Order-4 dodecahedral honeycomb
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {5,3,4}
{5,31,1}
Coxeter diagram
Cells {5,3} (dodecahedron)
Faces {5} (pentagon)
Edge figure {4} (square)
Vertex figure
octahedron
Dual Order-5 cubic honeycomb
Coxeter group BH3, [4,3,5]
DH3, [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 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 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

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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

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It has a half symmetry construction, {5,31,1}, with two types (colors) of dodecahedra in the Wythoff construction.             .

Images

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

 
A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model

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

Four regular compact honeycombs in H3
 
{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.

[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
figure
     
 
{3,3}
     
 
{3,4}
     
 
{3,5}
     
 
{3,6}
     
 
{3,7}
     
 
{3,8}
     
 
{3,∞}
     

Rectified order-4 dodecahedral honeycomb

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Rectified order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{5,3,4}
r{5,31,1}
Coxeter diagram        
            
Cells r{5,3}  
{3,4}  
Faces triangle {3}
pentagon {5}
Vertex figure  
square prism
Coxeter group  , [4,3,5]
 , [5,31,1]
Properties Vertex-transitive, edge-transitive

The rectified order-4 dodecahedral honeycomb,        , has alternating 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}
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There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H3
Image        
Symbols r{5,3,4}
       
r{4,3,5}
       
r{3,5,3}
       
r{5,3,5}
       
Vertex
figure
       

Truncated order-4 dodecahedral honeycomb

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Truncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{5,3,4}
t{5,31,1}
Coxeter diagram        
            
Cells t{5,3}  
{3,4}  
Faces triangle {3}
decagon {10}
Vertex figure  
square pyramid
Coxeter group  , [4,3,5]
 , [5,31,1]
Properties Vertex-transitive

The truncated order-4 dodecahedral honeycomb,        , has 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:

 
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Four truncated regular compact honeycombs in H3
Image        
Symbols t{5,3,4}
       
t{4,3,5}
       
t{3,5,3}
       
t{5,3,5}
       
Vertex
figure
       

Bitruncated order-4 dodecahedral honeycomb

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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        
            
Cells t{3,5}  
t{3,4}  
Faces square {4}
pentagon {5}
hexagon {6}
Vertex figure  
digonal disphenoid
Coxeter group  , [4,3,5]
 , [5,31,1]
Properties Vertex-transitive

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

 

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Three bitruncated compact honeycombs in H3
Image      
Symbols 2t{4,3,5}
       
2t{3,5,3}
       
2t{5,3,5}
       
Vertex
figure
     

Cantellated order-4 dodecahedral honeycomb

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Cantellated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{5,3,4}
rr{5,31,1}
Coxeter diagram        
            
Cells rr{3,5}  
r{3,4}  
{}x{4}  
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure  
wedge
Coxeter group  , [4,3,5]
 , [5,31,1]
Properties Vertex-transitive

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

 

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Four cantellated regular compact honeycombs in H3
Image        
Symbols rr{5,3,4}
       
rr{4,3,5}
       
rr{3,5,3}
       
rr{5,3,5}
       
Vertex
figure
       

Cantitruncated order-4 dodecahedral honeycomb

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Cantitruncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{5,3,4}
tr{5,31,1}
Coxeter diagram        
            
Cells tr{3,5}  
t{3,4}  
{}x{4}  
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure  
mirrored sphenoid
Coxeter group  , [4,3,5]
 , [5,31,1]
Properties Vertex-transitive

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

 

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Four cantitruncated regular compact honeycombs in H3
Image        
Symbols tr{5,3,4}
       
tr{4,3,5}
       
tr{3,5,3}
       
tr{5,3,5}
       
Vertex
figure
       

Runcinated order-4 dodecahedral honeycomb

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

Runcitruncated order-4 dodecahedral honeycomb

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Runcitruncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{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  
isosceles-trapezoidal pyramid
Coxeter group  , [4,3,5]
Properties Vertex-transitive

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

 

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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
figure
       

Runcicantellated order-4 dodecahedral honeycomb

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

Omnitruncated order-4 dodecahedral honeycomb

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

See also

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References

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  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (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