| Alternated order-4 hexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (3.4)4 |
| Schläfli symbol | h{6,4} or (3,4,4) |
| Wythoff symbol | 4 | 3 4 |
| Coxeter diagram |    or    |
| Symmetry group | [(4,4,3)], (*443) |
| Dual | Order-4-4-3_t0 dual tiling |
| Properties | Vertex-transitive |
In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}.
Uniform constructions
There are four uniform constructions, with some of lower ones which can be seen with two colors of triangles:
| *443 | 3333 | *3232 | 3*22 |
|---|---|---|---|
=  |    =   |  =   =  | =  |
| |  | ||
| (4,4,3) = h{6,4} | hr{6,6} = h{6,4}1⁄2 | ||
Related polyhedra and tiling
| Uniform tetrahexagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||
 =  =  = | = | = = = | = | =  = =  | = | | |||||
 |  |  |  |  |  |  | |||||
| {6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
| Uniform duals | |||||||||||
 | | | | | | | |||||
 |  |  |  |  |  |  | |||||
| V64 | V4.12.12 | V(4.6)2 | V6.8.8 | V46 | V4.4.4.6 | V4.8.12 | |||||
| Alternations | |||||||||||
| [1+,6,4] (*443) | [6+,4] (6*2) | [6,1+,4] (*3222) | [6,4+] (4*3) | [6,4,1+] (*662) | [(6,4,2+)] (2*32) | [6,4]+ (642) | |||||
| = | =  | =   | = | =  | =  | | |||||
 |  |  |  |  |  |  | |||||
| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} | |||||
| Uniform hexahexagonal tilings | ||||||
|---|---|---|---|---|---|---|
| Symmetry: [6,6], (*662) | ||||||
=   = | = = | = = | =  = | = = | = = | = = |
 |  |  |  |  |  |  |
| {6,6} = h{4,6} | t{6,6} = h2{4,6} | r{6,6} {6,4} | t{6,6} = h2{4,6} | {6,6} = h{4,6} | rr{6,6} r{6,4} | tr{6,6} t{6,4} |
| Uniform duals | ||||||
| | | | | | | |
 |  |  |  |  |  |  |
| V66 | V6.12.12 | V6.6.6.6 | V6.12.12 | V66 | V4.6.4.6 | V4.12.12 |
| Alternations | ||||||
| [1+,6,6] (*663) | [6+,6] (6*3) | [6,1+,6] (*3232) | [6,6+] (6*3) | [6,6,1+] (*663) | [(6,6,2+)] (2*33) | [6,6]+ (662) |
| = | | = | | =  | | |
| | | | | | | |
 | | | |  | ||
| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
| Uniform (4,4,3) tilings | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [(4,4,3)] (*443) | [(4,4,3)]+ (443) | [(4,4,3+)] (3*22) | [(4,1+,4,3)] (*3232) | |||||||
| | | | | | | | | | | |
| | | | | | | | | | | |
| |  |  |  |  |  |  |  | |  |  |
| h{6,4} t0(4,4,3) | h2{6,4} t0,1(4,4,3) | {4,6}1/2 t1(4,4,3) | h2{6,4} t1,2(4,4,3) | h{6,4} t2(4,4,3) | r{6,4}1/2 t0,2(4,4,3) | t{4,6}1/2 t0,1,2(4,4,3) | s{4,6}1/2 s(4,4,3) | hr{4,6}1/2 hr(4,3,4) | h{4,6}1/2 h(4,3,4) | q{4,6} h1(4,3,4) |
| Uniform duals | ||||||||||
 |  |  | | |||||||
| V(3.4)4 | V3.8.4.8 | V(4.4)3 | V3.8.4.8 | V(3.4)4 | V4.6.4.6 | V6.8.8 | V3.3.3.4.3.4 | V(4.4.3)2 | V66 | V4.3.4.6.6 |
| Similar H2 tilings in *3232 symmetry | ||||||||
|---|---|---|---|---|---|---|---|---|
| Coxeter diagrams | | | | | ||||
 | | | | | | | | |
| | | | | |||||
| Vertex figure | 66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
| Image |  | | |  | ||||
| Dual |  | | ||||||
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
See also
Wikimedia Commons has media related to Uniform tiling 3-4-3-4-3-4-3-4.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery Archived 2013-03-24 at the Wayback Machine
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch