Order-7 tetrahedral honeycomb
Order-7 tetrahedral honeycomb | |
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Type | Hyperbolic regular honeycomb |
Schläfli symbols | {3,3,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,3} ![]() |
Faces | {3} |
Edge figure | {7} |
Vertex figure | {3,7} ![]() |
Dual | {7,3,3} |
Coxeter group | [7,3,3] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
Images
![]() Poincaré disk model (cell-centered) | ![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
Related polytopes and honeycombs
It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.
{3,3,p} polytopes | |||||||||||
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Space | S3 | H3 | |||||||||
Form | Finite | Paracompact | Noncompact | ||||||||
Name | {3,3,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() | {3,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {3,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() | {3,3,6}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {3,3,7}![]() ![]() ![]() ![]() ![]() ![]() ![]() | {3,3,8}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ... {3,3,∞}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
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} |
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.
Order-8 tetrahedral honeycomb
Order-8 tetrahedral honeycomb | |
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Type | Hyperbolic regular honeycomb |
Schläfli symbols | {3,3,8} {3,(3,4,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,3} ![]() |
Faces | {3} |
Edge figure | {8} |
Vertex figure | {3,8} ![]() {(3,4,3)} ![]() |
Dual | {8,3,3} |
Coxeter group | [3,3,8] [3,((3,4,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
![]() Poincaré disk model (cell-centered) | ![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,8,1+] = [3,((3,4,3))].
Infinite-order tetrahedral honeycomb
Infinite-order tetrahedral honeycomb | |
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Type | Hyperbolic regular honeycomb |
Schläfli symbols | {3,3,∞} {3,(3,∞,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,3} ![]() |
Faces | {3} |
Edge figure | {∞} |
Vertex figure | {3,∞} ![]() {(3,∞,3)} ![]() |
Dual | {∞,3,3} |
Coxeter group | [∞,3,3] [3,((3,∞,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,∞}. It has infinitely many tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
![]() Poincaré disk model (cell-centered) | ![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,∞,3)}, Coxeter diagram, =
, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1+] = [3,((3,∞,3))].
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) [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
- 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]