Order-5 dodecahedral honeycomb

Order-5 dodecahedral honeycomb
Perspective projection view from center of Poincaré disk model
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol	Template:Math
Coxeter-Dynkin diagram	Template:CDD
Cells	Template:Math (regular dodecahedron)
Faces	Template:Math (pentagon)
Edge figure	Template:Math (pentagon)
Vertex figure	icosahedron
Dual	Self-dual
Coxeter group	Template:Math
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 Template:Math it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

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}

Related polytopes and honeycombs

There are four regular compact honeycombs in 3D hyperbolic space: Template:Regular compact H3 honeycombs

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}, Template:CDD, of this honeycomb has all truncated icosahedron cells. Template:535 family

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: Template:Icosahedral vertex figure tessellations

This honeycomb is a part of a sequence of regular polytopes and honeycombs with dodecahedral cells: Template:Dodecahedral tessellations small

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	Template:CDD
Cells	r{5,3} {3,5}
Faces	triangle {3} pentagon {5}
Vertex figure	pentagonal prism
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive, edge-transitive

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

Related tilings and honeycomb

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

There are four rectified compact regular honeycombs: Template:Rectified compact H3 honeycombs

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	Template:CDD
Cells	t{5,3} {3,5}
Faces	triangle {3} decagon {10}
Vertex figure	pentagonal pyramid
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

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

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	Template:CDD
Cells	t{3,5}
Faces	pentagon {5} hexagon {6}
Vertex figure	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, Template:CDD, has truncated icosahedron cells, with a tetragonal disphenoid vertex figure.

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	Template:CDD
Cells	rr{5,3} r{3,5} {}x{5}
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, Template:CDD, 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	Template:CDD
Cells	tr{5,3} t{3,5} {}x{5}
Faces	square {4} pentagon {5} hexagon {6} decagon {10}
Vertex figure	mirrored sphenoid
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

The cantitruncated order-5 dodecahedral honeycomb, Template:CDD, has truncated icosidodecahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.

File:H3 535-1110 center ultrawide.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	Template:CDD
Cells	{5,3} {}x{5} Error creating thumbnail:
Faces	square {4} pentagon {5}
Vertex figure	Error creating thumbnail: triangular antiprism
Coxeter group|$2\times {\bar{K}}_3$, [[5,3,5]]
Properties	Vertex-transitive, edge-transitive

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

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	Template:CDD
Cells	t{5,3} Error creating thumbnail: rr{5,3} Error creating thumbnail: {}x{5} Error creating thumbnail: {}x{10} File:Decagonal prism.png
Faces	triangle {3} square {4} pentagon {5} decagon {10}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	${\bar{K}}_3$, [5,3,5]
Properties	Vertex-transitive

The runcitruncated order-5 dodecahedral honeycomb, Template:CDD, 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	Template:CDD
Cells	tr{5,3} Error creating thumbnail: {}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, Template:CDD, 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. Template:Isbn. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 Template:Isbn (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