Order-5 dodecahedral honeycomb

The order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

Order-5 dodecahedral honeycomb

Perspective projection view
from center of Poincaré disk model
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol{5,3,5}
Coxeter-Dynkin diagram
Cells{5,3}
Facespentagon {5}
Edge figurepentagon {5}
Vertex figure
icosahedron
DualSelf-dual
Coxeter group, [5,3,5]
PropertiesRegular

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 Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.

Images
It is analogous to the 2D hyperbolic order-5 pentagonal tiling, {5,5}

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 is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells.

There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, , of this honeycomb has all truncated icosahedron cells.

The Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb.

This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:

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

Rectified order-5 dodecahedral honeycomb
Rectified order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{5,3,5}
Coxeter diagram
Cellsr{5,3}
{3,5}
Facestriangle {3}
pentagon {5}
Vertex figure
pentagonal prism
Coxeter group, [5,3,5]
PropertiesVertex-transitive, edge-transitive

The rectified order-5 dodecahedral honeycomb, , has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure.

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

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

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

 r{7,3,5}
 ... r{∞,3,5}

Image
Cells

{3,5}
r{3,3}
r{4,3}
r{5,3}
r{6,3}
r{7,3}
r{∞,3}

Truncated order-5 dodecahedral honeycomb
Truncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{5,3,5}
Coxeter diagram
Cellst{5,3}
{3,5}
Facestriangle {3}

decagon {10}

Vertex figure
pentagonal pyramid
Coxeter group, [5,3,5]
PropertiesVertex-transitive

The truncated order-5 dodecahedral honeycomb, , has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure.

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-5 dodecahedral honeycomb
Bitruncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbol2t{5,3,5}
Coxeter diagram
Cellst{3,5}
Facespentagon {5}
hexagon {6}
Vertex figure
tetragonal disphenoid
Coxeter group, [[5,3,5]]
PropertiesVertex-transitive, edge-transitive, cell-transitive

The bitruncated order-5 dodecahedral honeycomb, , has truncated icosahedron cells, with a tetragonal 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-5 dodecahedral honeycomb
Cantellated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{5,3,5}
Coxeter diagram
Cellsrr{5,3}
r{3,5}
{}x{5}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group, [5,3,5]
PropertiesVertex-transitive

The cantellated order-5 dodecahedral honeycomb, , has rhombicosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.

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

The cantitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated icosahedron, and pentagonal prism 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-5 dodecahedral honeycomb
Runcinated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,3{5,3,5}
Coxeter diagram
Cells{5,3}
{}x{5}
Facessquare {4}
pentagon {5}
Vertex figure
triangular antiprism
Coxeter group, [[5,3,5]]
PropertiesVertex-transitive, edge-transitive

The runcinated order-5 dodecahedral honeycomb, , has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure.

Three runcinated regular compact honeycombs in H3
Image
Symbols t0,3{4,3,5}
 t0,3{3,5,3}
 t0,3{5,3,5}
Vertex
figure

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

The runcitruncated order-5 dodecahedral honeycomb, , has truncated dodecahedron, rhombicosidodecahedron, pentagonal prism, and decagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

The runcicantellated order-5 dodecahedral honeycomb is equivalent to the runcitruncated order-5 dodecahedral honeycomb.

Omnitruncated order-5 dodecahedral honeycomb
Omnitruncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,2,3{5,3,5}
Coxeter diagram
Cellstr{5,3}
{}x{10}
Facessquare {4}
hexagon {6}
decagon {10}
Vertex figure
phyllic disphenoid
Coxeter group, [[5,3,5]]
PropertiesVertex-transitive

The omnitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron and decagonal prism cells, with a phyllic disphenoid vertex figure.

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)
  • 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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