Order-5-4 square honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Order-4-5 square honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | self-dual |
| Coxeter group | [4,5,4] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-4 square honeycomb (or
4,5,4 honeycomb) a regular space-filling
tessellation (or
honeycomb) with
Schläfli symbol .
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 pentagonal tiling vertex figure.
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs :
Order-5-5 pentagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-5 pentagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | self-dual |
| Coxeter group | [5,5,5] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-5 pentagonal honeycomb (or
5,5,5 honeycomb) a regular space-filling
tessellation (or
honeycomb) with
Schläfli symbol .
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-5 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
Order-5-6 hexagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-6 hexagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols |
|
| Coxeter diagrams |
= |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure |
|
| Dual | self-dual |
| Coxeter group | [6,5,6]
[6,((5,3,5))] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-6 hexagonal honeycomb (or
6,5,6 honeycomb) is a regular space-filling
tessellation (or
honeycomb) with
Schläfli symbol . It has six
order-5 hexagonal tilings,, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an
order-6 pentagonal tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram,, with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1<sup>+</sup>] = [6,((5,3,5))].
Order-5-7 heptagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-7 hexagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | self-dual |
| Coxeter group | [7,5,7] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-7 heptagonal honeycomb (or
7,5,7 honeycomb) is a regular space-filling
tessellation (or
honeycomb) with
Schläfli symbol . It has seven order-5 heptagonal tilings,, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling
vertex arrangement.
Order-5-infinite apeirogonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-infinite apeirogonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols |
|
| Coxeter diagrams |
↔ |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure |
|
| Dual | self-dual |
| Coxeter group | [∞,5,∞]
[∞,((5,∞,5))] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-infinite apeirogonal honeycomb (or
∞,5,∞ honeycomb) is a regular space-filling
tessellation (or
honeycomb) with
Schläfli symbol . It has infinitely many
order-5 apeirogonal tilings around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an
infinite-order pentagonal tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram,, with alternating types or colors of cells.
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
