Order-5 octahedral honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Order-5 octahedral honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | |
| 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 . It has five octahedra 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.
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with octahedral cells:
Order-6 octahedral honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-6 octahedral honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols |
|
| Coxeter diagrams |
= |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure |
|
| Dual | |
| 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 . It has six octahedra,, 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.
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram,, with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,6,1<sup>+</sup>] = [3,((4,3,4))].
Order-7 octahedral honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-7 octahedral honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | |
| 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 . It has seven octahedra,, 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.
Order-8 octahedral honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-8 octahedral honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | |
| 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 . It has eight octahedra,, 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.
Infinite-order octahedral honeycomb
| bgcolor=#e7dcc3 colspan=2 | Infinite-order octahedral honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols |
|
| Coxeter diagrams |
= |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure |
|
| Dual | |
| 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 . It has infinitely many octahedra,, 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.
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram, =, with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,∞,1<sup>+</sup>] = [3,((4,∞,4))].
See also
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications,, (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I, II)
- George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) http://www.sciencedirect.com/science/article/pii/0021869382903180
- Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)https://arxiv.org/abs/1310.8608
- Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
