Order-4 dodecahedral honeycomb Regular tiling of hyperbolic 3-space
Order-4 dodecahedral honeycomb
Type Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol {5,3,4}1,1 }
Coxeter diagram
Cells {5,3} (dodecahedron )
Faces {5} (pentagon )
Edge figure {4} (square )
Vertex figure octahedron
Dual Order-5 cubic honeycomb
Coxeter group BH 3 , [4,3,5]DH 3 , [5,31,1 ]
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 {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
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 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.
There are four regular compact honeycombs in 3D hyperbolic space:
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:
{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
{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}1,1 }
Coxeter diagram
Cells r{5,3} {3,4}
Faces triangle {3}pentagon {5}
Vertex figure square prism
Coxeter group
B
H
¯
3
{\displaystyle {\overline {BH}}_{3}}
D
H
¯
3
{\displaystyle {\overline {DH}}_{3}}
1,1 ]
Properties Vertex-transitive, edge-transitive
The rectified order-4 dodecahedral honeycomb , octahedron and icosidodecahedron cells, with a square prism vertex figure .
It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling , r{5,4}
There are four rectified compact regular honeycombs:
Truncated order-4 dodecahedral honeycomb
Truncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{5,3,4}1,1 }
Coxeter diagram
Cells t{5,3} {3,4}
Faces triangle {3}decagon {10}
Vertex figure square pyramid
Coxeter group
B
H
¯
3
{\displaystyle {\overline {BH}}_{3}}
D
H
¯
3
{\displaystyle {\overline {DH}}_{3}}
1,1 ]
Properties Vertex-transitive
The truncated order-4 dodecahedral honeycomb , octahedron and truncated dodecahedron cells, with a square pyramid 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:
Bitruncated order-4 dodecahedral honeycomb
Bitruncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol 2t{5,3,4}1,1 }
Coxeter diagram
Cells t{3,5} t{3,4}
Faces square {4}pentagon {5}hexagon {6}
Vertex figure digonal disphenoid
Coxeter group
B
H
¯
3
{\displaystyle {\overline {BH}}_{3}}
D
H
¯
3
{\displaystyle {\overline {DH}}_{3}}
1,1 ]
Properties Vertex-transitive
The bitruncated order-4 dodecahedral honeycomb , or bitruncated order-5 cubic honeycomb , 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}1,1 }
Coxeter diagram
Cells rr{3,5} r{3,4} {}x{4}
Faces triangle {3}square {4}pentagon {5}
Vertex figure wedge
Coxeter group
B
H
¯
3
{\displaystyle {\overline {BH}}_{3}}
D
H
¯
3
{\displaystyle {\overline {DH}}_{3}}
1,1 ]
Properties Vertex-transitive
The cantellated order-4 dodecahedral honeycomb , rhombicosidodecahedron , cuboctahedron , and cube cells, with a wedge vertex figure .
Cantitruncated order-4 dodecahedral honeycomb
Cantitruncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{5,3,4}1,1 }
Coxeter diagram
Cells tr{3,5} t{3,4} {}x{4}
Faces square {4}hexagon {6}decagon {10}
Vertex figure mirrored sphenoid
Coxeter group
B
H
¯
3
{\displaystyle {\overline {BH}}_{3}}
D
H
¯
3
{\displaystyle {\overline {DH}}_{3}}
1,1 ]
Properties Vertex-transitive
The cantitruncated order-4 dodecahedral honeycomb , truncated icosidodecahedron , truncated octahedron , and cube cells, with a mirrored sphenoid vertex figure .
Runcitruncated order-4 dodecahedral honeycomb
The runcitruncated order-4 dodecahedral honeycomb , truncated dodecahedron , rhombicuboctahedron , decagonal prism , and cube cells, with an isosceles-trapezoidal pyramid vertex figure .
Four runcitruncated regular compact honeycombs in H3
Image
Symbols
t0,1,3 {5,3,4} t0,1,3 {4,3,5} t0,1,3 {3,5,3} t0,1,3 {5,3,5}
Vertex
References
Coxeter , Regular Polytopes ISBN 0-486-61480-8 294–296)Coxeter , The Beauty of Geometry: Twelve Essays , Dover Publications, 1999 ISBN 0-486-40919-8 Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 Norman Johnson Uniform Polytopes , Manuscript
N.W. Johnson : The Theory of Uniform Polytopes and Honeycombs , Ph.D. Dissertation, University of Toronto, 1966N.W. Johnson: Geometries and Transformations , (2018) Chapter 13: Hyperbolic Coxeter groups
