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Order-6 tetrahedral honeycomb
H3 336 CC center.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {3,3,6}
{3,3[3]}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png
Cells {3,3} Uniform polyhedron-33-t0.png
Faces Triangle {3}
Edge figure Hexagon {6}
Vertex figure Triangular tiling {3,6}
Uniform tiling 63-t2.png Uniform tiling 333-t1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Dual Hexagonal tiling honeycomb, {6,3,3}
Coxeter groups , [6,3,3]
, [3,3[3]]
Properties Regular, quasiregular

In the geometry of hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}. It has six tetrahedra {3,3} around each edge. All vertices are ideal vertices with infinitely many tetrahedra existing around each ideal vertex in a triangular tiling vertex arrangement.[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.

Contents

Symmetry constructionsEdit

It has a second construction as a uniform honeycomb, Schläfli symbol {3,3[3]}, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,6,1+] ↔ [3,((3,3,3))] or [3,3[3]]:             .

Related polytopes and honeycombsEdit

It is similar to the 2-dimensional hyperbolic tiling, infinite-order triangular tiling, {3,∞}, having all triangle faces, and all ideal vertices.

 

Related regular honeycombsEdit

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells and/or infinite vertex figures.

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}

633 honeycombsEdit

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual hexagonal tiling honeycomb, {6,3,3}.

Tetrahedral cell honeycombsEdit

It a part of a sequence of regular polychora and honeycombs with tetrahedral cells.

Triangular tiling vertex figuresEdit

It a part of a sequence of honeycombs with triangular tiling vertex figures.

Hyperbolic uniform honeycombs: {p,3,6} and {p,3[3]}

Form Paracompact Noncompact
Name {3,3,6}
{3,3[3]}
{4,3,6}
{4,3[3]}
{5,3,6}
{5,3[3]}
{6,3,6}
{6,3[3]}
{7,3,6}
{7,3[3]}
{8,3,6}
{8,3[3]}
... {∞,3,6}
{∞,3[3]}
       
     
       
     
       
     
       
     
       
     
       
     
       
     
       
     
Image              
Cells  
{3,3}
     
 
{4,3}
     
 
{5,3}
     
 
{6,3}
     
 
{7,3}
     
 
{8,3}
     
 
{∞,3}
     

Rectified order-6 tetrahedral honeycombEdit

Rectified order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbols r{3,3,6} or t1{3,3,6}
Coxeter diagrams        
            
Cells {3,4}  
{3,6}  
Vertex figure  
Hexagonal prism { }×{6}
     
Coxeter groups  , [6,3,3]
 , [3,3[3]]
Properties Vertex-transitive, edge-transitive

The rectified order-6 tetrahedral honeycomb, t1{3,3,6} has octahedral and triangular tiling cells connected in a hexagonal prism vertex figure.

  
Perspective projection view within Poincaré disk model

r{p,3,6}

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

Truncated order-6 tetrahedral honeycombEdit

Truncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{3,3,6} or t0,1{3,3,6}
Coxeter diagrams        
            
Cells t{3,3}  
{3,6}  
Vertex figure  
Hexagonal pyramid { }v{6}
Coxeter groups  , [6,3,3]
 , [3,3[3]]
Properties Vertex-transitive

The truncated order-6 tetrahedral honeycomb, t0,1{3,3,6} has truncated tetrahedra and triangular tiling cells connected in a hexagonal prism vertex figure.

 

Bitruncated order-6 tetrahedral honeycombEdit

 

Cantellated order-6 tetrahedral honeycombEdit

Cantellated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{3,3,6} or t0,2{3,3,6}
Coxeter diagrams        
            
Cells r{3,3}  
r{3,6}  
{}x{6}  
Vertex figure  
tetrahedron
Coxeter groups  , [6,3,3]
 , [3,3[3]]
Properties Vertex-transitive

The cantellated order-6 tetrahedral honeycomb, t0,2{3,3,6} has cuboctahedron and trihexagonal tiling cells connected in a tetrahedron vertex figure.

 

Cantitruncated order-6 tetrahedral honeycombEdit

Cantitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{3,3,6} or t0,1,2{3,3,6}
Coxeter diagrams        
            
Cells tr{3,3}  
r{3,6}  
{}x{6}  
Vertex figure  
tetrahedron
Coxeter groups  , [6,3,3]
 , [3,3[3]]
Properties Vertex-transitive

The cantitruncated order-6 tetrahedral honeycomb, t0,1,2{3,3,6} has truncated cuboctahedron and trihexagonal tiling cells connected in an octahedron vertex figure.

 

See alsoEdit

ReferencesEdit

  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, (2018) Chapter 13: Hyperbolic Coxeter groups