Uniform Colorings and Symmetry
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.
The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.
| Name | Regular hexahedron | Square prism | Cuboid | Trigonal trapezohedron |
|---|---|---|---|---|
| Coxeter-Dynkin | ||||
| Schläfli symbol | {4,3} | {4}×{} | {}×{}×{} | |
| Wythoff symbol | 3 | 4 2 | 4 2 | 2 | 2 2 2 | | |
| Symmetry | Oh (*432) |
D4h (*422) |
D2h (*222) |
D3d (2*3) |
| Symmetry order | 24 | 16 | 8 | 12 |
| Image (uniform coloring) |
(111) |
(112) |
(123) |
(111), (112), (122), and (222) |
Read more about this topic: Cube
Famous quotes containing the words uniform and/or symmetry:
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—William Howard Taft (1857–1930)
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