Order-4 dodecahedral honeycomb

Order-4 dodecahedral honeycomb
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Type	Hyperbolic regular honeycomb
Schläfli symbol	{5,3,4} {5,31,1}
Coxeter diagram	↔
Cells	{5,3}
Faces	pentagon {5}
Edge figure	square {4}
Vertex figure	80px octahedron
Dual	Order-5 cubic honeycomb
Coxeter group	BH3, [5,3,4] DH3, [5,31,1]
Properties	Regular, Quasiregular 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.

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 a half symmetry construction, {5,3^1,1}, with two types (colors) of hexagonal tilings in the Wythoff construction. ↔ .

Images

It contains 2D hyperbolic order-4 pentagonal tiling, {5,4}

Beltrami-Klein model

Related polytopes and honeycombs

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}
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	100px	100px	100px	100px	100px	100px
{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 [5,3^1,1] 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

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	80px cube
Coxeter group	BH3, [5,3,4] DH3, [5,31,1]
Properties	Vertex-transitive, edge-transitive

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

240px180px

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

Related honeycombs

There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	r{5,3,4}	r{4,3,5}	r{3,5,3}	r{5,3,5}
Vertex figure	100px	100px	100px	100px

Truncated order-4 dodecahedral honeycomb

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	80px Square pyramid
Coxeter group	BH3, [5,3,4] DH3, [5,31,1]
Properties	Vertex-transitive

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

480px

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

Related honeycombs

Four truncated regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	t{5,3,4}	t{4,3,5}	t{3,5,3}	t{5,3,5}
Vertex figure	100px	100px	100px	100px

Bitruncated order-4 dodecahedral honeycomb

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	triangle {3} square {4} hexagon {6}
Vertex figure	80px tetrahedron
Coxeter group	BH3, [5,3,4] DH3, [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 tetrahedron vertex figure.

480px

Related honeycombs

Three bitruncated regular compact honeycombs in H³

Image	100px	100px	100px
Symbols	2t{4,3,5}	2t{3,5,3}	2t{5,3,5}
Vertex figure	100px

Cantellated order-4 dodecahedral honeycomb

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} cube
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	80px Triangular prism
Coxeter group	BH3, [5,3,4] DH3, [5,31,1]
Properties	Vertex-transitive

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

480px

Cantitruncated order-4 dodecahedral honeycomb

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} cube
Faces	square {4} hexagon {6} decagon {10}
Vertex figure	80px mirrored sphenoid
Coxeter group	BH3, [5,3,4] DH3, [5,31,1]
Properties	Vertex-transitive

The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and mirrored sphenoid vertex figure.

480px

Related honeycombs

Four cantitruncated regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	tr{5,3,4}	tr{4,3,5}	tr{3,5,3}	tr{5,3,5}
Vertex figure	100px	100px	100px	100px

Runcitruncated order-4 dodecahedral honeycomb

Runcitruncated order-4 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,3{5,3,4}
Coxeter diagram
Cells	t{5,3} rr{3,4} {}x{10} {}x{4}
Faces	triangle {3} square {4} decagon {10}
Vertex figure	80px quad pyramid
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

The runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and a quadrilateral pyramid vertex figure.

480px

See also

• Convex uniform honeycombs in hyperbolic space
• Poincaré homology sphere Poincaré dodecahedral space
• Seifert–Weber space Seifert–Weber dodecahedral space
• List of regular polytopes

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, (2015) Chapter 13: Hyperbolic Coxeter groups