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