Order-6 hexagonal tiling honeycomb

From Wikipedia, the free encyclopedia
Order-6 hexagonal tiling honeycomb
H3 636 FC boundary.png
Perspective projection view
from center of Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol {6,3,6}
{6,3[3]}
Coxeter diagram CDel 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.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png
CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u.pngCDel splitplit2u.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node.png
Cells {6,3} Uniform tiling 63-t0.svg
Faces hexagon {6}
Edge figure hexagon {6}
Vertex figure {3,6} or {3[3]}
Uniform tiling 63-t2.svg Uniform tiling 333-t1.svg
Dual Self-dual
Coxeter group , [6,3,6]
, [6,3[3]]
Properties Regular, quasiregular

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.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling is {3,6}, the vertex figure of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.[1]

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.

Related tilings[]

The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.

H2 tiling 2ii-4.png

It contains CDel node 1.pngCDel 3.pngCDel node 1.pngCDel ultra.pngCDel node.png and CDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node 1.png that tile 2-hypercycle surfaces, which are similar to the paracompact tilings CDel node 1.pngCDel 3.pngCDel node 1.pngCDel infin.pngCDel node.png and CDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node 1.png (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively):

H2 tiling 23i-6.png H2-I-3-dual.svg

Symmetry[]

Subgroup relations:
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node h0.pngCDel node c1.pngCDel 6.pngCDel node c2.pngCDel split1.pngCDel branch c3.png

The order-6 hexagonal tiling honeycomb has a half-symmetry construction: CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png.

It also has an index-6 subgroup, [6,3*,6], with a non-simplex fundamental domain. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u.pngCDel splitplit2u.pngCDel node.png.

Related polytopes and honeycombs[]

The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space.

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 nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form.

[6,3,6] family honeycombs
{6,3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{6,3,6}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
t{6,3,6}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
rr{6,3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t0,3{6,3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
2t{6,3,6}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
tr{6,3,6}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t0,1,3{6,3,6}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
t0,1,2,3{6,3,6}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.png
H3 636 FC boundary.png H3 636 boundary 0100.png H3 636-1100.png H3 636-1010.png H3 636-1001.png H3 636-0110.png H3 636-1110.png H3 636-1011.png H3 636-1111.png

This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry: 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.

The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:

Hyperbolic uniform honeycombs: {p,3,6}
Form Paracompact Noncompact
Name {3,3,6} {4,3,6} {5,3,6} {6,3,6} {7,3,6} {8,3,6} ... {∞,3,6}
Image H3 336 CC center.png H3 436 CC center.png H3 536 CC center.png H3 636 FC boundary.png Hyperbolic honeycomb 7-3-6 poincare.png Hyperbolic honeycomb 8-3-6 poincare.png Hyperbolic honeycomb i-3-6 poincare.png
Cells Tetrahedron.png
{3,3}
Hexahedron.png
{4,3}
Dodecahedron.png
{5,3}
Uniform tiling 63-t0.svg
{6,3}
Heptagonal tiling.svg
{7,3}
H2-8-3-dual.svg
{8,3}
H2-I-3-dual.svg
{∞,3}

It is also part of a sequence of regular polychora and honeycombs with 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.png 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 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png CDel 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.png CDel 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

It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures:

{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{∞,3,∞}
Image Stereographic polytope 5cell.png Cubic honeycomb.png H3 535 CC center.png H3 636 FC boundary.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 8-3-8 poincare.png Hyperbolic honeycomb i-3-i poincare.png
Cells Tetrahedron.png
{3,3}
Hexahedron.png
{4,3}
Dodecahedron.png
{5,3}
Uniform tiling 63-t0.svg
{6,3}
Heptagonal tiling.svg
{7,3}
H2-8-3-dual.svg
{8,3}
H2-I-3-dual.svg
{∞,3}
Vertex
figure
5-cell verf.png
{3,3}
Cubic honeycomb verf.png
{3,4}
Order-5 dodecahedral honeycomb verf.png
{3,5}
Uniform tiling 63-t2.svg
{3,6}
Order-7 triangular tiling.svg
{3,7}
H2-8-3-primal.svg
{3,8}
H2 tiling 23i-4.png
{3,∞}

Rectified order-6 hexagonal tiling honeycomb[]

Rectified order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{6,3,6} or t1{6,3,6}
Coxeter diagrams CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel split1.pngCDel branch.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png
CDel branch 11.pngCDel splitcross.pngCDel branch.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h1.png
Cells {3,6} Uniform tiling 63-t2.svg
r{6,3} Uniform tiling 63-t1.svg
Faces triangle {3}
hexagon {6}
Vertex figure Rectified order-6 hexagonal tiling honeycomb verf.png
hexagonal prism
Coxeter groups , [6,3,6]
, [6,3[3]]
, [3[3,3]]
Properties Vertex-transitive, edge-transitive

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

it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel branch 11.pngCDel splitcross.pngCDel branch.png.

H3 636 boundary 0100.png

It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface.

H2 tiling 2ii-2.png

Related honeycombs[]

The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures:

r{p,3,6}
Space H3
Form Paracompact Noncompact
Name r{3,3,6}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{4,3,6}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{5,3,6}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{6,3,6}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{7,3,6}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
... r{∞,3,6}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Image H3 336 CC center 0100.png H3 436 CC center 0100.png H3 536 CC center 0100.png H3 636 boundary 0100.png
Cells
Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Uniform polyhedron-33-t1.png
r{3,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cuboctahedron.png
r{4,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.svg
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Triheptagonal tiling.svg
r{7,3}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-2.png
r{∞,3}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png

It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q}

Euclidean/hyperbolic(paracompact/noncompact) quarter honeycombs q{p,3,q}
p \ q 4 6 8 ... ∞
4 Bitruncated alternated cubic tiling.png
q{4,3,4}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel nodes 10r.pngCDel splitcross.pngCDel nodes 10l.pngCDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png
q{4,3,6}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel nodes 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel node 1.pngCDel split1.pngCDel branch 10luru.pngCDel split2.pngCDel node.png


CDel node h1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel nodes 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel label4.png


CDel node h1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel nodes 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel labelinfin.png
6 q{6,3,4}
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel branch 10r.pngCDel splitcross.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel branch 10luru.pngCDel split2.pngCDel node 1.png
H3 636 boundary 0100.png
q{6,3,6}
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.png

CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel label4.png

CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel labelinfin.png
8 q{8,3,4}
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel label4.pngCDel branch 10r.pngCDel splitcross.pngCDel nodes 10l.png
q{8,3,6}
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel label4.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.png
q{8,3,8}
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel label4.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel label4.png

CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel labelinfin.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel labelinfin.png
... q{∞,3,4}
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel labelinfin.pngCDel branch 10r.pngCDel splitcross.pngCDel nodes 10l.png
q{∞,3,6}
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel labelinfin.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.png

CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel labelinfin.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel label4.png

CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel labelinfin.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 10l.pngCDel labelinfin.png

Truncated order-6 hexagonal tiling honeycomb[]

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

The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.[2]

H3 636-1100.png

Bitruncated order-6 hexagonal tiling honeycomb[]

Bitruncated order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol bt{6,3,6} or t1,2{6,3,6}
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cells t{3,6} Uniform tiling 63-t12.svg
Faces hexagon {6}
Vertex figure Bitruncated order-6 hexagonal tiling honeycomb verf.png
tetrahedron
Coxeter groups , [[6,3,6]]
, [6,3[3]]
, [3,3,6]
Properties Regular

The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png. It contains hexagonal tiling facets, with a tetrahedron vertex figure.

H3 636-0110.png

Cantellated order-6 hexagonal tiling honeycomb[]

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

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

H3 636-1010.png

Cantitruncated order-6 hexagonal tiling honeycomb[]

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

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

H3 636-1110.png

Runcinated order-6 hexagonal tiling honeycomb[]

Runcinated order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,6}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u 11.pngCDel splitplit2u.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node 1.png
Cells {6,3} Uniform tiling 63-t0.svgUniform tiling 333-t012.svg
{}×{6} Hexagonal prism.png
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Runcinated order-6 hexagonal tiling honeycomb verf.png
triangular antiprism
Coxeter groups , [[6,3,6]]
Properties Vertex-transitive, edge-transitive

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

H3 636-1001.png

It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png with square and hexagonal faces:

H2 tiling 266-5.png

Runcitruncated order-6 hexagonal tiling honeycomb[]

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

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

H3 636-1011.png

Omnitruncated order-6 hexagonal tiling honeycomb[]

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

The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3{6,3,6}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.png has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure.

H3 636-1111.png

Alternated order-6 hexagonal tiling honeycomb[]

Alternated order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h{6,3,6}
Coxeter diagrams 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
Cells {3,6} Uniform tiling 63-t2.svg
{3[3]} Uniform tiling 333-t0.svg
Faces triangle {3}
Vertex figure Uniform tiling 63-t0.svg
hexagonal tiling
Coxeter groups , [6,3[3]]
Properties Regular, quasiregular

The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb, 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. It contains triangular tiling facets in a hexagonal tiling vertex figure.

Cantic order-6 hexagonal tiling honeycomb[]

Cantic order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h2{6,3,6}
Coxeter diagrams 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
Cells t{3,6} Uniform tiling 63-t12.svg
r{6,3} Uniform tiling 63-t1.svg
h2{6,3} Uniform tiling 333-t01.png
Faces triangle {3}
hexagon {6}
Vertex figure Rectified triangular tiling honeycomb verf.png
triangular prism
Coxeter groups , [6,3[3]]
Properties Vertex-transitive, edge-transitive

The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, 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, with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure.

Runcic order-6 hexagonal tiling honeycomb[]

Runcic order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h3{6,3,6}
Coxeter diagrams 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
Cells rr{3,6} Uniform tiling 63-t02.svg
{6,3} Uniform tiling 63-t0.svg
{3[3]} Uniform tiling 333-t0.svg
{3}x{} Triangular prism.png
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Runcic order-6 hexagonal tiling honeycomb verf.png
triangular cupola
Coxeter groups , [6,3[3]]
Properties Vertex-transitive

The runcic hexagonal tiling honeycomb, h3{6,3,6}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png, or CDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node 1.png, has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure.

Runicantic order-6 hexagonal tiling honeycomb[]

Runcicantic order-6 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h2,3{6,3,6}
Coxeter diagrams 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
Cells tr{6,3} Uniform tiling 63-t012.png
t{6,3} Uniform tiling 63-t01.svg
h2{6,3} Uniform tiling 333-t01.png
{}x{3} Triangular prism.png
Faces triangle {3}
square {4}
hexagon {6}
dodecagon {12}
Vertex figure Runcicantic order-6 hexagonal tiling honeycomb verf.png
rectangular pyramid
Coxeter groups , [6,3[3]]
Properties Vertex-transitive

The runcicantic order-6 hexagonal tiling honeycomb, h2,3{6,3,6}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.png, or CDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node 1.png, contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure.

See also[]

  • Convex uniform honeycombs in hyperbolic space
  • Regular tessellations of hyperbolic 3-space
  • Paracompact uniform honeycombs

References[]

  1. ^ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
  2. ^ Twitter Rotation around 3 fold axis
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
  • Norman Johnson Uniform Polytopes, Manuscript
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
    • N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
Retrieved from ""