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Regular tiling of the hyperbolic plane From Wikipedia, the free encyclopedia
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.
Order-4 hexagonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 64 |
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 |
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 [6*,4], 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 |
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 | [6,4] (*642) ![]() ![]() ![]() ![]() ![]() |
[6,6] (*662) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[(6,6,3)] = [6,6,1+] (*663) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[1+,6,6,1+] (*3333) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||
Symbol | {6,4} | r{6,6} = {6,4}1/2 | r(6,3,6) = r{6,6}1/2 | r{6,6}1/4 | |||
Coxeter diagram |
![]() ![]() ![]() ![]() ![]() |
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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}π, an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.
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.
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.
Symmetry mutation of quasiregular tilings: 6.n.6.n | |||||||||||
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Symmetry *6n2 [n,6] |
Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||||||
*632 [3,6] |
*642 [4,6] |
*652 [5,6] |
*662 [6,6] |
*762 [7,6] |
*862 [8,6]... |
*∞62 [∞,6] |
[iπ/λ,6] | ||||
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.∞ |
Uniform tetrahexagonal tilings | |||||||||||
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Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*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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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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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: [6,6], (*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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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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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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Coxeter diagrams |
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Vertex figure |
66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
Image | ![]() |
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Dual | ![]() |
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Uniform tilings in symmetry *3222 | ||||
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