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Order-5 cubic honeycomb
H3 435 CC center.png
Poincaré disk models
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {4,3,5}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces square {4}
Edge figure pentagon {5}
Vertex figure Order-5 cubic honeycomb verf.png
icosahedron
Coxeter group BH3, [5,3,4]
Dual Order-4 dodecahedral honeycomb
Properties Regular

The order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral 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

 
It is analogous to the 2D hyperbolic order-5 square tiling, {4,5}
 
One cell, centered in Poincare ball model
 
Main cells
 
Cells with extended edges to ideal boundary

SymmetryEdit

It a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.

Related polytopes and honeycombsEdit

It has a related alternation honeycomb, represented by             , having icosahedron and tetrahedron cells.

Compact regular 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}

543 honeycombsEdit

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}
       

Polytopes with icosahedral vertex figuresEdit

It is in a sequence of polychora and honeycomb with icosahedron vertex figures:

Related polytopes and honeycombs with cubic cellsEdit

It in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.

Rectified order-5 cubic honeycombEdit

Rectified order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{4,3,5} or 2r{5,3,4}
2r{5,31,1}
Coxeter diagram        
            
Cells r{4,3}  
{3,5}  
Faces triangle {3}
square {4}
Vertex figure  
pentagonal prism
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb,        , has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.

 

Related honeycombEdit

 
It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{4,5} with square and pentagonal faces

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
       
r{p,3,5}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name r{3,3,5}
       
r{4,3,5}
       
     
r{5,3,5}
       
r{6,3,5}
       
     
r{7,3,5}
       
... r{∞,3,5}
       
      
Image        
Cells
 
{3,5}
     
 
r{3,3}
     
 
r{4,3}
     
 
r{5,3}
     
 
r{6,3}
     
 
r{7,3}
     
 
r{∞,3}
     

Truncated order-5 cubic honeycombEdit

Truncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{4,3,5}
Coxeter diagram        
Cells t{4,3}  
{3,5}  
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure  
pentagonal pyramid
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The truncated order-5 cubic honeycomb,        , has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.

 

It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5} with truncated square and pentagonal faces:

 

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, with octahedral cells at the truncated vertices.

 

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-5 cubic honeycombEdit

Same as Bitruncated order-4 dodecahedral honeycomb

Cantellated order-5 cubic honeycombEdit

Cantellated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{4,3,5}
Coxeter diagram        
Cells rr{4,3}  
r{3,5}  
{}x{5}  
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure  
wedge
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The cantellated order-5 cubic honeycomb,        , has rhombicuboctahedron and icosidodecahedron cells, with a wedge vertex figure.

 

Related honeycombsEdit

It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:

 

Cantitruncated order-5 cubic honeycombEdit

Cantitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{4,3,5}
Coxeter diagram        
Cells tr{4,3}  
t{3,5}  
Faces square {4}
pentagon {5}
hexagon {6}
octahedron {8}
Vertex figure  
Mirrored sphenoid
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The cantitruncated order-5 cubic honeycomb,        , has rhombicuboctahedron and icosidodecahedron cells, with a mirrored sphenoid vertex figure.

 

Related honeycombsEdit

It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:

 
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-5 cubic honeycombEdit

Runcinated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbol t0,3{4,3,5}
Coxeter diagram        
Cells {4,3}  
{5,3}  
{}x{5}  
Faces Square {4}
Pentagon {5}
Vertex figure  
octahedron
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb        , has cube, dodecahedron, and pentagonal prism cells, with an octahedron vertex figure.

 

It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5},       with square and pentagonal faces:

 

Related honeycombsEdit

It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:

 
Three runcinated regular compact honeycombs in H3
Image      
Symbols t0,3{4,3,5}
       
t0,3{3,5,3}
       
t0,3{5,3,5}
       
Vertex
figure
     

Runcitruncated order-5 cubic honeycombEdit

Runctruncated order-5 cubic honeycomb
Runcicantellated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{4,3,5}
Coxeter diagram        
Cells t{4,3}  
rr{5,3}  
{}x{5}  
{}x{8}  
Faces Triangle {3}
Square {4}
Pentagon {5}
Octagon {8}
Vertex figure  
quad-pyramid
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb        , has cube, dodecahedron, and pentagonal prism cells, with a quad-pyramid vertex figure.

 

Related honeycombsEdit

It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:

 

Omnitruncated order-5 cubic honeycombEdit

Omnitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbol t0,1,2,3{4,3,5}
Coxeter diagram        
Cells tr{5,3}  
tr{4,3}  
{10}x{}  
{8}x{}  
Faces Square {4}
Hexagon {6}
Octagon {8}
Decagon {10}
Vertex figure  
tetrahedron
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb has Coxeter diagram        .

 

Related honeycombsEdit

It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:

 

Alternated order-5 cubic honeycombEdit

Alternated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h{4,3,5}
Coxeter diagram             
Cells {3,3}  
{3,5}  
Faces triangle {3}
pentagon {5}
Vertex figure  
icosidodecahedron
Coxeter group DH3, [5,31,1]
Properties quasiregular

In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.

 

Related honeycombsEdit

It has 3 related forms: the cantic order-5 cubic honeycomb,        , the runcic order-5 cubic honeycomb,        , and the runcicantic order-5 cubic honeycomb,        .

Cantic order-5 cubic honeycombEdit

Cantic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h2{4,3,5}
Coxeter diagram             
Cells r{5,3}  
t{3,5}  
t{3,3}  
Faces Triangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure  
Rectangular pyramid
Coxeter group DH3, [5,31,1]
Properties Vertex-transitive

The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h2{4,3,5} and a rectangular pyramid vertex figure.

 

Runcic order-5 cubic honeycombEdit

Runcic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h3{4,3,5}
Coxeter diagram             
Cells {5,3}  
rr{5,3}  
{3,3}  
Faces Triangle {3}
square {4}
pentagon {5}
Vertex figure  
triangular prism
Coxeter group DH3, [5,31,1]
Properties Vertex-transitive

The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h3{4,3,5} and a triangular prism vertex figure.

 

Runcicantic order-5 cubic honeycombEdit

Runcicantic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h2,3{4,3,5}
Coxeter diagram             
Cells t{5,3}  
tr{5,3}  
t{3,3}  
Faces Triangle {3}
square {4}
hexagon {6}
dodecagon {10}
Vertex figure  
mirrored sphenoid
Coxeter group DH3, [5,31,1]
Properties Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h2,3{4,3,5} and a mirrored sphenoid vertex figure.

 

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)
  • 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, (2015) Chapter 13: Hyperbolic Coxeter groups