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Order-4 hexagonal tiling honeycomb

Order-4 hexagonal tiling honeycomb
H3 634 FC boundary.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {6,3,4}
{6,31,1}
t0,1{(3,6)2}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel branch 11.pngCDel 6a6b.pngCDel branch.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node.png
CDel K6 636 11.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png
Cells {6,3} Uniform tiling 63-t0.png Uniform tiling 63-t12.png Uniform tiling 333-t012.png
Faces hexagon {6}
Edge figure square {4}
Vertex figure Order-4 hexagonal tiling honeycomb verf.png
octahedron, {3,4}
Dual Order-6 cubic honeycomb
Coxeter groups BV3, [6,3,4]
DV3, [6,31,1]
[(6,3)[2]]
Properties Regular, quasiregular honeycomb

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

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.

The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, 8 hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.[1]

Contents

ImagesEdit

 
Perspective projection
 
One cell, viewed from outside Poincare sphere
 
The vertices of a t{(3,∞,3)},      tiling exists as a 2-hypercycle within this honeycomb
 
It is analogous to the H2 order-4 apeirogonal tiling, {∞,4}, shown here with one green apeirogon outlined by its horocycle

SymmetryEdit

 
Subgroup relations

It has three reflective simplex symmetry construction. The uniform construction {6,31,1} has two types (colors) of hexagonal tilings in the Wythoff construction.              A quarter symmetry construction can have four colors of hexagonal tilings:      .

An additional two reflective symmetries exist with nonsimplex fundamental domains:Coxeter notation: [6,3*,4], index 6,        , and [6,(3,4)*], index 48, with a cube fundamental domain, and octahedral Coxeter diagram with three axial infinite branches:  . It can be seen with 8 colors of hexagonal tilings.

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

 

Related polytopes and honeycombsEdit

Regular 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}

634 honeycombsEdit

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb, {4,3,6}.

Quasiregular honeycombsEdit

It has a related alternation honeycomb, represented by             , having triangular tiling and octahedron cells.

Hexagonal tiling cellsEdit

It is a part of sequence of regular honeycombs with hexagonal tiling cells of the form {6,3,p}:

Octahedral vertex figuresEdit

This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb all which have octahedral vertex figures.

Rectified order-4 hexagonal tiling honeycombEdit

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

The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4},         has octahedral and trihexagonal tiling facets, with a square prism vertex figure.

 

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

 

Truncated order-4 hexagonal tiling honeycombEdit

Truncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{6,3,4} or t0,1{6,3,4}
Coxeter diagram        
            
Cells {3,4}  
t{6,3}  
Faces Triangle {3}
Dodecagon {12}
Vertex figure  
square pyramid
Coxeter groups BV3, [6,3,4]
DV3, [6,31,1]
Properties Vertex-transitive

The truncated order-4 hexagonal tiling honeycomb, t0,1{6,3,4},         has octahedron and truncated hexagonal tiling facets, with a square pyramid vertex figure.

 

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

 

Bitruncated order-4 hexagonal tiling honeycombEdit

Bitruncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{6,3,4} or t1,2{6,3,4}
Coxeter diagram        
            
            
            
Cells t{4,3}  
t{3,6}  
t{3,6}  
Faces Triangle {3}
hexagon {6}
octagon {8}
Vertex figure  
tetrahedron
Coxeter groups BV3, [6,3,4]
DV3, [6,31,1]
[4,3[3]]
[3[ ]×[3]]
Properties Vertex-transitive

The bitruncated order-4 hexagonal tiling honeycomb, t1,2{6,3,4},         has Truncated octahedron and hexagonal tiling cells, with a tetrahedral vertex figure.

 

Cantellated order-4 hexagonal tiling honeycombEdit

Cantellated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{6,3,4} or t0,2{6,3,4}
Coxeter diagram        
            
Cells r{3,4}  
rr{6,3}  
Faces Triangle {3}
square {4}
hexagon {6}
Vertex figure  
triangular prism
Coxeter groups BV3, [6,3,4]
DV3, [6,31,1]
Properties Vertex-transitive

The cantellated order-4 hexagonal tiling honeycomb, t0,2{6,3,4},         has cuboctahedron and rhombitrihexagonal tiling cells, with a triangular prism vertex figure.

 

Runcinated order-4 hexagonal tiling honeycombEdit

Runcinated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,4}
Coxeter diagram        
              
Cells {4,3}  
{6,3}  
{}x{6}  
Faces Triangle {3}
square {4}
hexagon {6}
Vertex figure  
triangular antiprism
Coxeter groups BV3, [6,3,4]
DV3, [6,31,1]
Properties Vertex-transitive

The runcinated order-4 hexagonal tiling honeycomb, t0,3{6,3,4},         has cube, hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure.

 

It contains the 2D hyperbolic rhombitetrahexagonal tiling, rr{4,6},       with square and hexagonal faces. It also has a half symmetry construction    .

   
            =    

Omnitruncated order-4 hexagonal tiling honeycombEdit

Omnitruncated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{6,3,4}
Coxeter diagram        
Cells tr{4,3}  
tr{6,3}  
{}x{6}  
{4,3}  
Faces square {4}
hexagon {6}
dodecagon {12}
Vertex figure  
tetrahedron
Coxeter groups BV3, [6,3,4]
Properties Vertex-transitive

The omnitruncated order-4 hexagonal tiling honeycomb, t0,1,2,3{6,3,4},         has truncated cuboctahedron, truncated trihexagonal tiling, hexagonal prism, and cube cells, with a tetrahedron vertex figure.

 

Alternated order-4 hexagonal tiling honeycombEdit

Alternated order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h{6,3,4}
Coxeter diagrams             
          
Cells
Faces Triangle {3}
Hexagon {6}
Vertex figure      
truncated octahedron
Coxeter groups BV3, [6,3,4]
Properties Vertex-transitive, edge-transitive

Quarter order-4 hexagonal tiling honeycombEdit

Quarter order-4 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol q{6,3,4}
Coxeter diagram             
Cells {3,6}  
{3,3}  
t{3,3}  
rr{3,6}  
Faces {3}, {6}
Vertex figure  
Triangular cupola
Coxeter groups  , [3[ ]x[ ]]
Properties Vertex-transitive

The quarter order-4 hexagonal tiling honeycomb, q{6,3,4},         or       with a triangular cupola 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