### - Art Gallery -

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 seven 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

 Cell-centered 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 S3 H3
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 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 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 eight 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, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, with alternating types or colors of cubic cells.
Infinite-order cubic honeycomb

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, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of cubic cells.

Convex uniform honeycombs in hyperbolic space
List of regular polytopes
Infinite-order hexagonal tiling honeycomb

References

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)

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]