Order-7 dodecahedral honeycomb
Order-7 dodecahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {5,3,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 is a regular space-filling tessellation (or honeycomb).
Geometry
[edit]With 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
[edit]It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}.
{5,3,p} polytopes | |||||||
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Space | S3 | H3 | |||||
Form | Finite | Compact | Paracompact | Noncompact | |||
Name | {5,3,3} | {5,3,4} | {5,3,5} | {5,3,6} | {5,3,7} | {5,3,8} | ... {5,3,∞} |
Image | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
Vertex figure | ![]() {3,3} | ![]() {3,4} | ![]() {3,5} | ![]() {3,6} | ![]() {3,7} | ![]() {3,8} | ![]() {3,∞} |
It a part of a sequence of honeycombs {5,p,7}.
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} |
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
Order-8 dodecahedral honeycomb
[edit]Order-8 dodecahedral honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {5,3,8} {5,(3,4,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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, , with alternating types or colors of dodecahedral cells.
Infinite-order dodecahedral honeycomb
[edit]Infinite-order dodecahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {5,3,∞} {5,(3,∞,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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, , with alternating types or colors of dodecahedral cells.
See also
[edit]- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
- Infinite-order hexagonal tiling honeycomb
References
[edit]- 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)
External links
[edit]- 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]
- {5,3,∞} Honeycomb in H^3 YouTube rotation of Poincare sphere