Order-6-4 triangular honeycomb
| Order-6-4 triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,6,4} |
| Coxeter diagrams |  File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h0.png = File:CDel 3.pngFile:CDel node.pngFile:CDel split1-66.pngFile:CDel nodes.png
|
| Cells | {3,6} File:Uniform tiling 63-t2-red.svg |
| Faces | {3} |
| Edge figure | {4} |
| Vertex figure | {6,4} File:H2 tiling 246-1.png r{6,6} File:H2 tiling 266-2.png |
| Dual | {4,6,3} |
| 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 {3,6,4}.
Geometry
[edit | edit source]It has four triangular tiling {3,6} 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.
| File:Hyperbolic honeycomb 3-6-4 poincare.png Poincaré disk model |
File:H3 364 UHS plane at infinity.png Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,61,1}, Coxeter diagram, File:CDel 3.pngFile:CDel node.pngFile:CDel split1-66.pngFile:CDel nodes.png, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,4,1+] = [3,61,1].
Related polytopes and honeycombs
[edit | edit source]It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,6,p}
Order-6-5 triangular honeycomb
[edit | edit source]| Order-6-5 triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {3,6,5} |
| Coxeter diagram | Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png |
| Cells | {3,6} File:Uniform tiling 63-t2-red.svg |
| Faces | {3} |
| Edge figure | {5} |
| Vertex figure | {6,5} File:H2 tiling 256-1.png |
| Dual | {5,6,3} |
| 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 {3,6,5}. It has five triangular tiling, {3,6}, 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.
| File:Hyperbolic honeycomb 3-6-5 poincare.png Poincaré disk model |
File:H3 365 UHS plane at infinity.png Ideal surface |
Order-6-6 triangular honeycomb
[edit | edit source]| Order-6-6 triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,6,6} {3,(6,3,6)} |
| Coxeter diagrams | Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node h0.png = Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel split1-66.pngFile:CDel branch.png |
| Cells | {3,6} File:Uniform tiling 63-t2-red.svg |
| Faces | {3} |
| Edge figure | {6} |
| Vertex figure | {6,6} File:H2 tiling 266-4.png {(6,3,6)} File:H2 tiling 366-1.png |
| Dual | {6,6,3} |
| 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 {3,6,6}. It has infinitely many triangular tiling, {3,6}, 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.
| File:Hyperbolic honeycomb 3-6-6 poincare.png Poincaré disk model |
File:H3 366 UHS plane at infinity.png Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,3,6)}, Coxeter diagram, Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node h0.png = Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel split1-66.pngFile:CDel branch.png, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,6,1+] = [3,((6,3,6))].
Order-6-infinite triangular honeycomb
[edit | edit source]| Order-6-infinite triangular honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,6,∞} {3,(6,∞,6)} |
| Coxeter diagrams | Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node h0.png = Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel split1-66.pngFile:CDel branch.pngFile:CDel labelinfin.png |
| Cells | {3,6} File:Uniform tiling 63-t2-red.svg |
| Faces | {3} |
| Edge figure | {∞} |
| Vertex figure | {6,∞} File:H2 tiling 26i-4.png {(6,∞,6)} File:H2 tiling 66i-4.png |
| Dual | {∞,6,3} |
| 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 {3,6,∞}. It has infinitely many triangular tiling, {3,6}, 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.
| File:Hyperbolic honeycomb 3-6-i poincare.png Poincaré disk model |
File:H3 36i UHS plane at infinity.png Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,∞,6)}, Coxeter diagram, Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node h0.png = Error creating thumbnail: File:CDel 3.pngFile:CDel node.pngFile:CDel split1-66.pngFile:CDel branch.pngFile:CDel labelinfin.png, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,∞,1+] = [3,((6,∞,6))].
See also
[edit | edit source]References
[edit | edit source]- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapters 16–17: Geometries on Three-manifolds I, II)
- 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]
- Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
[edit | edit source]- Spherical Video: {3,6,∞} honeycomb with parabolic Möbius transform YouTube, Roice Nelson
- John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
- Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]