| Order-6 hexagonal tiling | |
|---|---|
 Poincaré disk model of thehyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 66 |
| Schläfli symbol | {6,6} |
| Wythoff symbol | 6 | 6 2 |
| Coxeter diagram |    |
| Symmetry group | [6,6], (*662) |
| Dual | self dual |
| Properties | Vertex-transitive,edge-transitive,face-transitive |
Ingeometry, theorder-6 hexagonal tiling is aregular tiling of thehyperbolic plane. It hasSchläfli symbol of {6,6} and isself-dual.
This tiling represents a hyperbolickaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry byorbifold notation is called *333333 with 6 order-3 mirror intersections. InCoxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.
The even/odd fundamental domains of thiskaleidoscope can be seen in the alternating colorings of the
tiling:
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices withSchläfli symbol {n,6}, andCoxeter diagram
, progressing to infinity.
| Regular tilings {n,6} | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
 {2,6}  |  {3,6}  |  {4,6}  |  {5,6}  |  {6,6} |  {7,6}  |  {8,6}  | ... |  {∞,6}  |
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,∞} |
| 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 |  |  | ||||||
