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Order-7 cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,7}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces {4}
Edge figure {7}
Vertex figure {3,7}
Order-7 triangular tiling.svg
Dual {7,3,4}
Coxeter group [4,3,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,7}, it has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.

Contents

ImagesEdit

Poincaré disk model
 
Cell-centered
 
 
One cell at center
 
One cell with ideal surface

Related polytopes and honeycombsEdit

It is one of a series of regular polytopes and honeycombs with cubic cells: {4,3,p}:

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}
             

Order-8 cubic honeycombEdit

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

In the geometry of hyperbolic 3-space, the order-8 cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,8}. It has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-8 triangular tiling vertex arrangement.

 
Poincaré disk model
Cell-centered
 
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,4,3)}, Coxeter diagram,       , with alternating types or colors of cubic cells.

Infinite-order cubic honeycombEdit

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

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

 
Poincaré disk model
Cell-centered
 
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,∞,3)}, Coxeter diagram,       , with alternating types or colors of cubic cells.

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 (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)

External linksEdit