# Order-7 cubic honeycomb

(Redirected from Order-8 cubic honeycomb)
Order-7 cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,7}
Coxeter diagrams
Cells {4,3}
Faces {4}
Edge figure {7}
Vertex figure {3,7}
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.

## 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}:

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

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 modelCell-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

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 modelCell-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.

## 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)