Order-7-3 triangular honeycomb
| Order-7-4 triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,7,4} |
| Coxeter diagrams |       =  
|
| Cells | {3,7} 
|
| Faces | {3} |
| Edge figure | {4} |
| Vertex figure | {7,4}  r{7,7} 
|
| Dual | {4,7,3} |
| Coxeter group | [3,7,4] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,4}.
Geometry
It has four triangular tiling {3,7} 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.
 Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,71,1}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,7,4,1+] = [3,71,1].
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,7,p}
Order-7-5 triangular honeycomb
| Order-7-5 triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,7,5} |
| Coxeter diagrams | 
|
| Cells | {3,7}
|
| Faces | {3} |
| Edge figure | {5} |
| Vertex figure | {7,5} 
|
| Dual | {5,7,3} |
| Coxeter group | [3,7,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,5}. It has five order-5 hexagonal tiling, {3,7}, 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.
 Poincaré disk model |
 Ideal surface |
Order-7-6 triangular honeycomb
| Order-7-7 triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,7,6} {3,(7,3,7)} |
| Coxeter diagrams |  = 
|
| Cells | {3,7}
|
| Faces | {3} |
| Edge figure | {6} |
| Vertex figure | {7,6}  {(7,3,7)} 
|
| Dual | {7,7,3} |
| Coxeter group | [3,7,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,6}. It has infinitely many order-7 triangular tiling, {3,7}, 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 hexagonal tiling vertex arrangement.
 Poincaré disk model |
 Ideal surface |
Order-7-infinite triangular honeycomb
| Order-7-infinite triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,7,∞} {3,(7,∞,7)} |
| Coxeter diagrams |  = 
|
| Cells | {3,7}
|
| Faces | {3} |
| Edge figure | {∞} |
| Vertex figure | {7,∞}  {(7,∞,7)} 
|
| Dual | {∞,7,3} |
| Coxeter group | [∞,7,3] [3,((7,∞,7))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,∞}. It has infinitely many triangular tiling, {3,7}, 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.
 Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(7,∞,7)}, Coxeter diagram, = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,7,∞,1+] = [3,((7,∞,7))].
See also
References
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
- 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]