In the geometry of hyperbolic 3space, the order7 cubic honeycomb is a regular spacefilling tessellation (or honeycomb). With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultraideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order7 triangular tiling vertex arrangement.
Images
Cellcentered 

One cell at center 
One cell with ideal surface 
Related polytopes and honeycombs
It is one of a series of regular polytopes and honeycombs with cubic cells: {4,3,p}:
{4,3,p} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
Name  {4,3,3}  {4,3,4}  {4,3,5}  {4,3,6}  {4,3,7}  {4,3,8}  ... {4,3,∞} 
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 order7 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} 

Order8 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 3space, the order8 cubic honeycomb a regular spacefilling tessellation (or honeycomb). With Schläfli symbol {4,3,8}. It has eight cubes {4,3} around each edge. All vertices are ultraideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order8 triangular tiling vertex arrangement.
Poincaré disk model Cellcentered 
Poincaré disk model 
It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,4,3)}, Coxeter diagram, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, with alternating types or colors of cubic cells.
Infiniteorder cubic honeycomb
Infiniteorder 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 3space, the infiniteorder cubic honeycomb a regular spacefilling tessellation (or honeycomb). With Schläfli symbol {4,3,∞}. It has infinitely many cubes {4,3} around each edge. All vertices are ultraideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an infiniteorder triangular tiling vertex arrangement.
Poincaré disk model Cellcentered 
Poincaré disk model 
It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,∞,3)}, Coxeter diagram, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of cubic cells.
See also
Convex uniform honeycombs in hyperbolic space
List of regular polytopes
Infiniteorder hexagonal tiling honeycomb
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 9935678, ISBN 0486409198 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0824707095 (Chapters 16–17: Geometries on Threemanifolds I,II)
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,7897 (1982) [1]
Hao Chen, JeanPhilippe Labbé, Lorentzian Coxeter groups and BoydMaxwell ball packings, (2013)[2]
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World  Scientific Library
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