Order-7 tetrahedral honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Order-7 tetrahedral honeycomb |
|---|
| Type | Hyperbolic regular honeycomb |
| Schläfli symbols | |
| Coxeter diagrams | |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure | |
| Dual | |
| 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 . It has seven
tetrahedra 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.
Related polytopes and honeycombs
It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, .
It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, .
It is a part of a sequence of hyperbolic honeycombs, .
Order-8 tetrahedral honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-8 tetrahedral honeycomb |
|---|
| Type | Hyperbolic regular honeycomb |
| Schläfli symbols |
|
| Coxeter diagrams |
= |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure |
|
| Dual | |
| 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 . It has eight
tetrahedra 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.
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram,, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,8,1<sup>+</sup>] = [3,((3,4,3))].
Infinite-order tetrahedral honeycomb
| bgcolor=#e7dcc3 colspan=2 | Infinite-order tetrahedral honeycomb |
|---|
| Type | Hyperbolic regular honeycomb |
| Schläfli symbols |
|
| Coxeter diagrams |
= |
| Cells | |
| Faces | |
| Edge figure | |
| Vertex figure |
|
| Dual | |
| 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 . It has infinitely many
tetrahedra 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.
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram, =, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1<sup>+</sup>] = [3,((3,∞,3))].
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
