
Uniform polyhedron-33-t2.png
(3.3.3)
- - - Uniform tiling 63-t2.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Triangular tiling
H3 336 CC center.png
5 truncated hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,1{6,3,3} or t{6,3,3}
(1)
Uniform polyhedron-33-t0.png
(3.3.3)
- - (3)
Uniform tiling 63-t01.png
(3.12.12)
Truncated order-3 hexagonal tiling honeycomb verf.png
Triangular pyramid
H3 633-1100.png
6 cantellated hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2{6,3,3} or rr{6,3,3}
(1)
Uniform polyhedron-33-t1.png
3.3.3.3
(2)
Triangular prism.png
(4.4.3)
- (2)
Uniform tiling 63-t02.png
(3.4.6.4)
Cantellated order-3 hexagonal tiling honeycomb verf.png H3 633-1010.png
7 runcinated hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{6,3,3}
(1)
Uniform polyhedron-33-t2.png
(3.3.3)
(3)
Triangular prism.png
(4.4.3)
(3)
Hexagonal prism.png
(4.4.6)
(1)
Uniform tiling 63-t0.png
(6.6.6)
Runcinated order-3 hexagonal tiling honeycomb verf.png H3 633-1001.png
8 cantellated order-6 tetrahedral
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2{3,3,6} or rr{3,3,6}
(1)
Uniform polyhedron-33-t02.png
(3.4.3.4)
- (2)
Hexagonal prism.png
(4.4.6)
(2)
Uniform tiling 63-t1.png
(3.6.3.6)
Cantellated order-6 tetrahedral honeycomb verf.png H3 633-0101.png
9 bitruncated hexagonal
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{6,3,3} or 2t{6,3,3}
(2)
Uniform polyhedron-33-t01.png
(3.6.6)
- - (2)
Uniform tiling 63-t12.png
(6.6.6)
Bitruncated order-3 hexagonal tiling honeycomb verf.png H3 633-0110.png
10 truncated order-6 tetrahedral
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1{3,3,6} or t{3,3,6}
(6)
Uniform polyhedron-33-t12.png
(3.6.6)
- - (1)
Uniform tiling 63-t2.png
(3.3.3.3.3.3)
Truncated order-6 tetrahedral honeycomb verf.png H3 633-0011.png
11 cantitruncated hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2{6,3,3} or tr{6,3,3}
(1)
Uniform polyhedron-33-t01.png
(3.6.6)
(1)
Triangular prism.png
(4.4.3)
- (2)
Uniform tiling 63-t012.png
(4.6.12)
Cantitruncated order-3 hexagonal tiling honeycomb verf.png H3 633-1110.png
12 runcitruncated hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{6,3,3}
(1)
Uniform polyhedron-33-t02.png
(3.4.3.4)
(2)
Triangular prism.png
(4.4.3)
(1)
Dodecagonal prism.png
(4.4.12)
(1)
Uniform tiling 63-t01.png
(3.12.12)
Runcitruncated order-3 hexagonal tiling honeycomb verf.png H3 633-1101.png
13 runcitruncated order-6 tetrahedral
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,6}
(1)
Uniform polyhedron-33-t12.png
(3.6.6)
(1)
Hexagonal prism.png
(4.4.6)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Uniform tiling 63-t02.png
(3.4.6.4)
Runcitruncated order-6 tetrahedral honeycomb verf.png H3 633-1011.png
14 cantitruncated order-6 tetrahedral
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2{3,3,6} or tr{3,3,6}
(2)
Uniform polyhedron-33-t012.png
(4.6.6)
- (1)
Hexagonal prism.png
(4.4.6)
(1)
Uniform tiling 63-t12.png
(6.6.6)
Cantitruncated order-6 tetrahedral honeycomb verf.png H3 633-0111.png
15 omnitruncated hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{6,3,3}
(1)
Uniform polyhedron-33-t012.png
(4.6.6)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Dodecagonal prism.png
(4.4.12)
(1)
Uniform tiling 63-t012.png
(4.6.12)
Omnitruncated order-3 hexagonal tiling honeycomb verf.png H3 633-1111.png
Alternated forms
# Honeycomb name
Coxeter diagram: CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.pngCDel 3.pngCDel node n4.png
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
1
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 2.pngCDel node n4.png
4
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
Alt
[137] alternated hexagonal
(CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png) = CDel branch hh.pngCDel splitcross.pngCDel branch hh.png
- - (4)
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
(4)
Uniform polyhedron-33-t2.png
(3.3.3)
Uniform polyhedron-33-t01.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
(3.6.6)
[138] cantic hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
(1)
Uniform polyhedron-33-t1.png
(3.3.3.3)
- (2)
Uniform tiling 333-t01.png
(3.6.3.6)
(2)
Uniform polyhedron-33-t12.png
(3.6.6)
Cantic hexagonal tiling honeycomb verf.png
[139] runcic hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
(1)
Uniform polyhedron-33-t0.png
(4.4.4)
(1)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
(3)
Uniform polyhedron-33-t02.png
(3.4.3.4)
Runcic hexagonal tiling honeycomb verf.png
[140] runcicantic hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
(1)
Uniform polyhedron-33-t01.png
(3.10.10)
(1)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 333-t01.png
(3.6.3.6)
(2)
Uniform polyhedron-33-t012.png
(4.6.6)
Runcicantic hexagonal tiling honeycomb verf.png
Nonuniform snub rectified order-6 tetrahedral
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel branch hh.pngCDel split2.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3,6}
Uniform polyhedron-33-s012.png Uniform tiling 63-h12.png Tetrahedron.png
Irr. (3.3.3)
Nonuniform cantic snub order-6 tetrahedral
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr3{3,3,6}
Nonuniform omnisnub order-6 tetrahedral
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{6,3,3}
Uniform polyhedron-33-s012.png Uniform tiling 63-snub.png Tetrahedron.png
Irr. (3.3.3)

[6,3,4] family[edit]

There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 4.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 4.pngCDel node n4.png
2
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
16 (Regular) order-4 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{6,3,4}
- - - (8)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
(6.6.6)
Order-4 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
(3.3.3.3)
H3 634 FC boundary.png
17 rectified order-4 hexagonal
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t1{6,3,4} or r{6,3,4}
(2)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png
(3.3.3.3)
- - (4)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.png
(3.6.3.6)
Rectified order-4 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
(4.4.4)
H3 634 boundary 0100.png
18 rectified order-6 cubic
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t1{4,3,6} or r{4,3,6}
(6)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png
(3.4.3.4)
- - (2)
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t2.png
(3.3.3.3.3.3)
Rectified order-6 cubic honeycomb verf.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
(6.4.4)
H3 436 CC center 0100.png
19 order-6 cubic
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
{4,3,6}
(20)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png
(4.4.4)
- - - Uniform tiling 63-t2.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
(3.3.3.3.3.3)
H3 436 CC center.png
20 truncated order-4 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,1{6,3,4} or t{6,3,4}
(1)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png
(3.3.3.3)
- - (4)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t01.png
(3.12.12)
Truncated order-4 hexagonal tiling honeycomb verf.png H3 634-1100.png
21 bitruncated order-6 cubic
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t1,2{6,3,4} or 2t{6,3,4}
(2)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated octahedron.png
(4.6.6)
- - (2)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t12.png
(6.6.6)
Bitruncated order-4 hexagonal tiling honeycomb verf.png H3 634-0110.png
22 truncated order-6 cubic
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t0,1{4,3,6} or t{4,3,6}
(6)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png
(3.8.8)
- - (1)
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t2.png
(3.3.3.3.3.3)
Truncated order-6 cubic honeycomb verf.png H3 634-0011.png
23 cantellated order-4 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,2{6,3,4} or rr{6,3,4}
(1)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png
(3.4.3.4)
(2)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
(4.4.4)
- (2)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t02.png
(3.4.6.4)
Cantellated order-4 hexagonal tiling honeycomb verf.png H3 634-1010.png
24 cantellated order-6 cubic
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,2{4,3,6} or rr{4,3,6}
(2)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Small rhombicuboctahedron.png
(3.4.4.4)
- (2)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Hexagonal prism.png
(4.4.6)
(1)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.png
(3.6.3.6)
Cantellated order-6 cubic honeycomb verf.png H3 634-0101.png
25 runcinated order-6 cubic
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,3{6,3,4}
(1)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png
(4.4.4)
(3)
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png
Tetragonal prism.png
(4.4.4)
(3)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png
Hexagonal prism.png
(4.4.6)
(1)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
(6.6.6)
Runcinated order-4 hexagonal tiling honeycomb verf.png H3 634-1010.png
26 cantitruncated order-4 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,2{6,3,4} or tr{6,3,4}
(1)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated octahedron.png
(4.6.6)
(1)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
(4.4.4)
- (2)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t012.png
(4.6.12)
Cantitruncated order-4 hexagonal tiling honeycomb verf.png H3 634-1110.png
27 cantitruncated order-6 cubic
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t0,1,2{4,3,6} or tr{4,3,6}
(2)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Great rhombicuboctahedron.png
(4.6.8)
- (1)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Hexagonal prism.png
(4.4.6)
(1)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t12.png
(6.6.6)
Cantitruncated order-6 cubic honeycomb verf.png H3 634-0111.png
28 runcitruncated order-4 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,3{6,3,4}
(1)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png
Tetragonal prism.png
(4.4.4)
(2)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Dodecagonal prism.png
(4.4.12)
(1)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t01.png
(3.12.12)
Runcitruncated order-4 hexagonal tiling honeycomb verf.png H3 634-1101.png
29 runcitruncated order-6 cubic
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t0,1,3{4,3,6}
(1)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png
(3.8.8)
(2)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Octagonal prism.png
(4.4.8)
(1)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png
Hexagonal prism.png
(4.4.6)
(1)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t02.png
(3.4.6.4)
Runcitruncated order-6 cubic honeycomb verf.png H3 634-1011.png
30 omnitruncated order-6 cubic
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t0,1,2,3{6,3,4}
(1)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Great rhombicuboctahedron.png
(4.6.8)
(1)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Octagonal prism.png
(4.4.8)
(1)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Dodecagonal prism.png
(4.4.12)
(1)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t012.png
(4.6.12)
Omnitruncated order-4 hexagonal tiling honeycomb verf.png H3 634-1111.png
Alternated forms
# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 4.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 4.pngCDel node n4.png
2
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
Alt
[87] alternated order-6 cubic
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png
h{4,3,6}
Tetrahedron.png CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
(3.3.3)
    Uniform tiling 63-t2.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
(3.3.3.3.3.3)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.png
(3.6.3.6)
[88] cantic order-6 cubic
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png
h2{4,3,6}
(2)
Truncated tetrahedron.png
(3.6.6)
- - (1)
Uniform tiling 63-t1.png
(3.6.3.6)
(2)
Uniform tiling 63-t12.png
(6.6.6)
Cantic order-6 cubic honeycomb verf.png
[89] runcic order-6 cubic
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node 1.png
h3{4,3,6}
(1)
Tetrahedron.png
(3.3.3)
- - (1)
Uniform tiling 63-t0.png
(6.6.6)
(3)
Uniform tiling 63-t02.png
(3.4.6.4)
Runcic order-6 cubic honeycomb verf.png
[90] runcicantic order-6 cubic
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node 1.png
h2,3{4,3,6}
(1)
Truncated tetrahedron.png
(3.6.6)
- - (1)
Uniform tiling 63-t01.png
(3.12.12)
(2)
Uniform tiling 63-t012.png
(4.6.12)
Runcicantic order-6 cubic honeycomb verf.png
[141] alternated order-4 hexagonal
CDel node h1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node.pngCDel split1.pngCDel branch 10luru.pngCDel split2.pngCDel node.png
h{6,3,4}
- - Uniform tiling 333-t0.png
(3.3.3.3.3.3)
Uniform polyhedron-43-t2.png
(3.3.3.3)
Uniform polyhedron-43-t12.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
(4.6.6)
[142] cantic order-4 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel split1.pngCDel branch 10luru.pngCDel split2.pngCDel node 1.png
h1{6,3,4}
(1)
Uniform polyhedron-43-t1.png
(3.4.3.4)
- (2)
Uniform tiling 333-t01.png
(3.6.3.6)
(2)
Uniform polyhedron-43-t12.png
(4.6.6)
Cantic order-4 hexagonal tiling honeycomb verf.png
[143] runcic order-4 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png
h3{6,3,4}
(1)
Uniform polyhedron-43-t0.png
(4.4.4)
(1)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
(3)
Uniform polyhedron-43-t02.png
(3.4.4.4)
Runcic order-4 hexagonal tiling honeycomb verf.png
[144] runcicantic order-4 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
h2,3{6,3,4}
(1)
Uniform polyhedron-43-t01.png
(3.8.8)
(1)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 333-t01.png
(3.6.3.6)
(2)
Uniform polyhedron-43-t012.png
(4.6.8)
Runcicantic order-4 hexagonal tiling honeycomb verf.png
[151] quarter order-4 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel node 1.pngCDel split1.pngCDel branch 10luru.pngCDel split2.pngCDel node.png
q{6,3,4}
(3)
Uniform polyhedron-33-t01.png
(1)
Uniform polyhedron-33-t0.png
- (1)
Uniform tiling 333-t0.png
(3)
Uniform tiling 333-t02.png
Paracompact honeycomb DP3 1100 verf.png
Nonuniform bisnub order-6 cubic
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.pngCDel node h.pngCDel split1.pngCDel branch hh.pngCDel split2.pngCDel node h.png
2s{4,3,6}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Uniform polyhedron-43-h01.svg
(3.3.3.3.3.3)
- - CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-h12.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)
Nonuniform runcic bisnub order-6 cubic
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node h.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Nonuniform snub rectified order-6 cubic
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel branch hh.pngCDel split2.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{4,3,6}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Snub hexahedron.png
(3.3.3.3.3)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
Tetrahedron.png
(3.3.3)
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Trigonal antiprism.png
(3.3.3.4)
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)
Nonuniform runcic snub rectified order-6 cubic
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
sr3{4,3,6}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node 1.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node 1.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Nonuniform snub rectified order-4 hexagonal
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
sr{6,3,4}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Uniform polyhedron-43-h01.svg
(3.3.3.3.3.3)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
Tetrahedron.png
(3.3.3)
- CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)
Nonuniform runcisnub rectified order-4 hexagonal
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
sr3{6,3,4}
Nonuniform omnisnub rectified order-6 cubic
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
ht0,1,2,3{6,3,4}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
Snub hexahedron.png
(3.3.3.3.4)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png
Square antiprism.png
(3.3.3.4)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Hexagonal antiprism.png
(3.3.3.6)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)

[6,3,5] family[edit]

# Honeycomb name
Coxeter diagram
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 5.pngCDel node n5.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 5.pngCDel node n5.png
2
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 2.pngCDel node n5.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
31 order-5 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{6,3,5}
- - - (20)
Uniform tiling 63-t0.png
(6)3
Order-5 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Icosahedron
H3 635 FC boundary.png
32 rectified order-5 hexagonal
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t1{6,3,5} or r{6,3,5}
(2)
Uniform polyhedron-53-t2.png
(3.3.3.3.3)
- - (5)
Uniform tiling 63-t1.png
(3.6)2
Rectified order-5 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png
(5.4.4)
H3 635 boundary 0100.png
33 rectified order-6 dodecahedral
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t1{5,3,6} or r{5,3,6}
(5)
Uniform polyhedron-53-t1.png
(3.5.3.5)
- - (2)
Uniform tiling 63-t2.png
(3)6
Rectified order-6 dodecahedral honeycomb verf.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
(6.4.4)
H3 536 CC center 0100.png
34 order-6 dodecahedral
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
{5,3,6}
Uniform polyhedron-53-t0.png
(5.5.5)
- - - (∞)
Uniform tiling 63-t2.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
(3)6
H3 536 CC center.png
35 truncated order-5 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t0,1{6,3,5} or t{6,3,5}
(1)
Uniform polyhedron-53-t2.png
(3.3.3.3.3)
- - (5)
Uniform tiling 63-t01.png
3.12.12
Truncated order-5 hexagonal tiling honeycomb verf.png H3 635-1100.png
36 cantellated order-6 dodecahedral
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t0,2{6,3,5} or rr{6,3,5}
(1)
Uniform polyhedron-53-t1.png
(3.5.3.5)
(2)
Pentagonal prism.png
(5.4.4)
- (2)
Uniform tiling 63-t02.png
3.4.6.4
Cantellated order-5 hexagonal tiling honeycomb verf.png H3 635-1010.png
37 runcinated order-6 dodecahedral
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,3{6,3,5}
(1)
Uniform polyhedron-53-t0.png
(5.5.5)
- (6)
Hexagonal prism.png
(6.4.4)
(1)
Uniform tiling 63-t0.png
(6)3
Runcinated order-5 hexagonal tiling honeycomb verf.png H3 635-1001.png
38 cantellated order-6 dodecahedral
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,2{5,3,6} or rr{5,3,6}
(2)
Uniform polyhedron-53-t02.png
(4.3.4.5)
- (2)
Hexagonal prism.png
(6.4.4)
(1)
Uniform tiling 63-t1.png
(3.6)2
Cantellated order-6 dodecahedral honeycomb verf.png H3 635-0101.png
39 bitruncated order-6 dodecahedral
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t1,2{6,3,5} or 2t{6,3,5}
(2)
Uniform polyhedron-53-t12.png
(5.6.6)
- - (2)
Uniform tiling 63-t12.png
(6)3
Bitruncated order-5 hexagonal tiling honeycomb verf.png H3 635-0110.png
40 truncated order-6 dodecahedral
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t0,1{5,3,6} or t{5,3,6}
(6)
Uniform polyhedron-53-t01.png
(3.10.10)
- - (1)
Uniform tiling 63-t2.png
(3)6
Truncated order-6 dodecahedral honeycomb verf.png H3 635-0011.png
41 cantitruncated order-5 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t0,1,2{6,3,5} or tr{6,3,5}
(1)
Uniform polyhedron-53-t12.png
(5.6.6)
(1)
Pentagonal prism.png
(5.4.4)
- (2)
Uniform tiling 63-t012.png
4.6.10
Cantitruncated order-5 hexagonal tiling honeycomb verf.png H3 635-1110.png
42 runcitruncated order-5 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,1,3{6,3,5}
(1)
Uniform polyhedron-53-t02.png
(4.3.4.5)
(1)
Pentagonal prism.png
(5.4.4)
(2)
Dodecagonal prism.png
(12.4.4)
(1)
Uniform tiling 63-t01.png
3.12.12
Runcitruncated order-5 hexagonal tiling honeycomb verf.png H3 635-1011.png
43 runcitruncated order-6 dodecahedral
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t0,1,3{5,3,6}
(1)
Uniform polyhedron-53-t01.png
(3.10.10)
(1)
Decagonal prism.png
(10.4.4)
(2)
Hexagonal prism.png
(6.4.4)
(1)
Uniform tiling 63-t02.png
3.4.6.4
Runcitruncated order-6 dodecahedral honeycomb verf.png H3 635-1011.png
44 cantitruncated order-6 dodecahedral
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t0,1,2{5,3,6} or tr{5,3,6}
(1)
Uniform polyhedron-53-t012.png
(4.6.10)
- (2)
Hexagonal prism.png
(6.4.4)
(1)
Uniform tiling 63-t12.png
(6)3
Cantitruncated order-6 dodecahedral honeycomb verf.png H3 635-0111.png
45 omnitruncated order-6 dodecahedral
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t0,1,2,3{6,3,5}
(1)
Uniform polyhedron-53-t012.png
(4.6.10)
(1)
Decagonal prism.png
(10.4.4)
(1)
Dodecagonal prism.png
(12.4.4)
(1)
Uniform tiling 63-t012.png
4.6.12
Omnitruncated order-5 hexagonal tiling honeycomb verf.png H3 635-1111.png
Alternated forms
# Honeycomb name
Coxeter diagram
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 5.pngCDel node n5.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 5.pngCDel node n5.png
2
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 2.pngCDel node n5.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
Alt
[145] alternated order-5 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
h{6,3,5}
- - - (20)
Uniform tiling 333-t1.png
(3)6
(12)
Icosahedron.png
(3)5
Uniform polyhedron-53-t12.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
(5.6.6)
[146] cantic order-5 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node.png
h2{6,3,5}
(1)
Uniform polyhedron-53-t1.png
(3.5.3.5)
- (2)
Uniform tiling 333-t01.png
(3.6.3.6)
(2)
Uniform polyhedron-53-t12.png
(5.6.6)
Cantic order-5 hexagonal tiling honeycomb verf.png
[147] runcic order-5 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node 1.png
h3{6,3,5}
(1)
Uniform polyhedron-53-t0.png
(5.5.5)
(1)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
(3)
Uniform polyhedron-53-t02.png
(3.4.5.4)
Runcic order-5 hexagonal tiling honeycomb verf.png
[148] runcicantic order-5 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node 1.png
h2,3{6,3,5}
(1)
Uniform polyhedron-53-t01.png
(3.10.10)
(1)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 333-t01.png
(3.6.3.6)
(2)
Uniform polyhedron-53-t012.png
(4.6.10)
Runcicantic order-5 hexagonal tiling honeycomb verf.png
Nonuniform snub rectified order-6 dodecahedral
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel branch hh.pngCDel split2.pngCDel node h.pngCDel 5.pngCDel node h.png
sr{5,3,6}
Uniform polyhedron-53-s012.png
(3.3.5.3.5)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
- Trigonal antiprism.png
(3.3.3.3)
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Uniform tiling 63-h12.png
(3.3.3.3.3.3)
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-t0.png
irr. tet
Nonuniform omnisnub order-5 hexagonal
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
ht0,1,2,3{6,3,5}
Uniform polyhedron-53-s012.png
(3.3.5.3.5)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 5.pngCDel node h.png
Pentagonal antiprism.png
(3.3.3.5)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 5.pngCDel node h.png
Hexagonal antiprism.png
(3.3.3.6)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.6.3.6)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-t0.png
irr. tet

[6,3,6] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png

# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 6.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 6.pngCDel node n4.png
2
CDel node n1.pngCDel 6.pngCDel node n3.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
46 order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{6,3,6}
- - - (20)
Uniform tiling 63-t0.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
(6.6.6)
Uniform tiling 63-t2.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
(3.3.3.3.3.3)
H3 636 FC boundary.png
47 rectified order-6 hexagonal
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
t1{6,3,6} or r{6,3,6}
(2)
Uniform tiling 63-t2.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png
(3.3.3.3.3.3)
- - (6)
Uniform tiling 63-t1.png
(3.6.3.6)
Rectified order-6 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png
(6.4.4)
H3 636 boundary 0100.png
48 truncated order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
t0,1{6,3,6} or t{6,3,6}
(1)
Uniform tiling 63-t2.png
(3.3.3.3.3.3)
- - (6)
Uniform tiling 63-t01.png
(3.12.12)
Truncated order-6 hexagonal tiling honeycomb verf.png H3 636-1100.png
49 cantellated order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t0,2{6,3,6} or rr{6,3,6}
(1)
Uniform tiling 63-t1.png
(3.6.3.6)
(2)
Hexagonal prism.png
(4.4.6)
- (2)
Uniform tiling 63-t012.png
(3.6.4.6)
Cantellated order-6 hexagonal tiling honeycomb verf.png H3 636-1010.png
50 Runcinated order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
t0,3{6,3,6}
(1)
Uniform tiling 63-t0.png
(6.6.6)
(3)
Hexagonal prism.png
(4.4.6)
(3)
Hexagonal prism.png
(4.4.6)
(1)
Uniform tiling 63-t0.png
(6.6.6)
Runcinated order-6 hexagonal tiling honeycomb verf.png H3 636-1001.png
51 cantitruncated order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t0,1,2{6,3,6} or tr{6,3,6}
(1)
Uniform tiling 63-t12.png
(6.6.6)
(1)
Hexagonal prism.png
(4.4.6)
- (2)
Uniform tiling 63-t012.png
(4.6.12)
Cantitruncated order-6 hexagonal tiling honeycomb verf.png H3 636-1110.png
52 runcitruncated order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
t0,1,3{6,3,6}
(1)
Uniform tiling 63-t012.png
(3.6.4.6)
(1)
Hexagonal prism.png
(4.4.6)
(2)
Decagonal prism.png
(4.4.12)
(1)
Uniform tiling 63-t01.png
(3.12.12)
Runcitruncated order-6 hexagonal tiling honeycomb verf.png H3 636-1011.png
53 omnitruncated order-6 hexagonal
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.png
t0,1,2,3{6,3,6}
(1)
Uniform tiling 63-t012.png
(4.6.12)
(1)
Decagonal prism.png
(4.4.12)
(1)
Decagonal prism.png
(4.4.12)
(1)
Uniform tiling 63-t012.png
(4.6.12)
Omnitruncated order-6 hexagonal tiling honeycomb verf.png H3 636-1111.png
[1] bitruncated order-6 hexagonal
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel branch 11.pngCDel splitcross.pngCDel branch 11.png
t1,2{6,3,6} or 2t{6,3,6}
(2)
Uniform tiling 63-t12.png
(6.6.6)
- - (2)
Uniform tiling 63-t12.png
(6.6.6)
Bitruncated order-6 hexagonal tiling honeycomb verf.png H3 636-0110.png
Alternated forms
# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
CDel node n2.pngCDel 3.pngCDel node n3.pngCDel 6.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 6.pngCDel node n4.png
2
CDel node n1.pngCDel 6.pngCDel node n3.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 6.pngCDel node n2.pngCDel 3.pngCDel node n3.png
Alt
[47] rectified order-6 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel node.pngCDel splitsplit1.pngCDel branch4 11.pngCDel splitsplit2.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png
q{6,3,6} = r{6,3,6}
(2)
Uniform tiling 63-t2.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png
(3.3.3.3.3.3)
- - (6)
Uniform tiling 63-t1.png
(3.6.3.6)
Rectified order-6 hexagonal tiling honeycomb verf.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png
(6.4.4)
H3 636 boundary 0100.png
[54] triangular
(CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png) = CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
h{6,3,6} = {3,6,3}
- - - CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 63-t2.png
(3.3.3.3.3.3)
Uniform tiling 63-t0.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{6,3}
H3 363 FC boundary.png
[55] cantic order-6 hexagonal
( CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png) = CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
h2{6,3,6} = r{3,6,3}
(1)
Uniform tiling 63-t1.png
(3.6.3.6)
- (2)
Uniform tiling 63-t12.png
(6.6.6)
(2)
Uniform tiling 333-t01.png
(3.6.3.6)
Cantic order-6 hexagonal tiling honeycomb verf.png H3 363 boundary 0100.png
[149] runcic order-6 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node 1.png
h3{6,3,6}
(1)
Uniform tiling 63-t0.png
(6.6.6)
(1)
Triangular prism.png
(4.4.3)
(3)
Uniform tiling 63-t02.png
(3.4.6.4)
(1)
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
Runcic order-6 hexagonal tiling honeycomb verf.png
[150] runcicantic order-6 hexagonal
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node 1.png
h2,3{6,3,6}
(1)
Uniform tiling 63-t01.png
(3.12.12)
(1)
Triangular prism.png
(4.4.3)
(2)
Uniform tiling 63-t012.png
(4.6.12)
(1)
Uniform tiling 333-t01.png
(3.6.3.6)
Runcicantic order-6 hexagonal tiling honeycomb verf.png
[137] alternated hexagonal
(CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h0.pngCDel node h1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel branch hh.pngCDel splitcross.pngCDel branch hh.png) = CDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
2s{6,3,6} = h{6,3,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
Uniform tiling 63-h12.png
(3.3.3.3.6)
- - CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-h12.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)
Uniform polyhedron-33-t01.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
(3.6.6)
Nonuniform snub rectified order-6 hexagonal
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
sr{6,3,6}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
Uniform tiling 63-h12.png
(3.3.3.3.3.3)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
Trigonal antiprism.png
(3.3.3.3)
- CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)
Nonuniform alternated runcinated order-6 hexagonal
CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.png
ht0,3{6,3,6}
CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.png
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Trigonal antiprism.png
(3.3.3.3)
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Trigonal antiprism.png
(3.3.3.3)
CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 333-t0.png
(3.3.3.3.3.3)
Tetrahedron.png
+(3.3.3)
Nonuniform omnisnub order-6 hexagonal
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h.png
ht0,1,2,3{6,3,6}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node h.png
Hexagonal antiprism.png
(3.3.3.6)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Hexagonal antiprism.png
(3.3.3.6)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
Tetrahedron.png
+(3.3.3)

[3,6,3] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: [3,6,3] or CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
CDel node n2.pngCDel 6.pngCDel node n3.pngCDel 3.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 6.pngCDel node n3.png
54 triangular
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{3,6,3}
- - - (∞)
Uniform tiling 63-t2.png
{3,6}
Uniform tiling 63-t0.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{6,3}
H3 363 FC boundary.png
55 rectified triangular
CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
t1{3,6,3} or r{3,6,3}
(2)
Uniform tiling 63-t0.png
(6)3
- - (3)
Uniform tiling 63-t1.png
(3.6)2
Rectified triangular tiling honeycomb verf.png
(3.4.4)
H3 363 boundary 0100.png
56 cantellated triangular
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2{3,6,3} or rr{3,6,3}
(1)
Uniform tiling 63-t1.png
(3.6)2
(2)
Triangular prism.png
(4.4.3)
- (2)
Uniform tiling 63-t02.png
(3.6.4.6)
Cantellated triangular tiling honeycomb verf.png H3 363-1010.png
57 runcinated triangular
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,6,3}
(1)
Uniform tiling 63-t2.png
(3)6
(6)
Triangular prism.png
(4.4.3)
(6)
Triangular prism.png
(4.4.3)
(1)
Uniform tiling 63-t2.png
(3)6
Runcinated triangular tiling honeycomb verf.png H3 363-1001.png
58 bitruncated triangular
CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{3,6,3} or 2t{3,6,3}
(2)
Uniform tiling 63-t01.png
(3.12.12)
- - (2)
Uniform tiling 63-t01.png
(3.12.12)
Bitruncated triangular tiling honeycomb verf.png H3 363-0110.png
59 cantitruncated triangular
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2{3,6,3} or tr{3,6,3}
(1)
Uniform tiling 63-t01.png
(3.12.12)
(1)
Triangular prism.png
(4.4.3)
- (2)
Uniform tiling 63-t012.png
(4.6.12)
Cantitruncated triangular tiling honeycomb verf.png H3 363-1110.png
60 runcitruncated triangular
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,6,3}
(1)
Uniform tiling 63-t02.png
(3.6.4.6)
(1)
Triangular prism.png
(4.4.3)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Uniform tiling 63-t01.png
(6)3
Runcitruncated triangular tiling honeycomb verf.png H3 363-1101.png
61 omnitruncated triangular
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,6,3}
(1)
Uniform tiling 63-t012.png
(4.6.12)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Uniform tiling 63-t012.png
(4.6.12)
Omnitruncated triangular tiling honeycomb verf.png H3 363-1111.png
[1] truncated triangular
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel branch 11.pngCDel splitcross.pngCDel branch 11.png
t0,1{3,6,3} or t{3,6,3} = {6,3,3}
(1)
Uniform tiling 63-t0.png
(6)3
- - (3)
Uniform tiling 63-t12.png
(6)3
Truncated triangular tiling honeycomb verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3}
H3 363-1100.png
Alternated forms
# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
CDel node n2.pngCDel 6.pngCDel node n3.pngCDel 3.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 6.pngCDel node n3.png
Alt
[56] cantellated triangular
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
s2{3,6,3}
(1)
Uniform tiling 63-t1.png
(3.6)2
CDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
- - (2)
Rhombitrihexagonal tiling snub edge coloring.png
(3.6.4.6)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.png
Triangular prism.png
(3.4.4)
Cantellated triangular tiling honeycomb verf.png H3 363-1010.png
[60] runcitruncated triangular
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
s2,3{3,6,3}
(1)
Uniform tiling 333-t012.png
(6)3
CDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
- (1)
Triangular prism.png
(4.4.3)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
(1)
Rhombitrihexagonal tiling snub edge coloring.png
(3.6.4.6)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.png
(2)
Hexagonal prism.png
(4.4.6)
Runcitruncated triangular tiling honeycomb verf.png H3 363-1101.png
[137] alternated hexagonal
( CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel branch hh.pngCDel splitcross.pngCDel branch hh.png ) = (CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png)
s{3,6,3}
Uniform tiling 333-t1.png
(3)6
CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
- - Uniform tiling 63-h12.png
(3)6
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
Tetrahedron.png
+(3)3
Uniform polyhedron-33-t01.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
(3.6.6)
Scaliform runcisnub triangular
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
s3{3,6,3}
Uniform tiling 333-t02.png
r{6,3}
CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
- Triangular prism.png
(3.4.4)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
Uniform tiling 333-t1.png
(3)6
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
Triangular cupola.png
tricup
Nonuniform omnisnub triangular tiling honeycomb
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{3,6,3}
Uniform tiling 63-snub.png
(3.3.3.3.6)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Octahedron.png
(3)4
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
Octahedron.png
(3)4
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
Uniform tiling 63-snub.png
(3.3.3.3.6)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h.png
Tetrahedron.png
+(3)3

[4,4,3] family[edit]

There are 15 forms, generated by ring permutations of the Coxeter group: [4,4,3] or CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
CDel node n2.pngCDel 4.pngCDel node n3.pngCDel 3.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 4.pngCDel node n3.png
62 square
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
{4,4,3}
- - - (6)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
Square tiling honeycomb verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cube
H3 443 FC boundary.png
63 rectified square
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
t1{4,4,3} or r{4,4,3}
(2)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png
- - (3)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t1.png
Rectified square tiling honeycomb verf.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Triangular prism
H3 443 boundary 0100.png
64 rectified order-4 octahedral
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t1{3,4,4} or r{3,4,4}
(4)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.png
- - (2)
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t2.png
Rectified order-4 octahedral honeycomb verf.png H3 344 CC center 0100.png
65 order-4 octahedral
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,4,4}
(∞)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t2.png
- - - Uniform tiling 44-t0.svg CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png H3 344 CC center.png
66 truncated square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1{4,4,3} or t{4,4,3}
(1)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png
- - (3)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
Truncated square tiling honeycomb verf.png H3 443-1100.png
67 truncated order-4 octahedral
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1{3,4,4} or t{3,4,4}
(4)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t12.png
- - (1)
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t2.png
Truncated order-4 octahedral honeycomb verf.png H3 443-0011.png
68 bitruncated square
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{4,4,3} or 2t{4,4,3}
(2)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t01.png
- - (2)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t12.png
Bitruncated square tiling honeycomb verf.png H3 443-0110.png
69 cantellated square
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2{4,4,3} or rr{4,4,3}
(1)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.png
(2)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Triangular prism.png
- (2)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.png
Cantellated square tiling honeycomb verf.png H3 443-1010.png
70 cantellated order-4 octahedral
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2{3,4,4} or rr{3,4,4}
(2)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t02.png
- (2)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Tetragonal prism.png
(1)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t1.png
Cantellated order-4 octahedral honeycomb verf.png H3 443-0101.png
71 runcinated square
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{4,4,3}
(1)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t2.png
(3)
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
Triangular prism.png
(3)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
Tetragonal prism.png
(1)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
Runcinated square tiling honeycomb verf.png H3 443-1001.png
72 cantitruncated square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2{4,4,3} or tr{4,4,3}
(1)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t01.png
(1)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Triangular prism.png
- (2)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t012.png
Cantitruncated square tiling honeycomb verf.png H3 443-1110.png
73 cantitruncated order-4 octahedral
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2{3,4,4} or tr{3,4,4}
(2)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t012.png
- (1)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Tetragonal prism.png
(1)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t12.png
Cantitruncated order-4 octahedral honeycomb verf.png H3 443-0111.png
74 runcitruncated square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{4,4,3}
(1)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t02.png
(1)
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
Triangular prism.png
(2)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Octagonal prism.png
(1)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
Runcitruncated square tiling honeycomb verf.png H3 443-1101.png
75 runcitruncated order-4 octahedral
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,4,4}
(1)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t12.png
(2)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Hexagonal prism.png
(1)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
Tetragonal prism.png
(1)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.png
Runcitruncated order-4 octahedral honeycomb verf.png H3 443-1011.png
76 omnitruncated square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{4,4,3}
(1)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t012.png
(1)
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Hexagonal prism.png
(1)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Octagonal prism.png
(1)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t012.png
Omnitruncated square tiling honeycomb verf.png H3 443-1111.png
Alternated forms
# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
CDel node n2.pngCDel 4.pngCDel node n3.pngCDel 3.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 3.pngCDel node n4.png
2
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 4.pngCDel node n3.png
Alt
[83] alternated square
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.png
h{4,4,3}
- - - CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png {4,3} Uniform polyhedron-43-t1.png
(4.3.4.3)
[84] cantic order-6 cubic
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node 1.pngCDel 3.pngCDel node.png
h2{4,4,3}
Uniform polyhedron-43-t1.png
(3.4.3.4)
- Uniform polyhedron-43-t01.png
(3.8.8)
Uniform tiling 44-t12.png
(4.8.8)
Cantic square tiling honeycomb verf.png
[85] runcic square
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node 1.png
h3{4,4,3}
Uniform polyhedron-43-t1.png
(3.4.3.4)
- Uniform polyhedron-43-t01.png
(3.8.8)
Uniform tiling 44-t12.png
(4.8.8)
Runcic square tiling honeycomb verf.png
[86] runcicantic square
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t12.png
(4.6.6)
- Uniform polyhedron-43-t012.png
(3.4.4.4)
Uniform tiling 44-t12.png
(4.8.8)
Runcicantic square tiling honeycomb verf.png
Nonsimplectic alternated rectified square
CDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10.pngCDel 2a2b-cross.pngCDel nodes 10ru.pngCDel split2.pngCDel node.png
hr{4,4,3}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png - - CDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.png {}x{3}
Scaliform snub order-4 octahedral
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel nodes.pngCDel split2-44.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel node.pngCDel split1-44.pngCDel nodes hh.pngCDel split2.pngCDel node h.png
s{3,4,4}
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png - - CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.png irr. {}v{4}
Scaliform runcisnub order-4 octahedral
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
s3{3,4,4}
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node 1.pngCDel 2.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node h.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.png cup-4
Nonuniform snub square
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png
s{4,4,3}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png - - CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png irr. {3,3}
Nonuniform snub rectified order-4 octahedral
CDel node.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,4,4}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png - CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png CDel node.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png irr. {3,3}
Nonuniform alternated runcitruncated square
CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,3{3,4,4}
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.png irr. {}v{4}
Nonuniform omnisnub square
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{4,4,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-43-s012.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
Octahedron.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png
Square antiprism.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 44-snub.png
irr. {3,3}

[4,4,4] family[edit]

There are 9 forms, generated by ring permutations of the Coxeter group: [4,4,4] or CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png.

# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Symmetry Vertex figure Picture
0
CDel node n2.pngCDel 4.pngCDel node n3.pngCDel 4.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 4.pngCDel node n4.png
2
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 4.pngCDel node n3.png
77 order-4 square
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
{4,4,4}
- - - CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
[4,4,4] CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Hexahedron.png
Cube
H3 444 FC boundary.png
78 truncated order-4 square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
t0,1{4,4,4} or t{4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
- - CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
[4,4,4] Truncated order-4 square tiling honeycomb verf.png H3 444-1100.png
79 bitruncated order-4 square
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
t1,2{4,4,4} or 2t{4,4,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
- - CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t12.png
[[4,4,4]] Bitruncated order-4 square tiling honeycomb verf.png H3 444-0110.png
80 runcinated order-4 square
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,3{4,4,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t2.png
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png
Tetragonal prism.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
Tetragonal prism.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
[[4,4,4]] Runcinated order-4 square tiling honeycomb verf.png H3 444-1001.png
81 runcitruncated order-4 square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,3{4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.png
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png
Tetragonal prism.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Octagonal prism.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
[4,4,4] Runcitruncated order-4 square tiling honeycomb verf.png H3 444-1101.png
82 omnitruncated order-4 square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t0,1,2,3{4,4,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t012.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Octagonal prism.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Octagonal prism.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t012.png
[[4,4,4]] Omnitruncated order-4 square tiling honeycomb verf.png H3 444-1111.png
[62] square
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
t1{4,4,4} or r{4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
- - CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t1.png
[4,4,4] Uniform tiling 44-t0.svg
Square tiling
H3 443 FC boundary.png
[63] rectified square
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
t0,2{4,4,4} or rr{4,4,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
- CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.png
[4,4,4] Cantellated order-4 square tiling honeycomb verf.png H3 444-1010.png
[66] truncated order-4 square
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
t0,1,2{4,4,4} or tr{4,4,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
- CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t012.png
[4,4,4] Cantitruncated order-4 square tiling honeycomb verf.png H3 444-0111.png
Alternated constructions
# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Symmetry Vertex figure Picture
0
CDel node n2.pngCDel 4.pngCDel node n3.pngCDel 4.pngCDel node n4.png
1
CDel node n1.pngCDel 2.pngCDel node n3.pngCDel 4.pngCDel node n4.png
2
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 2.pngCDel node n4.png
3
CDel node n1.pngCDel 4.pngCDel node n2.pngCDel 4.pngCDel node n4.png
Alt
[62] Square
( CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel nodes 11.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png ) = CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0.svg
(4.4.4.4)
- - Uniform tiling 44-t1.png
(4.4.4.4)
[1+,4,4,4]
=[4,4,4]
Bitruncated order-4 square tiling honeycomb verf.png H3 443 FC boundary.png
[63] rectified square
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
s2{4,4,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Tetragonal prism.png
- CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.png
[4+,4,4] Cantellated order-4 square tiling honeycomb verf.png H3 443 boundary 0100.png
[77] order-4 square
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.png
- - - CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.svg
[1+,4,4,4]
=[4,4,4]
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Hexahedron.png
Cube
H3 444 FC boundary.png
[78] truncated order-4 square
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.png
Uniform tiling 44-t12.png
(4.8.8)
- Uniform tiling 44-t12.png
(4.8.8)
- Uniform tiling 44-t1.png
(4.4.4.4)
[1+,4,4,4]
=[4,4,4]
Truncated order-4 square tiling honeycomb verf.png H3 444-1100.png
[79] bitruncated order-4 square
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel nodes 11.pngCDel split2-44.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t01.png
(4.8.8)
- - Uniform tiling 44-t01.png
(4.8.8)
Uniform tiling 44-t012.png
(4.8.8)
[1+,4,4,4]
=[4,4,4]
Bitruncated order-4 square tiling honeycomb verf.png H3 444-0110.png
[81] runcitruncated order-4 square tiling
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
s2,3{4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t02.png
CDel node 1.pngCDel 2.pngCDel node.png