Related Polyhedra and Tilings
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.)
| Symmetry:, (*632) | +, (632) | , (*333) | , (3*3) | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| {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} | h{6,3} | h1,2{6,3} | |
| Uniform duals | ||||||||||
| V6.6.6 | V3.12.12 | V3.6.3.6 | V6.6.6 | V3.3.3.3.3.3 | V3.4.12.4 | V.4.6.12 | V3.3.3.3.6 | V3.3.3.3.3.3 | ||
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
| Symmetry *n32 |
Spherical | Euclidean | Hyperbolic | |||||
|---|---|---|---|---|---|---|---|---|
| *232 D3h |
*332 Td |
*432 Oh |
*532 Ih |
*632 P6m |
*732 |
*832 |
*∞32 |
|
| Coxeter Schläfli |
t0,1,2{2,3} |
t0,1,2{3,3} |
t0,1,2{4,3} |
t0,1,2{5,3} |
t0,1,2{6,3} |
t0,1,2{7,3} |
t0,1,2{8,3} |
t0,1,2{∞,3} |
| Omnitruncated figure |
||||||||
| Vertex figure | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ |
| Dual figures | ||||||||
| Coxeter | ||||||||
| Omnitruncated duals |
||||||||
| Face configuration |
V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ |
Read more about this topic: Truncated Trihexagonal Tiling
Famous quotes containing the word related:
“One does not realize the historical sensation as a re-experiencing, but as an understanding that is closely related to the understanding of music, or rather of the world by means of music.”
—Johan Huizinga (1872–1945)