Order-4 hexagonal tiling
| Order-4 hexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane |
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| Type | Hyperbolic regular tiling |
| Vertex figure | 6.6.6.6 |
| Schläfli symbol | {6,4} |
| Wythoff symbol | 4 | 6 2 |
| Coxeter diagram |     |
| Symmetry group | [6,4], (*642) |
| Dual | Order-6 square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [1+,6,1+,4] (as 3*22), removing two of three mirrors (passing through the hexagon center, leaving an order-3 gyration point in the center of the hexagon) in the [6,4] symmetry. Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *3322 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry.
The kaleidoscopic domains can be seen as bicolored hexagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{6,6}, a quasiregular tiling and it can be called a hexahexagonal tiling.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram 
, progressing to infinity.
| Spherical | Euclidean | Hyperbolic tilings | ||||||
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 {6,2}  |
 {6,3}  |
{6,4} |
 {6,5}  |
 {6,6} |
 {6,7}  |
 {6,8}  |
... |  {6,∞}  |
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
| Spherical | Euclidean | Hyperbolic tilings | ||||||
|---|---|---|---|---|---|---|---|---|
 {2,4} |
 {3,4} |
 {4,4} |
 {5,4} |
{6,4} |
 {7,4} |
 {8,4} |
... |  {∞,4} |
| Symmetry: [6,4], (*642) | ||||||
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| {6,4} | t0,1{6,4} | t1{6,4} | t1,2{6,4} | t2{6,4} | t0,2{6,4} | t0,1,2{6,4} |
| Uniform duals | ||||||
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| 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) |
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| h0{6,4} | h0,1{6,4} | h1{6,4} | h1,2{6,4} | h2{6,4} | h0,2{6,4} | s{6,4} |
| Alternation duals | ||||||
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| V(3.4)4 | V3.(3.6)2 | V(3.4.4)2 | V3.3.(3.4)2 | V66 | V3.44 | V3.3.4.3.6 |
| Symmetry: [6,6], (*662) | ||||||||||
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| {6,6} | t0,1{6,6} |
t1{6,6} | t1,2{6,6} | t2{6,6} | t0,2{6,6} | t0,1,2{6,6} | ||||
| Uniform duals | ||||||||||
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| 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) |
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| h0{6,6} | h1{6,6} | h0,1{6,6} | h1,2{6,6} | h2{6,6} | h0,2{6,6} | s{6,6} | ||||
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| V(3.6)6 | V3.3.3.6.3.6 | V(3.4)4 | V3.3.3.6.3.6 | V(3.6)6 | V(3.4.4)2 | V3.3.6.3.6 | ||||
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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.
