Wikipedia

Order-7 tetrahedral honeycomb

Order-7 tetrahedral honeycomb
Type Hyperbolic regular honeycomb
Schläfli symbols {3,3,7}
Coxeter diagrams
Cells {3,3}
Faces {3}
Edge figure {7}
Vertex figure {3,7}
Dual {7,3,3}
Coxeter group [7,3,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.

Images

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Poincaré disk model (cell-centered) 		 
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
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It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.

{3,3,p} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3}
        		{3,3,4}
       
      		{3,3,5}
        		{3,3,6}
       
      		{3,3,7}
        		{3,3,8}
       
       		... {3,3,∞}
       
      
Image              
Vertex
figure 		 
{3,3}
      		 
{3,4}
     
    		 
{3,5}
      		 
{3,6}
     
    		 
{3,7}
      		 
{3,8}
     
     		 
{3,∞}
     
    

It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.

{3,3,7} {4,3,7} {5,3,7} {6,3,7} {7,3,7} {8,3,7} {∞,3,7}
             

It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.

Order-8 tetrahedral honeycomb

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Order-8 tetrahedral honeycomb
Type Hyperbolic regular honeycomb
Schläfli symbols {3,3,8}
{3,(3,4,3)}
Coxeter diagrams        
        =       
Cells {3,3}  
Faces {3}
Edge figure {8}
Vertex figure {3,8}  
{(3,4,3)}  
Dual {8,3,3}
Coxeter group [3,3,8]
[3,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.

 
Poincaré disk model (cell-centered) 		 
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram,       , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,8,1 ] = [3,((3,4,3))].

Infinite-order tetrahedral honeycomb

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Infinite-order tetrahedral honeycomb
Type Hyperbolic regular honeycomb
Schläfli symbols {3,3,∞}
{3,(3,∞,3)}
Coxeter diagrams        
        =       
Cells {3,3}  
Faces {3}
Edge figure {∞}
Vertex figure {3,∞}  
{(3,∞,3)}  
Dual {∞,3,3}
Coxeter group [∞,3,3]
[3,((3,∞,3))]
Properties Regular

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

 
Poincaré disk model (cell-centered) 		 
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

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

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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