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Order-4 dodecahedral honeycomb
H3 534 CC center.png
Type Hyperbolic regular honeycomb
Schläfli symbol {5,3,4}
{5,31,1}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces pentagon {5}
Edge figure square {4}
Vertex figure Order-4 dodecahedral honeycomb verf.png
octahedron
Dual Order-5 cubic honeycomb
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Regular, Quasiregular honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling 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 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.

Contents

DescriptionEdit

The dihedral angle of a 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 dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

SymmetryEdit

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

ImagesEdit

 
It contains 2D hyperbolic order-4 pentagonal tiling, {5,4}

 
Beltrami-Klein model

Related polytopes and honeycombsEdit

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:

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 honeycombEdit

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  
cube
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive, edge-transitive

The rectified order-4 dodecahedral honeycomb,        , has alternating octahedron and icosidodecahedron cells, with a cube vertex figure.

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

Related honeycombsEdit

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 honeycombEdit

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 BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

The truncated order-4 dodecahedral honeycomb,        , has octahedron and truncated dodecahedron cells, with a cube 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:

 

Related honeycombsEdit

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 honeycombEdit

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 triangle {3}
square {4}
hexagon {6}
Vertex figure  
tetrahedron
Coxeter group BH3, [5,3,4]
DH3, [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 tetrahedron vertex figure.

 

Related honeycombsEdit

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 honeycombEdit

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} cube  
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure  
Triangular prism
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

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

 

Related honeycombsEdit

Cantitruncated order-4 dodecahedral honeycombEdit

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} cube  
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure  
mirrored sphenoid
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a         coxeter diagram, and mirrored sphenoid vertex figure.

 

Related honeycombsEdit

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
       

Runcitruncated order-4 dodecahedral honeycombEdit

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  
quad pyramid
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a         coxeter diagram, and a quadrilateral pyramid vertex figure.

 

Related honeycombsEdit

See alsoEdit

ReferencesEdit

  • 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