# Order-4 hexagonal tiling

Order-4 hexagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic regular tiling |
| Vertex configuration | 6$^4$ |
| 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 [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 |

## 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.
Uniform constructions of 6.6.6.6 :
| | 1 color | 2 colors | 3 and 2 colors | 3 and 2 colors | 4, 3 and 2 colors | 4, 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) = | [(6,6,3)] = [6,6,1$^+$] (*663) = | [1$^+$,6,6,1$^+$] (*3333) = = | [1$^+$,6,6,1$^+$] (*3333) = = | [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(6,3,6) = r{6,6}$^1$/$_2$ | r{6,6}$^1$/$_4$ | r{6,6}$^1$/$_4$ | r{6,6}$^1$/$_4$ |
| Coxeter diagram | | = | = | = | = = | = = | = = |

## 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}$^π$, 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} vte :
| Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | 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} vte :
| Spherical | Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| 2$^4$ | 3$^4$ | 4$^4$ | 5$^4$ | 6$^4$ | 7$^4$ | 8$^4$ | ...∞$^4$ |
Symmetry mutation of quasiregular tilings: 6.n.6.n vte :
| Symmetry *6n2 [n,6] | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *6n2 [n,6] | *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 vte :
| 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 |||||||
| V6$^4$ | V4.12.12 | V(4.6)$^2$ | V6.8.8 | V4$^6$ | 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 vte :
| Symmetry: [6,6], (*662) |||||||
| {6,6} = h{4,6} | t{6,6} = h$_2${4,6} | r{6,6} {6,4} | t{6,6} = h$_2${4,6} | {6,6} = h{4,6} | rr{6,6} r{6,4} | tr{6,6} t{6,4} |
| Uniform duals |||||||
| V6$^6$ | V6.12.12 | V6.6.6.6 | V6.12.12 | V6$^6$ | 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} |
Similar H2 tilings in *3232 symmetry vte :
| Coxeter diagrams | | | | | | | | |
| Vertex figure | 6$^6$ | 6$^6$ | (3.4.3.4)$^2$ | (3.4.3.4)$^2$ | 3.4.6.6.4 | 3.4.6.6.4 | 6.4.6.4 | 6.4.6.4 |
| Image | | | | | | | | |
| Dual | | | | | | | | |
Uniform tilings in symmetry *3222 vte :
| 6$^4$ | 6.6.4.4 | (3.4.4)$^2$ | 4.3.4.3.3.3 |
| 6.6.4.4 | 6.4.4.4 | 3.4.4.4.4 |
| (3.4.4)$^2$ | 3.4.4.4.4 | 4$^6$ |