# Order-4 hexagonal tiling

Order-4 hexagonal tiling

Poincaré disk model of the hyperbolic plane
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.

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## 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

{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}
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
{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
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)
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
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
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
{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
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)
h0{6,6} h1{6,6} h0,1{6,6} h1,2{6,6} h2{6,6} h0,2{6,6} s{6,6}
Alternation duals
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
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## References

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## See also

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Last modified on 19 March 2013, at 22:50