Order-5 dodecahedral honeycomb

Order-5 dodecahedral honeycomb
File:H3 535 CC center.png Perspective projection view from center of Poincaré disk model
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol	{5,3,5} t0{5,3,5}
Coxeter-Dynkin diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Cells	{5,3} (regular dodecahedron) File:Uniform polyhedron-53-t0.png
Faces	{5} (pentagon)
Edge figure	{5} (pentagon)
Vertex figure	File:Order-5 dodecahedral honeycomb verf.png icosahedron
Dual	Self-dual
Coxeter group	K3, [5,3,5]
Properties	Regular

In hyperbolic geometry, 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. 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

File:H2 tiling 255-1.png

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

File:Order 5 dodecahedral honeycomb.png

Related polytopes and honeycombs

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

Four regular compact honeycombs in H³

File:H3 534 CC center.png {5,3,4}

File:H3 435 CC center.png {4,3,5}

File:H3 353 CC center.png {3,5,3}

File:H3 535 CC center.png {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, t_1,2{5,3,5}, File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, 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
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	r{5,3,5} t1{5,3,5}
Coxeter diagram	File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Cells	r{5,3} File:Uniform polyhedron-53-t1.png {3,5} File:Uniform polyhedron-53-t2.png
Faces	triangle {3} pentagon {5}
Vertex figure	File:Rectified order-5 dodecahedral honeycomb verf.png pentagonal prism
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive, edge-transitive

The rectified order-5 dodecahedral honeycomb, File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png, has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure.

File:H3 535 CC center 0100.png

Related tilings and honeycomb

File:H2 tiling 255-2.png

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	File:H3 534 CC center 0100.png	File:H3 435 CC center 0100.png	File:H3 353 CC center 0100.png	File:H3 535 CC center 0100.png
Symbols	r{5,3,4} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	r{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{3,5,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	r{5,3,5} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Rectified order-4 dodecahedral honeycomb verf.png	File:Rectified order-5 cubic honeycomb verf.png	File:Rectified icosahedral honeycomb verf.png	File:Rectified order-5 dodecahedral honeycomb verf.png

r{p,3,5}

Space	S3	H3
Form	Finite	Compact		Paracompact	Noncompact
Name	r{3,3,5} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{5,3,5} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{6,3,5} File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{7,3,5} File:CDel node.pngFile:CDel 7.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	... r{∞,3,5} File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Image	File:Stereographic rectified 600-cell.png	File:H3 435 CC center 0100.png	File:H3 535 CC center 0100.png	File:H3 635 boundary 0100.png
Cells File:Icosahedron.png {3,5} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	File:Uniform polyhedron-33-t1.svg r{3,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Cuboctahedron.png r{4,3} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Icosidodecahedron.png r{5,3} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Uniform tiling 63-t1.svg r{6,3} File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Triheptagonal tiling.svg r{7,3} File:CDel node.pngFile:CDel 7.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:H2 tiling 23i-2.png r{∞,3} File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png

Truncated order-5 dodecahedral honeycomb

Truncated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t{5,3,5} t0,1{5,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Cells	t{5,3} File:Uniform polyhedron-53-t01.png {3,5} File:Uniform polyhedron-53-t2.png
Faces	triangle {3} decagon {10}
Vertex figure	File:Truncated order-5 dodecahedral honeycomb verf.png pentagonal pyramid
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

The truncated order-5 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png, has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure. File:H3 535-0011 center ultrawide.png

Related honeycombs

Four truncated regular compact honeycombs in H³

Image	File:H3 435-0011 center ultrawide.png	File:H3 534-0011 center ultrawide.png	File:H3 353-0011 center ultrawide.png	File:H3 535-0011 center ultrawide.png
Symbols	t{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	t{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	t{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	t{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Truncated order-4 dodecahedral honeycomb verf.png	File:Truncated order-5 cubic honeycomb verf.png	File:Truncated icosahedral honeycomb verf.png	File:Truncated order-5 dodecahedral honeycomb verf.png

Bitruncated order-5 dodecahedral honeycomb

Bitruncated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	2t{5,3,5} t1,2{5,3,5}
Coxeter diagram	File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
Cells	t{3,5} File:Uniform polyhedron-53-t12.png
Faces	pentagon {5} hexagon {6}
Vertex figure	File:Bitruncated order-5 dodecahedral honeycomb verf.png tetragonal disphenoid
Coxeter group	$2\times {\bar{K}}_3$, [[5,3,5]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated order-5 dodecahedral honeycomb, File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, has truncated icosahedron cells, with a tetragonal disphenoid vertex figure. File:H3 535-0110 center ultrawide.png

Related honeycombs

Three bitruncated compact honeycombs in H³

Image	File:H3 534-0110 center ultrawide.png	File:H3 353-0110 center ultrawide.png	File:H3 535-0110 center ultrawide.png
Symbols	2t{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png	2t{3,5,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	2t{5,3,5} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Bitruncated order-5 cubic honeycomb verf.png	File:Bitruncated icosahedral honeycomb verf.png	File:Bitruncated order-5 dodecahedral honeycomb verf.png

Cantellated order-5 dodecahedral honeycomb

Cantellated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	rr{5,3,5} t0,2{5,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
Cells	rr{5,3} File:Uniform polyhedron-53-t02.png r{3,5} File:Uniform polyhedron-53-t1.png {}x{5} File:Pentagonal prism.png
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	File:Cantellated order-5 dodecahedral honeycomb verf.png wedge
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

The cantellated order-5 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, has rhombicosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure. File:H3 535-1010 center ultrawide.png

Cantitruncated order-5 dodecahedral honeycomb

Cantitruncated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr{5,3,5} t0,1,2{5,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
Cells	tr{5,3} File:Uniform polyhedron-53-t012.png t{3,5} File:Uniform polyhedron-53-t12.png {}x{5} File:Pentagonal prism.png
Faces	square {4} pentagon {5} hexagon {6} decagon {10}
Vertex figure	File:Cantitruncated order-5 dodecahedral honeycomb verf.png mirrored sphenoid
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

The cantitruncated order-5 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, has truncated icosidodecahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure. File:H3 535-1110 center ultrawide.png

Related honeycombs

Four cantitruncated regular compact honeycombs in H³

Image	File:H3 534-1110 center ultrawide.png	File:H3 534-0111 center ultrawide.png	File:H3 353-1110 center ultrawide.png	File:H3 535-1110 center ultrawide.png
Symbols	tr{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	tr{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png	tr{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	tr{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Cantitruncated order-4 dodecahedral honeycomb verf.png	File:Cantitruncated order-5 cubic honeycomb verf.png	File:Cantitruncated icosahedral honeycomb verf.png	File:Cantitruncated order-5 dodecahedral honeycomb verf.png

Runcinated order-5 dodecahedral honeycomb

Runcinated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,3{5,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png
Cells	{5,3} File:Uniform polyhedron-53-t0.png {}x{5} File:Pentagonal prism.png
Faces	square {4} pentagon {5}
Vertex figure	File:Runcinated order-5 dodecahedral honeycomb verf.png triangular antiprism
Coxeter group|$2\times {\bar{K}}_3$, [[5,3,5]]
Properties	Vertex-transitive, edge-transitive

The runcinated order-5 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png, has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure. File:H3 535-1001 center ultrawide.png

Related honeycombs

Three runcinated regular compact honeycombs in H³

Image	File:H3 534-1001 center ultrawide.png	File:H3 353-1001 center ultrawide.png	File:H3 535-1001 center ultrawide.png
Symbols	t0,3{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png	t0,3{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	t0,3{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png
Vertex figure	File:Runcinated order-5 cubic honeycomb verf.png	File:Runcinated icosahedral honeycomb verf.png	File:Runcinated order-5 dodecahedral honeycomb verf.png

Runcitruncated order-5 dodecahedral honeycomb

Runcitruncated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,3{5,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png
Cells	t{5,3} File:Uniform polyhedron-53-t01.png rr{5,3} File:Uniform polyhedron-53-t02.png {}x{5} File:Pentagonal prism.png {}x{10} File:Decagonal prism.png
Faces	triangle {3} square {4} pentagon {5} decagon {10}
Vertex figure	File:Runcitruncated order-5 dodecahedral honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

The runcitruncated order-5 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png, 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. File:H3 535-1101 center ultrawide.png

Omnitruncated order-5 dodecahedral honeycomb

Omnitruncated order-5 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,2,3{5,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.png
Cells	tr{5,3} File:Uniform polyhedron-53-t012.png {}x{10} File:Dodecagonal prism.png
Faces	square {4} hexagon {6} decagon {10}
Vertex figure	File:Omnitruncated order-5 dodecahedral honeycomb verf.png phyllic disphenoid
Coxeter group|$2\times {\bar{K}}_3$, [[5,3,5]]
Properties	Vertex-transitive

The omnitruncated order-5 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.png, has truncated icosidodecahedron and decagonal prism cells, with a phyllic disphenoid vertex figure. File:H3 535-1111 center ultrawide.png

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• 57-cell - An abstract regular polychoron which shared the {5,3,5} symbol.

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