| Order-4 hexagonal tiling | |
|---|---|
 Poincaré disk model of thehyperbolic 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 |
Ingeometry, theorder-4 hexagonal tiling is aregular tiling of thehyperbolic plane. It hasSchläfli symbol of {6,4}.
This tiling represents a hyperbolickaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry byorbifold notation is called *222222 with 6 order-2 mirror intersections. InCoxeter 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 distinctuniform colorings for the order-4 hexagonal tiling. They are similar to 7 of theuniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions andCoxeter 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 | |  = |  = | = = | |||
Theregular 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 apetrial octahedron, {3,4}π, an abstract polyhedron with vertices and edges of anoctahedron, but instead connected by 4Petrie polygon faces.
This tiling is topologically related as a part of sequence of regular tilings withhexagonal faces, starting with thehexagonal tiling, withSchläfli symbol {6,n}, andCoxeter 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 theoctahedron, withSchläfli symbol {n,4}, and Coxeter diagram, with n progressing to infinity.
| *n42 symmetry mutation of regular tilings: {n,4} | |||||||
|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | |||||
 |  |  |  | |  |  |  |
| 24 | 34 | 44 | 54 | 64 | 74 | 84 | ...∞4 |
| Symmetry mutation of quasiregular tilings: (6.n)2 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | |||||||||||
 | | | | | | | |||||
 |  | |  |  |  |  | |||||
| 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) | |||||
 =  |  =   | =   | = | =  | =  | | |||||
 |  |  |  |  |  |  | |||||
| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} | |||||
| Uniform hexahexagonal tilings | ||||||
|---|---|---|---|---|---|---|
| Symmetry:[6,6], (*662) | ||||||
=  = | = = | = = | = = | = = | = = | = = |
 |  | |  | |  |  |
| {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 | ||||||
| | | | | | |
 |  |  |  |  |  |  |
| 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) |
| = | | = | | = | | |
| | | | | | |
 |  | | |  | ||
| 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 | ||||||||
|---|---|---|---|---|---|---|---|---|
| Coxeter diagrams | | | | | ||||
 | | | | | | | | |
| | | | |||||
| Vertex figure | 66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
| Image |  | |  |  | ||||
| Dual |  | | ||||||
| Uniform tilings in symmetry *3222 | ||||
|---|---|---|---|---|
64 |  6.6.4.4.png) |  (3.4.4)2 | 4.3.4.3.3.3 | |
6.6.4.4 | 6.4.4.4 | 3.4.4.4.4.png) | ||
| (3.4.4)2 | 3.4.4.4.4 | 46 | ||
