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