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 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
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, , 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, , with alternating types or colors of cubic cells.
See also
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)
- 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)
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.