Order-7 cubic honeycomb

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.

Images

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

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}

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 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.

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)
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

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