Hexagonal tiling honeycomb

Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {6,3,3}
t{3,6,3}
2t{6,3,6}
2t{6,3[3]}
t{3[3,3]}
Coxeter diagrams




Cells {6,3}
Faces hexagon {6}
Edge figure triangle {3}
Vertex figure
tetrahedron {3,3}
Dual Order-6 tetrahedral honeycomb
Coxeter groups , [3,3,6]
, [3,6,3]
, [6,3,6]
, [6,3[3]]
, [3[3,3]]
Properties Regular

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.[1]

Images

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Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.

{6,3,3} {∞,3}
   
One hexagonal tiling cell of the hexagonal tiling honeycomb An order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

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Subgroup relations

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular:         [6,3,3],         [3,6,3],         [6,3,6],       [6,3[3]] and [3[3,3]]    , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are        ,        ,        ,       and    , representing different types (colors) of hexagonal tilings in the Wythoff construction.

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The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

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}

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.

[6,3,3] family honeycombs
{6,3,3} r{6,3,3} t{6,3,3} rr{6,3,3} t0,3{6,3,3} tr{6,3,3} t0,1,3{6,3,3} t0,1,2,3{6,3,3}
               
             
{3,3,6} r{3,3,6} t{3,3,6} rr{3,3,6} 2t{3,3,6} tr{3,3,6} t0,1,3{3,3,6} t0,1,2,3{3,3,6}

It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.

{p,3,3} honeycombs
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ... {∞,3,3}
Image              
Coxeter diagrams
 
1                                                        
4                                
6                                
12                          
24            
Cells
{p,3}
     
 
{3,3}
     
 
{4,3}
     
     
   
 
{5,3}
     
 
{6,3}
     
     
   
 
{7,3}
     
 
{8,3}
     
     
    
 
{∞,3}
     
     
    

It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:

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

Rectified hexagonal tiling honeycomb

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Rectified hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{6,3,3} or t1{6,3,3}
Coxeter diagrams        
            
Cells {3,3}  
r{6,3}   or  
Faces triangle {3}
hexagon {6}
Vertex figure  
triangular prism
Coxeter groups  , [3,3,6]
 , [3,3[3]]
Properties Vertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t1{6,3,3},         has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The       half-symmetry construction alternates two types of tetrahedra.

 

Hexagonal tiling honeycomb
       
Rectified hexagonal tiling honeycomb
        or      
   
Related H2 tilings
Order-3 apeirogonal tiling
     
Triapeirogonal tiling
      or     
    

Truncated hexagonal tiling honeycomb

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Truncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{6,3,3} or t0,1{6,3,3}
Coxeter diagram        
Cells {3,3}  
t{6,3}  
Faces triangle {3}
dodecagon {12}
Vertex figure  
triangular pyramid
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The truncated hexagonal tiling honeycomb, t0,1{6,3,3},         has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.

 

It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:

 

Bitruncated hexagonal tiling honeycomb

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Bitruncated hexagonal tiling honeycomb
Bitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{6,3,3} or t1,2{6,3,3}
Coxeter diagram        
            
Cells t{3,3}  
t{3,6}  
Faces triangle {3}
hexagon {6}
Vertex figure  
digonal disphenoid
Coxeter groups  , [3,3,6]
 , [3,3[3]]
Properties Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3},         has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

 

Cantellated hexagonal tiling honeycomb

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Cantellated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{6,3,3} or t0,2{6,3,3}
Coxeter diagram        
Cells r{3,3}  
rr{6,3}  
{}×{3}  
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure  
wedge
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The cantellated hexagonal tiling honeycomb, t0,2{6,3,3},         has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

 

Cantitruncated hexagonal tiling honeycomb

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Cantitruncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{6,3,3} or t0,1,2{6,3,3}
Coxeter diagram        
Cells t{3,3}  
tr{6,3}  
{}×{3}  
Faces triangle {3}
square {4}
hexagon {6}
dodecagon {12}
Vertex figure  
mirrored sphenoid
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3},         has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

 

Runcinated hexagonal tiling honeycomb

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Runcinated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,3}
Coxeter diagram        
Cells {3,3}  
{6,3}  
{}×{6} 
{}×{3}  
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure  
irregular triangular antiprism
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The runcinated hexagonal tiling honeycomb, t0,3{6,3,3},         has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

 

Runcitruncated hexagonal tiling honeycomb

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Runcitruncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,3{6,3,3}
Coxeter diagram        
Cells rr{3,3}  
{}x{3}  
{}x{12}  
t{6,3}  
Faces triangle {3}
square {4}
dodecagon {12}
Vertex figure  
isosceles-trapezoidal pyramid
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3},         has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

 

Runcicantellated hexagonal tiling honeycomb

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Runcicantellated hexagonal tiling honeycomb
runcitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,2,3{6,3,3}
Coxeter diagram        
Cells t{3,3}  
{}x{6}  
rr{6,3}  
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure  
isosceles-trapezoidal pyramid
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3},         has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

 

Omnitruncated hexagonal tiling honeycomb

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Omnitruncated hexagonal tiling honeycomb
Omnitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{6,3,3}
Coxeter diagram        
Cells tr{3,3}  
{}x{6}  
{}x{12}  
tr{6,3}  
Faces square {4}
hexagon {6}
dodecagon {12}
Vertex figure  
irregular tetrahedron
Coxeter groups  , [3,3,6]
Properties Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3},         has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.

 

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 Archived 2016-06-10 at the Wayback Machine) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [1] [2]
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [3]
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