| 释义 |
- Images
- Related polytopes and honeycombs Order-8 cubic honeycomb Infinite-order cubic honeycomb
- See also
- References
- External links
| Order-7 cubic honeycomb | | Type | Regular honeycomb | | Schläfli symbols | {4,3,7} | | Coxeter diagrams | node_1|4|node|3|node|7|node}} | | 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 Poincaré disk model
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}: {{Cubic tessellations small}}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 | node_1|4|node|3|node|8|node}}
{{CDD|node_1|4|node|3|node|8|node_h0}} = {{CDD|node_1|4|node|split1|branch|label4}} | | 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 model
Cell-centered |
Poincaré disk model |
It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,4,3)}, Coxeter diagram, {{CDD|node_1|4|node|split1|branch|label4}}, 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 | node_1|4|node|3|node|infin|node}}
{{CDD|node_1|4|node|3|node|infin|node_h0}} = {{CDD|node_1|4|node|split1|branch|labelinfin}} | | 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, {{CDD|node_1|4|node|split1|branch|labelinfin}}, with alternating types or colors of cubic cells. See also - 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|99035678}}, {{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)[https://arxiv.org/abs/1310.8608]
- [https://arxiv.org/abs/1511.02851 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 : Honeycombs (geometry)|Honeycombs (geometry)|Infinite-order tilings |