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
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 groupBH3, [5,3,4]
DH3, [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 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 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


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
cube
Coxeter groupBH3, [5,3,4]
DH3, [5,31,1]
PropertiesVertex-transitive, edge-transitive

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

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 groupBH3, [5,3,4]
DH3, [5,31,1]
PropertiesVertex-transitive

The truncated order-4 dodecahedral honeycomb, , has octahedron and truncated dodecahedron cells, with a cube 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}
Facestriangle {3}
square {4}
hexagon {6}
Vertex figure
tetrahedron
Coxeter groupBH3, [5,3,4]
DH3, [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 tetrahedron 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} cube
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
Triangular prism
Coxeter groupBH3, [5,3,4]
DH3, [5,31,1]
PropertiesVertex-transitive

The cantellated order-4 dodecahedral honeycomb,, has rhombicosidodecahedron and cuboctahedron, and cube cells, with a triangular prism 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} cube
Facessquare {4}
hexagon {6}
decagon {10}
Vertex figure
mirrored sphenoid
Coxeter groupBH3, [5,3,4]
DH3, [5,31,1]
PropertiesVertex-transitive

The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and 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

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
quad pyramid
Coxeter groupBH3, [5,3,4]
PropertiesVertex-transitive

The runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and a quadrilateral pyramid 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)
  • 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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