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Order-4 dodecahedral honeycomb - Wikipedia

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Regular tiling of hyperbolic 3-space

Order-4 dodecahedral honeycomb Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb Schläfli symbol {5,3,4}
{5,31,1} Coxeter diagram
↔ Cells {5,3} (dodecahedron)
Faces {5} (pentagon) Edge figure {4} (square) Vertex figure
octahedron Dual Order-5 cubic honeycomb Coxeter group BH3, [4,3,5]
DH3, [5,31,1] Properties Regular, Quasiregular honeycomb

In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) 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.

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 dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

It has a half symmetry construction, {5,31,1}, with two types (colors) of dodecahedra in the Wythoff construction. ↔ .

It can be seen as analogous to the 2D hyperbolic order-4 pentagonal tiling, {5,4}


A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model

There are four regular compact honeycombs in 3D hyperbolic space:

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form.

[5,3,4] family honeycombs {5,3,4}
r{5,3,4}
t{5,3,4}
rr{5,3,4}
t0,3{5,3,4}
tr{5,3,4}
t0,1,3{5,3,4}
t0,1,2,3{5,3,4}
{4,3,5}
r{4,3,5}
t{4,3,5}
rr{4,3,5}
2t{4,3,5}
tr{4,3,5}
t0,1,3{4,3,5}
t0,1,2,3{4,3,5}

There are eleven uniform honeycombs in the bifurcating [5,31,1] Coxeter group family, including this honeycomb in its alternated form. This construction can be represented by alternation (checkerboard) with two colors of dodecahedral cells.

This honeycomb is also related to the 16-cell, cubic honeycomb, and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures:

{p,3,4} regular honeycombs Space S3 E3 H3 Form Finite Affine Compact Paracompact Noncompact Name {3,3,4}

{4,3,4}



{5,3,4}

{6,3,4}



{7,3,4}

{8,3,4}



... {∞,3,4}



Image Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{∞,3}

This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

{5,3,p} Space S3 H3 Form Finite Compact Paracompact Noncompact Name {5,3,3}
{5,3,4}

{5,3,5}
{5,3,6}

{5,3,7}
{5,3,8}

... {5,3,∞}

Image Vertex
figure

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,∞}
Rectified order-4 dodecahedral honeycomb[edit] Rectified order-4 dodecahedral honeycomb Type Uniform honeycombs in hyperbolic space Schläfli symbol r{5,3,4}
r{5,31,1} Coxeter diagram
↔ Cells r{5,3}
{3,4} Faces triangle {3}
pentagon {5} Vertex figure
square prism Coxeter group B H ¯ 3 {\displaystyle {\overline {BH}}_{3}} , [4,3,5]
D H ¯ 3 {\displaystyle {\overline {DH}}_{3}} , [5,31,1] Properties Vertex-transitive, edge-transitive

The rectified order-4 dodecahedral honeycomb, , has alternating octahedron and icosidodecahedron cells, with a square prism vertex figure.

It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{5,4}

There are four rectified compact regular honeycombs:

Truncated order-4 dodecahedral honeycomb[edit] Truncated order-4 dodecahedral honeycomb Type Uniform honeycombs in hyperbolic space Schläfli symbol t{5,3,4}
t{5,31,1} Coxeter diagram
↔ Cells t{5,3}
{3,4} Faces triangle {3}
decagon {10} Vertex figure
square pyramid Coxeter group B H ¯ 3 {\displaystyle {\overline {BH}}_{3}} , [4,3,5]
D H ¯ 3 {\displaystyle {\overline {DH}}_{3}} , [5,31,1] Properties Vertex-transitive

The truncated order-4 dodecahedral honeycomb, , has octahedron and truncated dodecahedron cells, with a square pyramid vertex figure.

It can be seen as analogous to the 2D hyperbolic truncated order-4 pentagonal tiling, t{5,4} with truncated pentagon and square faces:

Bitruncated order-4 dodecahedral honeycomb[edit] Bitruncated order-4 dodecahedral honeycomb
Bitruncated order-5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schläfli symbol 2t{5,3,4}
2t{5,31,1} Coxeter diagram
↔ Cells t{3,5}
t{3,4} Faces square {4}
pentagon {5}
hexagon {6} Vertex figure
digonal disphenoid Coxeter group B H ¯ 3 {\displaystyle {\overline {BH}}_{3}} , [4,3,5]
D H ¯ 3 {\displaystyle {\overline {DH}}_{3}} , [5,31,1] Properties Vertex-transitive

The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, , has truncated octahedron and truncated icosahedron cells, with a digonal disphenoid vertex figure.

Cantellated order-4 dodecahedral honeycomb[edit] Cantellated order-4 dodecahedral honeycomb Type Uniform honeycombs in hyperbolic space Schläfli symbol rr{5,3,4}
rr{5,31,1} Coxeter diagram
↔ Cells rr{3,5}
r{3,4}
{}x{4} Faces triangle {3}
square {4}
pentagon {5} Vertex figure
wedge Coxeter group B H ¯ 3 {\displaystyle {\overline {BH}}_{3}} , [4,3,5]
D H ¯ 3 {\displaystyle {\overline {DH}}_{3}} , [5,31,1] Properties Vertex-transitive

The cantellated order-4 dodecahedral honeycomb, , has rhombicosidodecahedron, cuboctahedron, and cube cells, with a wedge vertex figure.

Cantitruncated order-4 dodecahedral honeycomb[edit] Cantitruncated order-4 dodecahedral honeycomb Type Uniform honeycombs in hyperbolic space Schläfli symbol tr{5,3,4}
tr{5,31,1} Coxeter diagram
↔ Cells tr{3,5}
t{3,4}
{}x{4} Faces square {4}
hexagon {6}
decagon {10} Vertex figure
mirrored sphenoid Coxeter group B H ¯ 3 {\displaystyle {\overline {BH}}_{3}} , [4,3,5]
D H ¯ 3 {\displaystyle {\overline {DH}}_{3}} , [5,31,1] Properties Vertex-transitive

The cantitruncated order-4 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated octahedron, and cube cells, with a mirrored sphenoid vertex figure.

Runcinated order-4 dodecahedral honeycomb[edit]

The runcinated order-4 dodecahedral honeycomb is the same as the runcinated order-5 cubic honeycomb.

Runcitruncated order-4 dodecahedral honeycomb[edit]

The runcitruncated order-4 dodecahedral honeycomb, , has truncated dodecahedron, rhombicuboctahedron, decagonal prism, and cube cells, with an isosceles-trapezoidal pyramid vertex figure.

Four runcitruncated regular compact honeycombs in H3 Image Symbols t0,1,3{5,3,4}
t0,1,3{4,3,5}
t0,1,3{3,5,3}
t0,1,3{5,3,5}
Vertex
figure Runcicantellated order-4 dodecahedral honeycomb[edit]

The runcicantellated order-4 dodecahedral honeycomb is the same as the runcitruncated order-5 cubic honeycomb.

Omnitruncated order-4 dodecahedral honeycomb[edit]

The omnitruncated order-4 dodecahedral honeycomb is the same as the omnitruncated order-5 cubic honeycomb.

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