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: Cubes
Famous quotes containing the words uniform and/or symmetry:
“When a uniform exercise of kindness to prisoners on our part has been returned by as uniform severity on the part of our enemies, you must excuse me for saying it is high time, by other lessons, to teach respect to the dictates of humanity; in such a case, retaliation becomes an act of benevolence.”
—Thomas Jefferson (1743–1826)
“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)