Order-4 octahedral honeycomb

Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {3,4,4}
{3,41,1}
Coxeter diagrams


Cells {3,4}
Faces triangle {3}
Edge figure square {4}
Vertex figure square tiling, {4,4}
Dual Square tiling honeycomb, {4,4,3}
Coxeter groups , [3,4,4]
, [3,41,1]
Properties Regular

The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.[1]

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.

Symmetry

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A half symmetry construction, [3,4,4,1+], exists as {3,41,1}, with two alternating types (colors) of octahedral cells:             .

A second half symmetry is [3,4,1+,4]:             .

A higher index sub-symmetry, [3,4,4*], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]:      .

This honeycomb contains     and       that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings      and      , respectively:

 
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The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs.

11 paracompact regular honeycombs
 
{6,3,3}
 
{6,3,4}
 
{6,3,5}
 
{6,3,6}
 
{4,4,3}
 
{4,4,4}
 
{3,3,6}
 
{4,3,6}
 
{5,3,6}
 
{3,6,3}
 
{3,4,4}

There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form.

[4,4,3] family honeycombs
{4,4,3}
       
r{4,4,3}
       
t{4,4,3}
       
rr{4,4,3}
       
t0,3{4,4,3}
       
tr{4,4,3}
       
t0,1,3{4,4,3}
       
t0,1,2,3{4,4,3}
       
               
             
{3,4,4}
       
r{3,4,4}
       
t{3,4,4}
       
rr{3,4,4}
       
2t{3,4,4}
       
tr{3,4,4}
       
t0,1,3{3,4,4}
       
t0,1,2,3{3,4,4}
       

It is a part of a sequence of honeycombs with a square tiling vertex figure:

{p,4,4} honeycombs
Space E3 H3
Form Affine Paracompact Noncompact
Name {2,4,4} {3,4,4} {4,4,4} {5,4,4} {6,4,4} ..{∞,4,4}
Coxeter
       
       
       
       
     
     
     
       
     
     
       
     
     
     
       
     
     
 
       
     
     
     
       
     
     
     
Image            
Cells  
{2,4}
     
 
{3,4}
     
 
{4,4}
     
 
{5,4}
     
 
{6,4}
     
 
{∞,4}
     

It a part of a sequence of regular polychora and honeycombs with octahedral cells:

{3,4,p} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,4,3}
       
 
    
{3,4,4}
       
     
     
{3,4,5}
       
{3,4,6}
       
     
{3,4,7}
       
{3,4,8}
       
      
... {3,4,∞}
       
      
Image              
Vertex
figure
 
{4,3}
     
 
   
 
{4,4}
     
   
   
 
{4,5}
     
 
{4,6}
     
   
 
{4,7}
     
 
{4,8}
     
    
 
{4,∞}
     
    

Rectified order-4 octahedral honeycomb

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Rectified order-4 octahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{3,4,4} or t1{3,4,4}
Coxeter diagrams        
            
            
            
Cells r{4,3}  
{4,4} 
Faces triangle {3}
square {4}
Vertex figure  
square prism
Coxeter groups  , [3,4,4]
 , [3,41,1]
Properties Vertex-transitive, edge-transitive

The rectified order-4 octahedral honeycomb, t1{3,4,4},         has cuboctahedron and square tiling facets, with a square prism vertex figure.

 

Truncated order-4 octahedral honeycomb

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Truncated order-4 octahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{3,4,4} or t0,1{3,4,4}
Coxeter diagrams        
            
            
            
Cells t{3,4}  
{4,4} 
Faces square {4}
hexagon {6}
Vertex figure  
square pyramid
Coxeter groups  , [3,4,4]
 , [3,41,1]
Properties Vertex-transitive

The truncated order-4 octahedral honeycomb, t0,1{3,4,4},         has truncated octahedron and square tiling facets, with a square pyramid vertex figure.

 

Bitruncated order-4 octahedral honeycomb

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The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.

Cantellated order-4 octahedral honeycomb

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Cantellated order-4 octahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{3,4,4} or t0,2{3,4,4}
s2{3,4,4}
Coxeter diagrams        
       
            
Cells rr{3,4}  
{}x4  
r{4,4}  
Faces triangle {3}
square {4}
Vertex figure  
wedge
Coxeter groups  , [3,4,4]
 , [3,41,1]
Properties Vertex-transitive

The cantellated order-4 octahedral honeycomb, t0,2{3,4,4},         has rhombicuboctahedron, cube, and square tiling facets, with a wedge vertex figure.

 

Cantitruncated order-4 octahedral honeycomb

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Cantitruncated order-4 octahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{3,4,4} or t0,1,2{3,4,4}
Coxeter diagrams        
            
Cells tr{3,4}  
{}x{4}  
t{4,4}  
Faces square {4}
hexagon {6}
octagon {8}
Vertex figure  
mirrored sphenoid
Coxeter groups  , [3,4,4]
 , [3,41,1]
Properties Vertex-transitive

The cantitruncated order-4 octahedral honeycomb, t0,1,2{3,4,4},         has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.

 

Runcinated order-4 octahedral honeycomb

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The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.

Runcitruncated order-4 octahedral honeycomb

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Runcitruncated order-4 octahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{3,4,4}
Coxeter diagrams        
            
Cells t{3,4}  
{6}x{}  
rr{4,4}  
Faces square {4}
hexagon {6}
octagon {8}
Vertex figure  
square pyramid
Coxeter groups  , [3,4,4]
Properties Vertex-transitive

The runcitruncated order-4 octahedral honeycomb, t0,1,3{3,4,4},         has truncated octahedron, hexagonal prism, and square tiling facets, with a square pyramid vertex figure.

 

Runcicantellated order-4 octahedral honeycomb

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The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.

Omnitruncated order-4 octahedral honeycomb

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The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling honeycomb.

Snub order-4 octahedral honeycomb

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Snub order-4 octahedral honeycomb
Type Paracompact scaliform honeycomb
Schläfli symbols s{3,4,4}
Coxeter diagrams        
            
     
            
            
Cells square tiling
icosahedron
square pyramid
Faces triangle {3}
square {4}
Vertex figure
Coxeter groups [4,4,3+]
[41,1,3+]
[(4,4,(3,3)+)]
Properties Vertex-transitive

The snub order-4 octahedral honeycomb, s{3,4,4}, has Coxeter diagram        . It is a scaliform honeycomb, with square pyramid, square tiling, and icosahedron facets.

See also

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References

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  1. ^ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • 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, (2015) Chapter 13: Hyperbolic Coxeter groups
    • Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336