Related Polyhedra and Tilings
This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane.
(33) |
(43) |
(53) |
(63) tiling |
(73) tiling |
It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6.
| Symmetry | Spherical | Planar | Hyperbolic... | |||||
|---|---|---|---|---|---|---|---|---|
| *232 D3h |
*332 Td |
*432 Oh |
*532 Ih |
*632 P6m |
*732 |
*832 ... |
*∞32 |
|
| Order | 12 | 24 | 48 | 120 | ∞ | |||
| Truncated figures |
2.6.6 |
3.6.6 |
4.6.6 |
5.6.6 |
6.6.6 |
7.6.6 |
8.6.6 |
3.4.∞.4 |
| Coxeter Schläfli |
t0,1{3,2} |
t0,1{3,3} |
t0,1{3,4} |
t0,1{3,5} |
t0,1{3,6} |
t0,1{3,7} |
t0,1{3,8} |
t0,1{3,∞} |
| n-kis figures |
V2.6.6 |
V3.6.6 |
V4.6.6 |
V5.6.6 |
V6.6.6 |
V7.6.6 |
||
| Coxeter | ||||||||
This tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces. The sequence has two vertex figures (n.6.6) and (6,6,6).
| Polyhedra | Euclidean tiling | Hyperbolic tiling | |||
|---|---|---|---|---|---|
Cube |
Rhombic dodecahedron |
Rhombic triacontahedron |
Rhombille |
||
Alternate truncated cube |
Truncated rhombic dodecahedron |
Truncated rhombic triacontahedron |
Hexagonal tiling |
||
The hexagonal tiling can be considered an elongated rhombic tiling, where each vertex of the rhombic tiling is stretched into a new edge. This is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions.
Rhombic tiling |
Hexagonal tiling |
Fencing uses this relation |
Read more about this topic: Hexagonal Tiling
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