Wythoff Constructions From Hexagonal and Triangular Tilings
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
| Wythoff | 3 | 6 2 | 2 3 | 6 | 2 | 6 3 | 2 6 | 3 | 6 | 3 2 | 6 3 | 2 | 6 3 2 | | | 6 3 2 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Schläfli | {6,3} | t0,1{6,3} | t1{6,3} | t1,2{6,3} | t2{6,3} | t0,2{6,3} | t0,1,2{6,3} | s{6,3} | h0{6,3} | h1,2{6,3} | |
| Coxeter | |||||||||||
| Image Vertex figure |
6.6.6 |
3.12.12 |
3.6.3.6 |
6.6.6 |
{36} |
3.4.6.4 |
4.6.12 |
3.3.3.3.6 |
(3.3)3 |
3.3.3.3.3.3 |
|
| Uniform duals | |||||||||||
| Coxeter | |||||||||||
| Image Vertex figure |
V6.6.6 |
V3.12.12 |
V3.6.3.6 |
V6.6.6 |
V3.3.3.3.3.3 |
V3.4.6.4 |
V4.6.12 |
V3.3.3.3.6 |
V(3.3)3 |
||
| Wythoff | 3 | 3 3 | 3 3 | 3 | 3 | 3 3 | 3 3 | 3 | 3 | 3 3 | 3 3 | 3 | 3 3 3 | | | 3 3 3 |
|---|---|---|---|---|---|---|---|---|
| Coxeter | ||||||||
| Image Vertex figure |
(3.3)3 |
3.6.3.6 |
(3.3)3 |
3.6.3.6 |
(3.3)3 |
3.6.3.6 |
6.6.6 |
3.3.3.3.3.3 |
Read more about this topic: Hexagonal Tiling