Order-6 hexagonal tiling honeycomb

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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.png
Faces hexagon {6}
Edge figure hexagon {6}
Vertex figure {3,6} or {3[3]}
Uniform tiling 63-t2.png Uniform tiling 333-t1.png
Dual Self-dual
Coxeter group Z3, [6,3,6]
VP3, [6,3[3]]
Properties Regular, quasiregular

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. 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.

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.

Images[edit]

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

H2 tiling 2ii-4.png

This honeycomb contains CDel node 1.pngCDel 3.pngCDel node 1.pngCDel ultra.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node 1.png that tile 2-hypercycle surfaces, similar to the paracompact tilings, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel infin.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node 1.png:

H2 tiling 23i-6.png H2 tiling 23i-1.png

Symmetry[edit]

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

This honeycomb has a half symmetry construction is CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png, which looks identical by cells and needs faces colored by their symmetry position to be distinct. Another lower symmetry, [6,3*,6], index 6 exists with a nonsimplex fundamental domain. It has an octahedral Coxeter diagram with 6 order-3 branches, and 3 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[edit]

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

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

It has a related alternation honeycomb, represented by 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, having alternating triangular tiling cells, and a regular form as CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, called a triangular tiling honeycomb.

This honeycomb is a part of a sequence of 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 tiling 238-1.png
{8,3}
H2 tiling 23i-1.png
{∞,3}

Rectified order-6 hexagonal tiling honeycomb[edit]

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.png
r{6,3} Uniform tiling 63-t1.png
Faces Triangle {3}
Hexagon {6}
Vertex figure Rectified order-6 hexagonal tiling honeycomb verf.png
Hexagonal prism {}×{6}
Coxeter groups Z3, [6,3,6]
DV3, [6,3[3]]
[4,3[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 tetrahedral 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.png or CDel 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 are on the ideal surface.

H2 tiling 2ii-2.png

Related honeycombs[edit]

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

The order-6 hexagonal tiling honeycomb is related to a matrix of 3-dimensional honeycombs: q{2p,4,2q}

Truncated order-6 hexagonal tiling honeycomb[edit]

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.png
t{6,3} Uniform tiling 63-t01.png
Faces Triangle {3}
Dodecagon {12}
Vertex figure Truncated order-6 hexagonal tiling honeycomb verf.png
hexagonal pyramid
Coxeter groups Z3, [6,3,6]
DV3, [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 tetrahedral and truncated hexagonal tiling facets, with a tetrahedral vertex figure.

H3 636-1100.png

Bitruncated order-6 hexagonal tiling honeycomb[edit]

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.

Cantellated order-6 hexagonal tiling honeycomb[edit]

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.png
rr{6,3} Uniform tiling 63-t02.png
Faces Triangle {3}
square {4}
hexagon {6}
Vertex figure Cantellated order-6 hexagonal tiling honeycomb verf.png
triangular prism
Coxeter groups Z3, [6,3,6]
DV3, [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 3.pngCDel node.png has trihexagonal tiling and rhombitrihexagonal tiling cells, with a triangular prism vertex figure.

H3 636-1010.png

Cantitruncated order-6 hexagonal tiling honeycomb[edit]

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.png
Faces Triangle {3}
hexagon {6}
dodecagon {12}
Vertex figure Cantitruncated order-6 hexagonal tiling honeycomb verf.png
triangular prism
Coxeter groups Z3, [6,3,6]
DV3, [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 3.pngCDel node.png has hexagonal tiling and truncated trihexagonal tiling cells, with a triangular prism vertex figure.

H3 636-1110.png

Runcinated order-6 hexagonal tiling honeycomb[edit]

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.pngUniform tiling 333-t012.png
{}×{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 Z3, [[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[edit]

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.png
rr{6,3} Uniform tiling 63-t02.png
{}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
Coxeter groups Z3, [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.

H3 636-1011.png

Omnitruncated order-6 hexagonal tiling honeycomb[edit]

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 Z3, [[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 rhombitrihexagonal tiling and dodecagonal prism cells, with a tetrahedron vertex figure.

H3 636-1111.png

Alternated order-6 hexagonal tiling honeycomb[edit]

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.

Cantic order-6 hexagonal tiling honeycomb[edit]

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

Runcic order-6 hexagonal tiling honeycomb[edit]

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 {3,6} Uniform tiling 63-t0.png
{}x{3} Triangular prism.png
rr{3,6} Uniform tiling 63-t02.png
{3,6} Uniform tiling 333-t0.png
Faces Triangle {3}
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.

Runicantic order-6 hexagonal tiling honeycomb[edit]

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-t01.png
{}x{3} Triangular prism.png
tr{3,6} Uniform tiling 63-t012.svg
r{3,6} Uniform tiling 333-t01.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Runcicantic order-6 hexagonal tiling honeycomb verf.png
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.

See also[edit]

References[edit]

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