Order4 dodecahedral honeycomb
In the geometry of hyperbolic 3space, the order4 dodecahedral honeycomb is one of four compact regular spacefilling tessellations (or honeycombs). 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 order5 cubic honeycomb.
Order4 dodecahedral honeycomb  

Type  Hyperbolic regular honeycomb 
Schläfli symbol  {5,3,4} {5,3^{1,1}} 
Coxeter diagram  
Cells  {5,3} 
Faces  pentagon {5} 
Edge figure  square {4} 
Vertex figure  octahedron 
Dual  Order5 cubic honeycomb 
Coxeter group  BH_{3}, [5,3,4] DH_{3}, [5,3^{1,1}] 
Properties  Regular, Quasiregular honeycomb 
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.
Description
The dihedral angle of a dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3space. However in hyperbolic space a properly scaled 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.
Images
Related polytopes and honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
{5,3,4} 
{4,3,5} 
{3,5,3} 
{5,3,5} 
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form.
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 16cell, cubic honeycomb, and order4 hexagonal tiling honeycomb all which have octahedral vertex figures:
{p,3,4} regular honeycombs  

Space  S^{3}  E^{3}  H^{3}  
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:
Space  S^{3}  H^{3}  

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 figure 
{3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞} 
Rectified order4 dodecahedral honeycomb
Rectified order4 dodecahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  r{5,3,4} r{5,3^{1,1}} 
Coxeter diagram  
Cells  r{5,3} {3,4} 
Faces  triangle {3} pentagon {5} 
Vertex figure  cube 
Coxeter group  BH_{3}, [5,3,4] DH_{3}, [5,3^{1,1}] 
Properties  Vertextransitive, edgetransitive 
The rectified order4 dodecahedral honeycomb,
Related honeycombs
There are four rectified compact regular honeycombs:
Image  

Symbols  r{5,3,4} 
r{4,3,5} 
r{3,5,3} 
r{5,3,5} 
Vertex figure 
Truncated order4 dodecahedral honeycomb
Truncated order4 dodecahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  t{5,3,4} t{5,3^{1,1}} 
Coxeter diagram  
Cells  t{5,3} {3,4} 
Faces  triangle {3} decagon {10} 
Vertex figure  Square pyramid 
Coxeter group  BH_{3}, [5,3,4] DH_{3}, [5,3^{1,1}] 
Properties  Vertextransitive 
The truncated order4 dodecahedral honeycomb,
It can be seen as analogous to the 2D hyperbolic truncated order4 pentagonal tiling, t{5,4} with truncated pentagon and square faces:
Related honeycombs
Image  

Symbols  t{5,3,4} 
t{4,3,5} 
t{3,5,3} 
t{5,3,5} 
Vertex figure 
Bitruncated order4 dodecahedral honeycomb
Bitruncated order4 dodecahedral honeycomb Bitruncated order5 cubic honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  2t{5,3,4} 2t{5,3^{1,1}} 
Coxeter diagram  
Cells  t{3,5} t{3,4} 
Faces  triangle {3} square {4} hexagon {6} 
Vertex figure  tetrahedron 
Coxeter group  BH_{3}, [5,3,4] DH_{3}, [5,3^{1,1}] 
Properties  Vertextransitive 
The bitruncated order4 dodecahedral honeycomb, or bitruncated order5 cubic honeycomb,
Related honeycombs
Image  

Symbols  2t{4,3,5} 
2t{3,5,3} 
2t{5,3,5} 
Vertex figure 
Cantellated order4 dodecahedral honeycomb
Cantellated order4 dodecahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  rr{5,3,4} rr{5,3^{1,1}} 
Coxeter diagram  
Cells  rr{3,5} r{3,4} {}x{4} cube 
Faces  triangle {3} square {4} pentagon {5} 
Vertex figure  Triangular prism 
Coxeter group  BH_{3}, [5,3,4] DH_{3}, [5,3^{1,1}] 
Properties  Vertextransitive 
The cantellated order4 dodecahedral honeycomb,
Related honeycombs
Four cantellated regular compact honeycombs in H^{3}  


Cantitruncated order4 dodecahedral honeycomb
Cantitruncated order4 dodecahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  tr{5,3,4} tr{5,3^{1,1}} 
Coxeter diagram  
Cells  tr{3,5} t{3,4} {}x{4} cube 
Faces  square {4} hexagon {6} decagon {10} 
Vertex figure  mirrored sphenoid 
Coxeter group  BH_{3}, [5,3,4] DH_{3}, [5,3^{1,1}] 
Properties  Vertextransitive 
The cantitruncated order4 dodecahedral honeycomb, is a uniform honeycomb constructed with a
Related honeycombs
Image  

Symbols  tr{5,3,4} 
tr{4,3,5} 
tr{3,5,3} 
tr{5,3,5} 
Vertex figure 
Runcitruncated order4 dodecahedral honeycomb
Runcitruncated order4 dodecahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  t_{0,1,3}{5,3,4} 
Coxeter diagram  
Cells  t{5,3} rr{3,4} {}x{10} {}x{4} 
Faces  triangle {3} square {4} decagon {10} 
Vertex figure  quad pyramid 
Coxeter group  BH_{3}, [5,3,4] 
Properties  Vertextransitive 
The runcititruncated order4 dodecahedral honeycomb, is a uniform honeycomb constructed with a
Related honeycombs
Four runcitruncated regular compact honeycombs in H^{3}  


See also
 Convex uniform honeycombs in hyperbolic space
 Poincaré homology sphere Poincaré dodecahedral space
 Seifert–Weber space Seifert–Weber dodecahedral space
 List of regular polytopes
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