| 释义 |
- Geometry
- Related polytopes and honeycombs Order-8 dodecahedral honeycomb Infinite-order dodecahedral honeycomb
- See also
- References
- External links
| Order-7 dodecahedral honeycomb | | Type | Regular honeycomb | | Schläfli symbols | {5,3,7} | | Coxeter diagrams | node_1|5|node|3|node|7|node}} | | Cells | {5,3} | | Faces | {5} | | Edge figure | {7} | | Vertex figure | {3,7}
| | Dual | {7,3,5} | | Coxeter group | [5,3,7] | | Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). GeometryWith Schläfli symbol {5,3,7}, it has seven dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
Poincaré disk model
Cell-centered |
Poincaré disk model |
Ideal surface |
Related polytopes and honeycombs It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}. {{Dodecahedral tessellations small}}It a part of a sequence of honeycombs {5,p,7}. {5,3,7} | {5,4,7 | {5,5,7 | {5,6,7 | {5,7,7 | {5,8,7 | {5,&infin-> It a part of a sequence of honeycombs {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 dodecahedral honeycomb| Order-8 dodecahedral honeycomb | | Type | Regular honeycomb | | Schläfli symbols | {5,3,8}
{5,(3,4,3)} | | Coxeter diagrams | node_1|5|node|3|node|8|node}}
{{CDD|node_1|5|node|3|node|8|node_h0}} = {{CDD|node_1|5|node|split1|branch|label4}} | | Cells | {5,3} | | Faces | {5} | | Edge figure | {8} | | Vertex figure | {3,8}, {(3,4,3)}
| | Dual | {8,3,5} | | Coxeter group | [5,3,8]
[5,((3,4,3))] | | Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,8}, it has eight dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra 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 {5,(3,4,3)}, Coxeter diagram, {{CDD|node_1|5|node|split1|branch|label4}}, with alternating types or colors of dodecahedral cells. {{-}} Infinite-order dodecahedral honeycomb| Infinite-order dodecahedral honeycomb | | Type | Regular honeycomb | | Schläfli symbols | {5,3,∞}
{5,(3,∞,3)} | | Coxeter diagrams | node_1|5|node|3|node|infin|node}}
{{CDD|node_1|5|node|3|node|infin|node_h0}} = {{CDD|node_1|5|node|split1|branch|labelinfin}} | | Cells | {5,3} | | Faces | {5} | | Edge figure | {∞} | | Vertex figure | {3,∞}, {(3,∞,3)}
| | Dual | {∞,3,5} | | Coxeter group | [5,3,∞]
[5,((3,∞,3))] | | Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk model
Cell-centered |
Poincaré disk model |
Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, {{CDD|node_1|5|node|split1|branch|labelinfin}}, with alternating types or colors of dodecahedral 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.
- [https://www.youtube.com/watch?v=w-I4IeLQQAM {5,3,∞} Honeycomb in H^3] YouTube rotation of Poincare sphere
2 : Honeycombs (geometry)|Infinite-order tilings
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