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 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:
| Spherical polyhedra | Euclidean tiling | Hyperbolic tiling | |||||
|---|---|---|---|---|---|---|---|
| Symmetry | *332 Td |
*432 Oh |
*532 Ih |
*632 p6m |
*732 |
*832 |
*∞32 |
| Quasiregular figures |
Octahedron |
Cuboctahedron |
Icosidodecahedron |
Trihexagonal tiling |
Triheptagonal tiling |
Trioctagonal tiling |
|
| Vertex 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 | |||||||
The subset of this sequence in which k is an even number has (*n33) reflectional symmetry.
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.)
| 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 |
| 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: Kagome Lattice
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