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Order-6 cubic honeycomb
H3 436 CC center.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol {4,3,6}
{4,3[3]}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png
CDel node 1.pngCDel ultra.pngCDel node.pngCDel split1.pngCDel branch.pngCDel uaub.pngCDel nodes 11.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node.png
Cells {4,3} Hexahedron.png
Faces square {4}
Edge figure pentagon {6}
Vertex figure triangular tiling {3,6}
Uniform tiling 63-t2.png Uniform tiling 333-t1.png
Coxeter group BV3, [6,3,4]
BP3, [4,3[3]]
Dual Order-4 hexagonal tiling honeycomb
Properties Regular, quasiregular

The order-6 cubic honeycomb is a paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, it has six cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. It is dual is the order-4 hexagonal tiling 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.

ImagesEdit

 
One cell viewed outside of Poincare sphere model
 
It is similar to the 2D hyperbolic infinite-order square tiling, {4,∞} with square faces. All vertices are on the ideal surface.

SymmetryEdit

A half symmetry construction exists as {4,3[3]}, with alternating two types (colors) of cubic cells.             . Another lower symmetry, [4,3*,6], index 6 exists with a nonsimplex fundamental domain,        .

This honeycomb contains       that tile 2-hypercycle surfaces, similar to this paracompact tiling,      :

 

Related polytopes and honeycombsEdit

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or 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}

It is related to the regular (order-4) cubic honeycomb of Euclidean 3-space, order-5 cubic honeycomb in hyperbolic space, which have 4 and 5 cubes per edge respectively.

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

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form.

It in a sequence of regular polychora and honeycombs with cubic cells.

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

Hyperbolic uniform honeycombs: {p,3,6}
Form Paracompact Noncompact
Name {3,3,6} {4,3,6} {5,3,6} {6,3,6} {7,3,6} {8,3,6} ... {∞,3,6}
Image              
Cells  
{3,3}
 
{4,3}
 
{5,3}
 
{6,3}
 
{7,3}
 
{8,3}
 
{∞,3}

Rectified order-6 cubic honeycombEdit

Rectified order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{4,3,6} or t1{4,3,6}
Coxeter diagrams        
            
            
            
Cells r{3,4}  
{3,6}  
Faces Triangle {3}
Square {4}
Vertex figure  
hexagonal prism {}×{6}
Coxeter groups BV3, [6,3,4]
DV3, [6,31,1]
[4,3[3]]
[3[ ]×[3]]
Properties Vertex-transitive, edge-transitive

The rectified order-6 cubic honeycomb, r{4,3,6},         has cuboctahedral and triangular tiling facets, with a hexagonal prism vertex figure.

 

It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{4,∞},       alternating apeirogonal and square faces:

 

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 cubic honeycombEdit

Truncated order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{4,3,6} or t0,1{4,3,6}
Coxeter diagrams        
            
Cells t{4,3}  
{3,6} 
Faces Triangle {3}
octagon {8}
Vertex figure  
hexagonal pyramid
Coxeter groups BV3, [6,3,4]
[4,3[3]]
Properties Vertex-transitive

The truncated order-6 cubic honeycomb, t{4,3,6},         has truncated octahedron and triangular tiling facets, with a hexagonal pyramid vertex figure.

 

It is similar to the 2D hyperbolic truncated infinite-order square tiling, t{4,∞},       with apeirogonal and octagonal (truncated square) faces:

 

Cantellated order-6 cubic honeycombEdit

Cantellated order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{4,3,6} or t0,2{4,3,6}
Coxeter diagrams        
            
Cells rr{4,3}  
r{3,6}  
Faces Triangle {3}
square {4}
hexagon {6}
octagon {8}
Vertex figure  
triangular prism
Coxeter groups BV3, [6,3,4]
[4,3[3]]
Properties Vertex-transitive

The cantellated order-6 cubic honeycomb, rr{4,3,6},         has rhombicuboctahedron and trihexagonal tiling facets, with a triangular prism vertex figure.

 

Alternated order-6 cubic honeycombEdit

Alternated order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbol h{4,3,6}
Coxeter diagram             
                 
              
Cells {3,3}  
{3,6}  
Faces Triangle {3}
Vertex figure  
trihexagonal tiling
Coxeter group DV3, [6,31,1]
Properties Vertex-transitive, edge-transitive, quasiregular

In 3-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellations (or honeycombs). As an alternated order-6 cubic honeycomb and Schläfli symbol h{4,3,6}, with Coxeter diagram         or      . It can be considered a quasiregular honeycomb, alternating triangular tiling and tetrahedron around each vertex in a trihexagonal tiling vertex figure.

SymmetryEdit

A half symmetry construction exists from {4,3[3]}, with alternating two types (colors) of cubic cells.             . Another lower symmetry, [4,3*,6], index 6 exists with a nonsimplex fundamental domain,        .

Related honeycombsEdit

It has 3 related form cantic order-6 cubic honeycomb, h2{4,3,6},        , runcic order-6 cubic honeycomb, h3{4,3,6},        , runcicantic order-6 cubic honeycomb, h2,3{4,3,6},        .

Cantic order-6 cubic honeycombEdit

Cantic order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol h2{4,3,6}
Coxeter diagram             
                 
Cells t{3,3}  
r{6,3}  
{6,3}  
Faces Triangle {3}
hexagon {6}
Vertex figure
Coxeter group DV3, [6,31,1]
Properties Vertex-transitive

The cantic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs) with Schläfli symbol h2{4,3,6}.

Runcic order-6 cubic honeycombEdit

Runcic order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol h3{4,3,6}
Coxeter diagram             
Cells {3,3}  
{6,3}  
rr{6,3}  
Faces Triangle {3}
hexagon {6}
Vertex figure triangular prism
Coxeter group DV3, [6,31,1]
Properties Vertex-transitive

The runcic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs). With Schläfli symbol h3{4,3,6}, with a triangular prism vertex figure.

Runcicantic order-6 cubic honeycombEdit

Runcicantic order-6 cubic honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol h2,3{4,3,6}
Coxeter diagram             
Cells {6,3}  
tr{6,3}  
{3,3}  
Faces Triangle {3}
square {4}
Vertex figure tetrahedron
Coxeter group DV3, [6,31,1]
Properties Vertex-transitive

The runcicantic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs). With Schläfli symbol h2,3{4,3,6}, with a tetrahedral 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)
  • 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