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Square tiling honeycomb
H3 443 FC boundary.png
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {4,4,3}
r{4,4,4}
{41,1,1}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
CDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.pngCDel 3g.pngCDel node g.png
Cells {4,4} Square tiling uniform coloring 1.png Square tiling uniform coloring 9.png Square tiling uniform coloring 7.png
Faces Square {4}
Edge figure Triangle {3}
Vertex figure Square tiling honeycomb verf.png
cube, {4,3}
Dual Order-4 octahedral honeycomb
Coxeter groups [4,4,3]
[41,1,1] ↔ [4,4,3*]
Properties Regular

In the geometry of hyperbolic 3-space, the square tiling honeycomb, is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, has three square tilings, {4,4} around each edge, and 6 square tilings around each vertex in a cubic {4,3} 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.

Contents

Rectified order-4 square tilingEdit

It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:

{4,4,4} r{4,4,4} = {4,4,3}
                =        
   

SymmetryEdit

It has three reflective symmetry constructions,         as a regular honeycomb, a half symmetry              and lastly              with 3 types (colors) of checkered square tilings. [4,4,3*] ↔ [41,1,1], index 6, and a final radial subgroup [4,(4,3)*], index 48, with a right dihedral angled octahedral fundamental domain, and 4 pairs of ultraparallel mirrors:        .

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}

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

[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}
       

This honeycomb is related to the 24-cell, {3,4,3}, with a cubic vertex figure.

It is a part of a sequence of honeycombs with square tiling cells:

Rectified square tiling honeycombEdit

Rectified square tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbols r{4,4,3} or t1{4,4,3}
2r{3,41,1}
r{41,1,1}
Coxeter diagrams        
            
            
              
Cells {4,3}  
r{4,4} 
Faces square {4}
Vertex figure  
Coxeter groups [4,4,3]
[41,1,1] ↔ [4,4,3*]
Properties Vertex-transitive, edge-transitive

The rectified square tiling honeycomb, t1{4,4,3},         has cube and square tiling facets, with a triangular prism vertex figure.

 

It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with triangle and apeirogonal faces.

 

Truncated square tiling honeycombEdit

Truncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{4,4,3} or t0,1{4,4,3}
Coxeter diagrams        
       
            
            
Cells {4,3}  
t{4,4} 
Faces square {4}
octagon {8}
Vertex figure  
triangular pyramid
Coxeter groups [4,4,3]
Properties Vertex-transitive

The truncated square tiling honeycomb, t{4,4,3},         has cube and truncated square tiling facets, with a triangular pyramid vertex figure. It is the same as the cantitruncated order-4 square tiling honeycomb, tr{4,4,4},        .

 

Bitruncated square tiling honeycombEdit

Bitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols 2t{4,4,3} or t1,2{4,4,3}
Coxeter diagram        
Cells t{4,3}  
t{4,4} 
Faces triangular {3}
square {4}
octagon {8}
Vertex figure  
digonal disphenoid
Coxeter groups [4,4,3]
Properties Vertex-transitive

The bitruncated square tiling honeycomb, 2t{4,4,3},         has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.

 

Cantellated square tiling honeycombEdit

Cantellated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{4,4,3} or t0,2{4,4,3}
Coxeter diagrams        
            
Cells r{4,3}  
rr{4,4} 
Faces triangular {3}
square {4}
Vertex figure  
triangular prism
Coxeter groups [4,4,3]
Properties Vertex-transitive

The cantellated square tiling honeycomb, rr{4,4,3},         has cube and truncated square tiling facets, with a triangular prism vertex figure.

 

Cantitruncated square tiling honeycombEdit

Cantitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{4,4,3} or t0,1,2{4,4,3}
Coxeter diagram        
Cells t{4,3}  
tr{4,4} 
{}x{3}  
Faces triangular {3}
square {4}
octagon {8}
Vertex figure  
tetrahedron
Coxeter groups [4,4,3]
Properties Vertex-transitive

The cantitruncated square tiling honeycomb, tr{4,4,3},         has truncated cube and truncated square tiling facets, with a tetrahedron vertex figure.

 

Runcinated square tiling honeycombEdit

Runcinated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{4,4,3}
Coxeter diagrams        
            
Cells {3,4}  
{4,4} 
{}x{4}  
{}x{3}  
Faces triangle {3}
square {4}
Vertex figure  
triangular antiprism
Coxeter groups [4,4,3]
Properties Vertex-transitive

The runcinated square tiling honeycomb, t0,3{4,4,3},         has octahedron, triangular prism, cube, and square tiling facets, with a triangular antiprism vertex figure.

 

Runcitruncated square tiling honeycombEdit

Runcitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{4,4,3}
s2,3{3,4,4}
Coxeter diagrams        
       
Cells rr{4,3}  
t{4,4} 
{}x{3}  
{}x{8}  
Faces triangle {3}
square {4}
Vertex figure  
trapezoidal pyramid
Coxeter groups [4,4,3]
Properties Vertex-transitive

The runcitruncated square tiling honeycomb, t0,1,3{4,4,3},         has rhombicuboctahedron, octagonal prism, triangular prism and truncated square tiling facets, with a trapezoidal pyramid vertex figure.

 

Omnitruncated square tiling honeycombEdit

Omnitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{4,4,3}
Coxeter diagram        
Cells {4,4}  
{}x{6}  
{}x{8}  
tr{4,3}  
Faces Square {4}
Hexagon {6}
Octagon {8}
Vertex figure  
tetrahedron
Coxeter groups [4,4,3]
Properties Vertex-transitive

The omnitruncated square tiling honeycomb, t0,1,2,3{4,4,3},         has truncated square tiling, truncated cuboctahedron, hexagonal prism, octagonal prism facets, with a tetrahedron vertex figure.

 

Omnisnub square tiling honeycombEdit

Omnisnub square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol h(t0,1,2,3{4,4,3})
Coxeter diagram        
Cells sr{4,4}  
sr{2,3}  
sr{2,4}  
sr{4,3}  
Faces Triangular {3}
Square {4}
Vertex figure Irr. tetrahedron
Coxeter group [4,4,3]+
Properties Nonuniform vertex-transitive

The alternated omnitruncated square tiling honeycomb (or omnisnub square tiling honeycomb), h(t0,1,2,3{4,4,3}),         has snub square tiling, snub cube, triangular antiprism, square antiprism, and tetrahedron cells, with an irregular vertex figure.

Alternated square tiling honeycombEdit

Alternated square tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbol h{4,4,3}
hr{4,4,4}
{(4,3,3,4)}
h{41,1,1}
Coxeter diagrams             
            
          
            
                   
Cells   (4.4.4.4)
  (4.4.4)
Faces Square {4}
Vertex figure   (4.3.4.3)
Coxeter groups [1+,4,4,3] ↔ [3,41,1]
[4,1+,4,4] ↔ [∞,4,4,∞]
[(4,4,3,3)]
[1+,41,1,1] ↔ [∞[6]]
Properties vertex-transitive, edge-transitive, quasiregular

The alternated square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space, composed of cube, and square tiling facets in a cuboctahedron vertex figure.

Alternated rectified square tiling honeycombEdit

Alternated rectified square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol hr{4,4,3}
Coxeter diagrams             
Cells
Faces
Vertex figure Triangular prism
Coxeter groups [4,1+,4,3] = [∞,3,3,∞]
Properties Nonsimplectic, vertex-transitive

The alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space.

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
    • Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336