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:
“The sugar maple is remarkable for its clean ankle. The groves of these trees looked like vast forest sheds, their branches stopping short at a uniform height, four or five feet from the ground, like eaves, as if they had been trimmed by art, so that you could look under and through the whole grove with its leafy canopy, as under a tent whose curtain is raised.”
—Henry David Thoreau (1817–1862)
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—George Gordon Noel Byron (1788–1824)