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Order-4-5 square honeycomb

Order-4-5 square honeycomb
Type Regular honeycomb
Schläfli symbols {4,4,5}
Coxeter diagrams
Cells {4,4}
Faces {4}
Edge figure {5}
Vertex figure {4,5}
Dual {5,4,4}
Coxeter group [4,4,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.

Images

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Poincaré disk model 		 
Ideal surface
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It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}

{4,4,p} honeycombs
Space E3 H3
Form Affine Paracompact Noncompact
Name {4,4,2} {4,4,3} {4,4,4} {4,4,5} {4,4,6} ...{4,4,∞}
Coxeter
       
       
        		       
     
      		       
      		       
     
     
      		       
      		       
     
     
      		       
     
     
      
Image          
Vertex
figure 		 
{4,2}
      		 
{4,3}
      		 
{4,4}
      		 
{4,5}
      		 
{4,6}
      		 
{4,∞}
     

Order-4-6 square honeycomb

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Order-4-6 square honeycomb
Type Regular honeycomb
Schläfli symbols {4,4,6}
{4,(4,3,4)}
Coxeter diagrams        
        =      
Cells {4,4}  
Faces {4}
Edge figure {6}
Vertex figure {4,6}  
{(4,3,4)}  
Dual {6,4,4}
Coxeter group [4,4,6]
[4,((4,3,4))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-6 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,6}. It has six square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-6 square tiling vertex arrangement.

 
Poincaré disk model 		 
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,3,4)}, Coxeter diagram,      , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,6,1+] = [4,((4,3,4))].

Order-4-infinite square honeycomb

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Order-4-infinite square honeycomb
Type Regular honeycomb
Schläfli symbols {4,4,∞}
{4,(4,∞,4)}
Coxeter diagrams        
        =       
Cells {4,4}  
Faces {4}
Edge figure {∞}
Vertex figure {4,∞}  
{(4,∞,4)}  
Dual {∞,4,4}
Coxeter group [∞,4,3]
[4,((4,∞,4))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-infinite square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,∞}. It has infinitely many square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an infinite-order square tiling vertex arrangement.

 
Poincaré disk model 		 
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,∞,4)}, Coxeter diagram,         =       , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,∞,1+] = [4,((4,∞,4))].

See also

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References

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  • 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 (Chapters 16–17: Geometries on Three-manifolds I, II)
  • George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
  • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
  • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
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