Order-4 dodecahedral honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs). 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.

Order-4 dodecahedral honeycomb
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol{5,3,4}
{5,31,1}
Coxeter diagram
↔
Cells{5,3}
Facespentagon {5}
Edge figuresquare {4}
Vertex figure
octahedron
DualOrder-5 cubic honeycomb
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesRegular, Quasiregular 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.

Description

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.

Symmetry

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

Images
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:

Four regular compact honeycombs in H3

{5,3,4}
{4,3,5}
{3,5,3}
{5,3,5}

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:

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
Rectified order-4 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{5,3,4}
r{5,31,1}
Coxeter diagram
↔
Cellsr{5,3}
{3,4}
Facestriangle {3}
pentagon {5}
Vertex figure
square prism
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesVertex-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:

Four rectified regular compact honeycombs in H3
Image
Symbols r{5,3,4}
 r{4,3,5}
 r{3,5,3}
 r{5,3,5}
Vertex
figure

Truncated order-4 dodecahedral honeycomb
Truncated order-4 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{5,3,4}
t{5,31,1}
Coxeter diagram
↔
Cellst{5,3}
{3,4}
Facestriangle {3}
decagon {10}
Vertex figure
square pyramid
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesVertex-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:

Four truncated regular compact honeycombs in H3
Image
Symbols t{5,3,4}
 t{4,3,5}
 t{3,5,3}
 t{5,3,5}
Vertex
figure

Bitruncated order-4 dodecahedral honeycomb
Bitruncated order-4 dodecahedral honeycomb
Bitruncated order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbol2t{5,3,4}
2t{5,31,1}
Coxeter diagram
↔
Cellst{3,5}
t{3,4}
Facessquare {4}
pentagon {5}
hexagon {6}
Vertex figure
digonal disphenoid
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesVertex-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.

Three bitruncated compact honeycombs in H3
Image
Symbols 2t{4,3,5}
 2t{3,5,3}
 2t{5,3,5}
Vertex
figure

Cantellated order-4 dodecahedral honeycomb
Cantellated order-4 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{5,3,4}
rr{5,31,1}
Coxeter diagram
↔
Cellsrr{3,5}
r{3,4}
{}x{4}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesVertex-transitive

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

Cantitruncated order-4 dodecahedral honeycomb
Cantitruncated order-4 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{5,3,4}
tr{5,31,1}
Coxeter diagram
↔
Cellstr{3,5}
t{3,4}
{}x{4}
Facessquare {4}
hexagon {6}
decagon {10}
Vertex figure
mirrored sphenoid
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesVertex-transitive

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

Four cantitruncated regular compact honeycombs in H3
Image
Symbols tr{5,3,4}
 tr{4,3,5}
 tr{3,5,3}
 tr{5,3,5}
Vertex
figure

Runcinated order-4 dodecahedral honeycomb

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

Runcitruncated order-4 dodecahedral honeycomb
Runcitruncated order-4 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,3{5,3,4}
Coxeter diagram
Cellst{5,3}
rr{3,4}
{}x{10}
{}x{4}
Facestriangle {3}
square {4}
decagon {10}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group, [4,3,5]
PropertiesVertex-transitive

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

Runcicantellated order-4 dodecahedral honeycomb

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

Omnitruncated order-4 dodecahedral honeycomb

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

See also

References
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
  • 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
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