Related Polyhedra and Tilings
A tiling with alternate large and small triangles is topologically identical to the trihexagonal tiling, but has a different symmetry group. The hexagons are distorted so 3 vertices are on the mid-edge of the larger triangles. As with the trihexagonal tiling, it has two uniform colorings:
The trihexagonal tiling is also one of eight uniform tilings that can be formed from the regular hexagonal tiling (or the dual triangular tiling) by a Wythoff construction. 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 | ||
| 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 |
The trihexagonal tiling forms the case k = 6 in a sequence of quasiregular polyhedra and tilings, each of which has a vertex figure with two k-gons and two triangles:
| Symmetry *n32 |
Spherical | Euclidean | Hyperbolic tiling | ||||
|---|---|---|---|---|---|---|---|
| *332 Td |
*432 Oh |
*532 Ih |
*632 p6m |
*732 |
*832 |
*∞32 |
|
| Quasiregular figures configuration |
3.3.3.3 |
3.4.3.4 |
3.5.3.5 |
3.6.3.6 |
3.7.3.7 |
3.8.3.8 |
3.∞.3.∞ |
| Coxeter diagram | |||||||
| Dual (rhombic) figures configuration |
V3.3.3.3 |
V3.4.3.4 |
V3.5.3.5 |
V3.6.3.6 |
V3.7.3.7 |
V3.8.3.8 |
V3.∞.3.∞ |
| Coxeter diagram | |||||||
The subset of this sequence in which k is an even number has (*n33) reflectional symmetry.
Read more about this topic: Trihexagonal Tiling
Famous quotes containing the word related:
“Becoming responsible adults is no longer a matter of whether children hang up their pajamas or put dirty towels in the hamper, but whether they care about themselves and others—and whether they see everyday chores as related to how we treat this planet.”
—Eda Le Shan (20th century)