Dodecahedralicosahedral honeycomb
Dodecahedralicosahedral honeycomb  

Type  Compact uniform honeycomb 
Schläfli symbol  {(3,5,3,5)} or {(5,3,5,3)} 
Coxeter diagram  or 
Cells 
{5,3} {3,5} r{5,3} 
Faces 
triangular {3} pentagon {5} 
Vertex figure 
rhombicosidodecahedron 
Coxeter group  [(5,3)^{[2]}] 
Properties  Vertextransitive, edgetransitive 
In the geometry of hyperbolic 3space, the dodecahedralicosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure.
A geometric honeycomb is a spacefilling of polyhedral or higherdimensional 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 nonEuclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Images
Wideangle perspective views:
Related honeycombs
There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group : , , , , .
Rectified dodecahedralicosahedral honeycomb
Rectified dodecahedralicosahedral honeycomb  

Type  Compact uniform honeycomb 
Schläfli symbol  r{(5,3,5,3)} 
Coxeter diagrams  or 
Cells 
r{5,3} rr{3,5} 
Faces 
triangular {3} decagon {10} 
Vertex figure 
cuboid 
Coxeter group  [[(5,3)^{[2]}]], 
Properties  Vertextransitive, edgetransitive 
The rectified dodecahedralicosahedral honeycomb is a compact uniform honeycomb, constructed from icosidodecahedron, and rhombicosidodecahedron cells, in a cuboid vertex figure. It has a Coxeter diagram .
 Perspective view from center of rhombicosidodecahedron
Cyclotruncated dodecahedralicosahedral honeycomb
Cyclotruncated dodecahedralicosahedral honeycomb  

Type  Compact uniform honeycomb 
Schläfli symbol  ct{(5,3,5,3)} 
Coxeter diagrams  or 
Cells 
t{5,3} {3,5} 
Faces 
triangular {3} decagon {10} 
Vertex figure 
square antiprism 
Coxeter group  [[(5,3)^{[2]}]], 
Properties  Vertextransitive, edgetransitive 
The cyclotruncated dodecahedralicosahedral honeycomb is a compact uniform honeycomb, constructed from truncated dodecahedron, icosahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram .
 Perspective view from center of icosahedron
Cyclotruncated icosahedraldodecahedral honeycomb
Cyclotruncated icosahedraldodecahedral honeycomb  

Type  Compact uniform honeycomb 
Schläfli symbol  ct{(3,5,3,5)} 
Coxeter diagrams  or 
Cells 
{5,3} t{3,5} 
Faces 
triangular {3} square {4} hexagon {6} 
Vertex figure 
triangular antiprism 
Coxeter group  [[(5,3)^{[2]}]], 
Properties  Vertextransitive, edgetransitive 
The cyclotruncated icosahedraldodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron, truncated icosahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .
 Perspective view from center of dodecahedron
It can be see as somewhat analogous to the pentahexagonal tiling with pentagonal and hexagonal faces:
Truncated dodecahedralicosahedral honeycomb
Truncated dodecahedralicosahedral honeycomb  

Type  Compact uniform honeycomb 
Schläfli symbol  t{(5,3,5,3)} 
Coxeter diagrams 
or or or 
Cells 
t{3,5} t{5,3} rr{3,5} tr{5,3} 
Faces 
triangular {3} square {4} hexagon {6} decagon {10} 
Vertex figure 
trapezoidal pyramid 
Coxeter group  [(5,3)^{[2]}] 
Properties  Vertextransitive 
The truncated dodecahedralicosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosahedron, truncated dodecahedron, rhombicosidodecahedron, truncated icosidodecahedron cells, in a trapezoidal pyramid vertex figure. It has a Coxeter diagram .
 Perspective view from center of truncated icosahedron
Omnitruncated dodecahedralicosahedral honeycomb
Omnitruncated dodecahedralicosahedral honeycomb  

Type  Compact uniform honeycomb 
Schläfli symbol  tr{(5,3,5,3)} 
Coxeter diagrams  
Cells  tr{3,5} 
Faces 
square {4} hexagon {6} decagon {10} 
Vertex figure 
Rhombic disphenoid 
Coxeter group  [(2,2)^{+}[(5,3)^{[2]}]], 
Properties  Vertextransitive, edgetransitive, celltransitive 
The omnitruncated dodecahedralicosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosidodecahedron cells, in a rhombic disphenoid vertex figure. It has a Coxeter diagram .
 Perspective view from center of truncated icosidodecahedron
See also
References
 Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
 Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0486409198 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212213)
 Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0824707095 (Chapter 1617: Geometries on Threemanifolds 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