Order-4 dodecahedral honeycomb

Order-4 dodecahedral honeycomb
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol	Template:Math
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	Template:Math(dodecahedron)
Faces	Template:Math (pentagon)
Edge figure	Template:Math (square)
Vertex figure	octahedron
Dual	Order-5 cubic honeycomb
Coxeter group	Template:Math
Properties	Regular, Quasiregular honeycomb

In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space. With Schläfli symbol Template:Math 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.

Description

The dihedral angle of a regular 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 regular 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, {5,3^1,1}, with two types (colors) of dodecahedra in the Wythoff construction. Template:CDD ↔ Template:CDD.

Images

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

File:Hyperbolic orthogonal dodecahedral honeycomb.png
A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model

Related polytopes and honeycombs

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

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form. Template:534 family

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: Template:Octahedral vertex figure tessellations

This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells: Template:Dodecahedral cell tessellations

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	r{5,3} {3,4} File:Uniform polyhedron-43-t2.png
Faces	triangle {3} pentagon {5}
Vertex figure	File:Rectified order-4 dodecahedral honeycomb verf.png square prism
Coxeter group	${\overline{BH}}_3$, [4,3,5] ${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive, edge-transitive

The rectified order-4 dodecahedral honeycomb, Template:CDD, has alternating octahedron and icosidodecahedron cells, with a square prism vertex figure.

Error creating thumbnail: File:Rectified order 4 dodecahedral honeycomb.png

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

Related honeycombs

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

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{5,3} File:Uniform polyhedron-53-t01.png {3,4} File:Uniform polyhedron-43-t2.png
Faces	triangle {3} decagon {10}
Vertex figure	Error creating thumbnail: square pyramid
Coxeter group	${\overline{BH}}_3$, [4,3,5] ${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive

The truncated order-4 dodecahedral honeycomb, Template:CDD, has octahedron and truncated dodecahedron cells, with a square pyramid vertex figure.

File:H3 435-0011 center ultrawide.png

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

File:H2-5-4-trunc-dual.svg

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{3,5} File:Uniform polyhedron-53-t12.png t{3,4} File:Uniform polyhedron-43-t12.png
Faces	square {4} pentagon {5} hexagon {6}
Vertex figure	File:Bitruncated order-4 dodecahedral honeycomb verf.png digonal disphenoid
Coxeter group	${\overline{BH}}_3$, [4,3,5] ${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive

The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, Template:CDD, has truncated octahedron and truncated icosahedron cells, with a digonal disphenoid vertex figure.

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	rr{3,5} Error creating thumbnail: r{3,4} Error creating thumbnail: {}x{4} File:Tetragonal prism.png
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	Error creating thumbnail: wedge
Coxeter group	${\overline{BH}}_3$, [4,3,5] ${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive

The cantellated order-4 dodecahedral honeycomb, Template:CDD, has rhombicosidodecahedron, cuboctahedron, and cube cells, with a wedge vertex figure.

File:H3 534-1010 center ultrawide.png

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	tr{3,5} File:Uniform polyhedron-53-t012.png t{3,4} File:Uniform polyhedron-43-t12.png {}x{4} File:Tetragonal prism.png
Faces	square {4} hexagon {6} decagon {10}
Vertex figure	File:Cantitruncated order-4 dodecahedral honeycomb verf.png mirrored sphenoid
Coxeter group	${\overline{BH}}_3$, [4,3,5] ${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive

The cantitruncated order-4 dodecahedral honeycomb, Template:CDD, has truncated icosidodecahedron, truncated octahedron, and cube cells, with a mirrored sphenoid vertex figure.

Error creating thumbnail:

Runcinated order-4 dodecahedral honeycomb

The runcinated order-4 dodecahedral honeycomb is the same as the runcinated order-5 cubic honeycomb.

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	Template:CDD
Cells	t{5,3} File:Uniform polyhedron-53-t01.png rr{3,4} Error creating thumbnail: {}x{10} {}x{4} File:Tetragonal prism.png
Faces	triangle {3} square {4} decagon {10}
Vertex figure	File:Runcitruncated order-4 dodecahedral honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter group	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The runcitruncated order-4 dodecahedral honeycomb, Template:CDD, has truncated dodecahedron, rhombicuboctahedron, decagonal prism, and cube cells, with an isosceles-trapezoidal pyramid vertex figure.

File:H3 534-1101 center ultrawide.png

Runcicantellated order-4 dodecahedral honeycomb

The runcicantellated order-4 dodecahedral honeycomb is the same as the runcitruncated order-5 cubic honeycomb.

Omnitruncated order-4 dodecahedral honeycomb

The omnitruncated order-4 dodecahedral honeycomb is the same as the omnitruncated order-5 cubic honeycomb.

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Poincaré homology sphere Poincaré dodecahedral space
• Seifert–Weber space Seifert–Weber dodecahedral space

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)
• Jeffrey R. Weeks The Shape of Space, 2nd edition Template:Isbn (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