Order-4 hexagonal tiling honeycomb

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Order-4 hexagonal tiling honeycomb
H3 634 FC boundary.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {6,3,4}
{6,31,1}
t0,1{(3,6)2}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel branch 11.pngCDel 6a6b.pngCDel branch.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node.png
CDel K6 636 11.png CDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png
Cells {6,3} Uniform tiling 63-t0.png Uniform tiling 63-t12.png Uniform tiling 333-t012.png
Faces hexagon {6}
Edge figure square {4}
Vertex figure Order-4 hexagonal tiling honeycomb verf.png
octahedron
Dual Order-6 cubic honeycomb
Coxeter groups , [4,3,6]
, [6,31,1]
, [(6,3)[2]]
PropertiesRegular, quasiregular

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

Contents

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, eight hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes. [1]

Images

Hyperbolic 3d order 4 hexagonal tiling.png
Perspective projection
Order-4 hexagonal tiling honeycomb cell.png
One cell, viewed from outside the Poincare sphere
H2 tiling 33i-7.png
The vertices of a t{(3,,3)}, CDel node 1.pngCDel split1.pngCDel branch 11.pngCDel labelinfin.png tiling exist as a 2-hypercycle within this honeycomb
Order-4 hexagonal tiling honeycomb one cell horocycle.png
The honeycomb is analogous to the H2 order-4 apeirogonal tiling, {,4}, shown here with one green apeirogon outlined by its horocycle

Symmetry

Subgroup relations Hyperbolic subgroup tree 36.png
Subgroup relations

The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.

The half-symmetry uniform construction {6,31,1} has two types (colors) of hexagonal tilings, with Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png. A quarter-symmetry construction also exists, with four colors of hexagonal tilings: CDel label6.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel label6.png.

An additional two reflective symmetries exist with non-simplectic fundamental domains: [6,3*,4], which is index 6, with Coxeter diagram CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.png; and [6,(3,4)*], which is index 48. The latter has a cubic fundamental domain, and an octahedral Coxeter diagram with three axial infinite branches: CDel K6 636 11.png . It can be seen as using eight colors to color the hexagonal tilings of the honeycomb.

The order-4 hexagonal tiling honeycomb contains CDel node 1.pngCDel 3.pngCDel node 1.pngCDel ultra.pngCDel node.png, which tile 2-hypercycle surfaces and are similar to the truncated infinite-order triangular tiling, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel infin.pngCDel node.png:

H2 tiling 23i-6.png

The order-4 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

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

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb.

[6,3,4] family honeycombs
{6,3,4} r{6,3,4} t{6,3,4} rr{6,3,4} t0,3{6,3,4} tr{6,3,4} t0,1,3{6,3,4} t0,1,2,3{6,3,4}
H3 634 FC boundary.png H3 634 boundary 0100.png H3 634-1100.png H3 634-1010.png H3 634-1001.png H3 634-1110.png H3 634-1101.png H3 634-1111.png
H3 436 CC center.png H3 436 CC center 0100.png H3 634-0011.png H3 634-0101.png H3 634-0110.png H3 634-0111.png H3 634-1011.png
{4,3,6} r{4,3,6} t{4,3,6} rr{4,3,6} 2t{4,3,6} tr{4,3,6} t0,1,3{4,3,6} t0,1,2,3{4,3,6}

The order-4 hexagonal tiling honeycomb has a related alternated honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png, with triangular tiling and octahedron cells.

It is a part of sequence of regular honeycombs of the form {6,3,p}, all of which are composed of hexagonal tiling cells:

{6,3,p} honeycombs
Space H3
Form Paracompact Noncompact
Name {6,3,3} {6,3,4} {6,3,5} {6,3,6} {6,3,7} {6,3,8} ... {6,3,}
Coxeter
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u.pngCDel splitplit2u.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
CDD 6-3star-infin.png
Image H3 633 FC boundary.png H3 634 FC boundary.png H3 635 FC boundary.png H3 636 FC boundary.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 6-3-8 poincare.png Hyperbolic honeycomb 6-3-i poincare.png
Vertex
figure
{3,p}
CDel node 1.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Octahedron.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
Icosahedron.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.png
Order-7 triangular tiling.svg
{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H2-8-3-primal.svg
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2 tiling 23i-4.png
{3,}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel labelinfin.png

This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.

{p,3,4} regular honeycombs
Space S3 E3 H3
FormFiniteAffineCompactParacompactNoncompact
Name {3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
CDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 11.png
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
{6,3,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.png
CDel node 1.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes 11.png
{7,3,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel nodes.png
{8,3,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.pngCDel uaub.pngCDel nodes.png
CDel node 1.pngCDel ultra.pngCDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.pngCDel uaub.pngCDel nodes 11.png
... {,3,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.pngCDel uaub.pngCDel nodes.png
CDel node 1.pngCDel ultra.pngCDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.pngCDel uaub.pngCDel nodes 11.png
Image Stereographic polytope 16cell.png Cubic honeycomb.png H3 534 CC center.png H3 634 FC boundary.png Hyperbolic honeycomb 7-3-4 poincare.png Hyperbolic honeycomb 8-3-4 poincare.png Hyperbolic honeycomb i-3-4 poincare.png
Cells Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.svg
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Heptagonal tiling.svg
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2-I-3-dual.svg
{,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

The aforementioned honeycombs are also quasiregular:

Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
SpaceEuclidean 4-spaceEuclidean 3-spaceHyperbolic 3-space
Name{3,3,4}
{3,31,1} =
{4,3,4}
{4,31,1} =
{5,3,4}
{5,31,1} =
{6,3,4}
{6,31,1} =
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
Image 16-cell nets.png Bicolor cubic honeycomb.png H3 534 CC center.png H3 634 FC boundary.png
Cells
{p,3}
Uniform polyhedron-33-t0.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t0.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-63-t0.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

Rectified order-4 hexagonal tiling honeycomb

Rectified order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{6,3,4} or t1{6,3,4}
Coxeter diagrams CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel node.pngCDel split1.pngCDel branch 11.pngCDel split2.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
Cells {3,4} Uniform polyhedron-43-t2.png
r{6,3} Uniform tiling 63-t1.png
Faces triangle {3}
hexagon {6}
Vertex figure Rectified order-4 hexagonal tiling honeycomb verf.png
square prism
Coxeter groups , [4,3,6]
, [4,3[3]]
, [6,31,1]
, [3[]×[]]
PropertiesVertex-transitive, edge-transitive

The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4}, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png has octahedral and trihexagonal tiling facets, with a square prism vertex figure.

H3 634 boundary 0100.png

It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{,4}, CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png which alternates apeirogonal and square faces:

H2 tiling 24i-2.png

Truncated order-4 hexagonal tiling honeycomb

Truncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{6,3,4} or t0,1{6,3,4}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
Cells {3,4} Uniform polyhedron-43-t2.png
t{6,3} Uniform tiling 63-t01.png
Faces triangle {3}
dodecagon {12}
Vertex figure Truncated order-4 hexagonal tiling honeycomb verf.png
square pyramid
Coxeter groups , [4,3,6]
, [6,31,1]
PropertiesVertex-transitive

The truncated order-4 hexagonal tiling honeycomb, t0,1{6,3,4}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png has octahedron and truncated hexagonal tiling facets, with a square pyramid vertex figure.

H3 634-1100.png

It is similar to the 2D hyperbolic truncated order-4 apeirogonal tiling, t{,4}, CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png with apeirogonal and square faces:

H2 tiling 24i-3.png

Bitruncated order-4 hexagonal tiling honeycomb

Bitruncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{6,3,4} or t1,2{6,3,4}
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel node 1.pngCDel split1.pngCDel branch 11.pngCDel split2.pngCDel node 1.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
Cells t{4,3} Uniform polyhedron-43-t12.png
t{3,6} Uniform tiling 63-t12.png
Faces square {4}
hexagon {6}
Vertex figure Bitruncated order-4 hexagonal tiling honeycomb verf.png
digonal disphenoid
Coxeter groups , [4,3,6]
, [4,3[3]]
, [6,31,1]
, [3[]×[]]
PropertiesVertex-transitive

The bitruncated order-4 hexagonal tiling honeycomb, t1,2{6,3,4}, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png has truncated octahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

H3 634-0110.png

Cantellated order-4 hexagonal tiling honeycomb

Cantellated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{6,3,4} or t0,2{6,3,4}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
Cells r{3,4} Uniform polyhedron-43-t1.png
{}x{4} Tetragonal prism.png
rr{6,3} Uniform tiling 63-t02.png
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Cantellated order-4 hexagonal tiling honeycomb verf.png
wedge
Coxeter groups , [4,3,6]
, [6,31,1]
PropertiesVertex-transitive

The cantellated order-4 hexagonal tiling honeycomb, t0,2{6,3,4}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png has cuboctahedron, cube, and rhombitrihexagonal tiling cells, with a wedge vertex figure.

H3 634-1010.png

Cantitruncated order-4 hexagonal tiling honeycomb

Cantitruncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{6,3,4} or t0,1,2{6,3,4}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
Cells t{3,4} Uniform polyhedron-43-t12.png
{}x{4} Tetragonal prism.png
tr{6,3} Uniform tiling 63-t012.png
Faces square {4}
hexagon {6}
dodecagon {12}
Vertex figure Cantitruncated order-4 hexagonal tiling honeycomb verf.png
mirrored sphenoid
Coxeter groups , [4,3,6]
, [6,31,1]
PropertiesVertex-transitive

The cantitruncated order-4 hexagonal tiling honeycomb, t0,1,2{6,3,4}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png has truncated octahedron, cube, and truncated trihexagonal tiling cells, with a mirrored sphenoid vertex figure.

H3 634-1110.png

Runcinated order-4 hexagonal tiling honeycomb

Runcinated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,4}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes 11.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node 1.png
Cells {4,3} Uniform polyhedron-43-t0.png
{}x{4} Tetragonal prism.png
{6,3} Uniform tiling 63-t0.png
{}x{6} Hexagonal prism.png
Faces square {4}
hexagon {6}
Vertex figure Runcinated order-4 hexagonal tiling honeycomb verf.png
irregular triangular antiprism
Coxeter groups , [4,3,6]
PropertiesVertex-transitive

The runcinated order-4 hexagonal tiling honeycomb, t0,3{6,3,4}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png has cube, hexagonal tiling and hexagonal prism cells, with an irregular triangular antiprism vertex figure.

H3 634-1001.png

It contains the 2D hyperbolic rhombitetrahexagonal tiling, rr{4,6}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png with square and hexagonal faces. The tiling also has a half symmetry construction CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png.

H2 tiling 246-5.png Uniform tiling 4.4.4.6.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node 1.png = CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png

Runcitruncated order-4 hexagonal tiling honeycomb

Runcitruncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,3{6,3,4}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Cells rr{3,4} Uniform polyhedron-43-t02.png
{}x{4} Tetragonal prism.png
{}x{12} Dodecagonal prism.png
t{6,3} Uniform tiling 63-t01.png
Faces triangle {3}
square {4}
dodecagon {12}
Vertex figure Runcitruncated order-4 hexagonal tiling honeycomb verf.png
isosceles-trapezoidal pyramid
Coxeter groups , [4,3,6]
PropertiesVertex-transitive

The runcitruncated order-4 hexagonal tiling honeycomb, t0,1,3{6,3,4}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png has rhombicuboctahedron, cube, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

H3 634-1101.png

Runcicantellated order-4 hexagonal tiling honeycomb

The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.

Omnitruncated order-4 hexagonal tiling honeycomb

Omnitruncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{6,3,4}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells tr{4,3} Uniform polyhedron-43-t012.png
tr{6,3} Uniform tiling 63-t012.svg
{}x{12} Dodecagonal prism.png
{}x{8} Octagonal prism.png
Faces square {4}
hexagon {6}
octagon {8}
dodecagon {12}
Vertex figure Omnitruncated order-4 hexagonal tiling honeycomb verf.png
irregular tetrahedron
Coxeter groups , [4,3,6]
PropertiesVertex-transitive

The omnitruncated order-4 hexagonal tiling honeycomb, t0,1,2,3{6,3,4}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png has truncated cuboctahedron, truncated trihexagonal tiling, dodecagonal prism, and octagonal prism cells, with an irregular tetrahedron vertex figure.

H3 634-1111.png

Alternated order-4 hexagonal tiling honeycomb

Alternated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbols h{6,3,4}
Coxeter diagrams CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
Cells {3[3]} Uniform tiling 333-t1.png
{3,4} Uniform polyhedron-43-t2.svg
Faces triangle {3}
Vertex figure Uniform polyhedron-43-t12.svg
truncated octahedron
Coxeter groups , [4,3[3]]
PropertiesVertex-transitive, edge-transitive, quasiregular

The alternated order-4 hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png, is composed of triangular tiling and octahedron cells, in a truncated octahedron vertex figure.

Cantic order-4 hexagonal tiling honeycomb

Cantic order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h2{6,3,4}
Coxeter diagrams CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png
Cells h2{6,3} Uniform tiling 333-t01.png
t{3,4} Uniform polyhedron-43-t12.svg
r{3,4} Uniform polyhedron-43-t1.svg
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Cantic order-4 hexagonal tiling honeycomb verf.png
wedge
Coxeter groups , [4,3[3]]
PropertiesVertex-transitive

The cantic order-4 hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png, is composed of trihexagonal tiling, truncated octahedron, and cuboctahedron cells, with a wedge vertex figure.

Runcic order-4 hexagonal tiling honeycomb

Runcic order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h3{6,3,4}
Coxeter diagrams 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
Cells {3[3]} Uniform tiling 333-t1.png
rr{3,4} Uniform polyhedron-43-t02.png
{4,3} Uniform polyhedron-43-t0.svg
{}x{3} Triangular prism.png
Faces triangle {3}
square {4}
Vertex figure Runcic order-4 hexagonal tiling honeycomb verf.png
triangular cupola
Coxeter groups , [4,3[3]]
PropertiesVertex-transitive

The runcic order-4 hexagonal tiling honeycomb, 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, is composed of triangular tiling, rhombicuboctahedron, cube, and triangular prism cells, with a triangular cupola vertex figure.

Runcicantic order-4 hexagonal tiling honeycomb

Runcicantic order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h2,3{6,3,4}
Coxeter diagrams 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
Cells h2{6,3} Uniform tiling 333-t01.png
tr{3,4} Uniform polyhedron-43-t012.png
t{4,3} Uniform polyhedron-43-t01.svg
{}x{3} Triangular prism.png
Faces triangle {3}
square {4}
hexagon {6}
octagon {8}
Vertex figure Runcicantic order-4 hexagonal tiling honeycomb verf.png
rectangular pyramid
Coxeter groups , [4,3[3]]
PropertiesVertex-transitive

The runcicantic order-4 hexagonal tiling honeycomb, 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, is composed of trihexagonal tiling, truncated cuboctahedron, truncated cube, and triangular prism cells, with a rectangular pyramid vertex figure.

Quarter order-4 hexagonal tiling honeycomb

Quarter order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol q{6,3,4}
Coxeter diagram 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
Cells {3[3]} Uniform tiling 333-t1.png
{3,3} Uniform polyhedron-33-t0.png
t{3,3} Uniform polyhedron-33-t01.png
h2{6,3} Uniform tiling 333-t01.png
Faces triangle {3}
hexagon {6}
Vertex figure Paracompact honeycomb DP3 1100 verf.png
triangular cupola
Coxeter groups , [3[]x[]]
PropertiesVertex-transitive

The quarter order-4 hexagonal tiling honeycomb, q{6,3,4}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png or CDel node 1.pngCDel split1.pngCDel branch 10luru.pngCDel split2.pngCDel node.png, is composed of triangular tiling, trihexagonal tiling, tetrahedron, and truncated tetrahedron cells, with a triangular cupola vertex figure.

See also

Related Research Articles

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The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

<span class="mw-page-title-main">Tetrahedral-octahedral honeycomb</span> Quasiregular space-filling tesselation

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

<span class="mw-page-title-main">Order-4 dodecahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

<span class="mw-page-title-main">Order-5 cubic honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

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<span class="mw-page-title-main">Uniform honeycombs in hyperbolic space</span> Tiling of hyperbolic 3-space by uniform polyhedra

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

<span class="mw-page-title-main">Hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Order-6 tetrahedral honeycomb</span>

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-6 cubic honeycomb</span>

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

<span class="mw-page-title-main">Order-6 dodecahedral honeycomb</span> Regular geometrical object in hyperbolic space

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-5 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Order-6 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Triangular tiling honeycomb</span>

The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

<span class="mw-page-title-main">Square tiling honeycomb</span>

In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

<span class="mw-page-title-main">Order-4 square tiling honeycomb</span>

In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.

<span class="mw-page-title-main">Order-4 octahedral honeycomb</span>

The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.

In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alteration of a hexagonal tiling honeycomb.

In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

References

  1. Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III