| Order-4 hexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 6 |
| Schläfli symbol | {6,4} |
| Wythoff symbol | 4 | 6 2 |
| Coxeter diagram |    
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| Symmetry group | , (*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 , removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry.
 *222222 |
 *443 |
 *3222 |
 *642 |
Uniform colorings
There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.
| 1 color | 2 colors | 3 and 2 colors | 4, 3 and 2 colors | ||||
|---|---|---|---|---|---|---|---|
| Uniform Coloring |
(1111) |
 (1212) |
 (1213) |
 (1113) |
 (1234) |
 (1123) |
 (1122) |
| Symmetry | (*642)   
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(*662)   = 
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= (*663)  =
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(*3333)  = =  
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| Symbol | {6,4} | r{6,6} = {6,4}/2 | r(6,3,6) = r{6,6}/2 | r{6,6}/4 | |||
| Coxeter diagram |
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Regular maps
The regular map {6,4}3 or {6,4}(4,0) can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a petrial octahedron, {3,4}, an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.
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.
| *n62 symmetry mutation of regular tilings: {6,n} | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
 {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.
| *n42 symmetry mutation of regular tilings: {n,4} | |||||||
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| Spherical | Euclidean | Hyperbolic tilings | |||||
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| 2 | 3 | 4 | 5 | 6 | 7 | 8 | ...∞ |
| Symmetry mutation of quasiregular tilings: (6.n) | |||||||||||
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| Symmetry *6n2 |
Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||||||
| *632 |
*642 |
*652 |
*662 |
*762 |
*862 ... |
*∞62 |
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| Quasiregular figures configuration |
 6.3.6.3 |
 6.4.6.4 |
 6.5.6.5 |
6.6.6.6 |
 6.7.6.7 |
 6.8.6.8 |
 6.∞.6.∞ |
6.∞.6.∞ | |||
| Dual figures | |||||||||||
| Rhombic figures configuration |
 V6.3.6.3 |
 V6.4.6.4 |
 V6.5.6.5 |
 V6.6.6.6 |
V6.7.6.7 |
 V6.8.6.8 |
 V6.∞.6.∞ |
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| Uniform tetrahexagonal tilings | |||||||||||
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| Symmetry: , (*642) (with (*662), (*443) , (*3222) index 2 subsymmetries) (And (*3232) index 4 subsymmetry) | |||||||||||
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| {6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
| Uniform duals | |||||||||||
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| V6 | V4.12.12 | V(4.6) | V6.8.8 | V4 | V4.4.4.6 | V4.8.12 | |||||
| Alternations | |||||||||||
(*443) |
(6*2) |
(*3222) |
(4*3) |
(*662) |
(2*32) |
(642) | |||||
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| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} | |||||
| Uniform hexahexagonal tilings | ||||||
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| Symmetry: , (*662) | ||||||
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| {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 | ||||||
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| V6 | V6.12.12 | V6.6.6.6 | V6.12.12 | V6 | V4.6.4.6 | V4.12.12 |
| Alternations | ||||||
(*663) |
(6*3) |
(*3232) |
(6*3) |
(*663) |
(2*33) |
(662) |
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| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
| Similar H2 tilings in *3232 symmetry | ||||||||
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| Vertex figure |
6 | (3.4.3.4) | 3.4.6.6.4 | 6.4.6.4 | ||||
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| Dual | 
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| Uniform tilings in symmetry *3222 | ||||
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6
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 6.6.4.4.png)
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 (3.4.4)
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4.3.4.3.3.3
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6.6.4.4
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6.4.4.4
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3.4.4.4.4.png)
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| (3.4.4)
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3.4.4.4.4
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See also
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.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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