page_title: Cube
section_title: Uniform colorings and symmetry
hierarchical_section_title: Cube / Uniform colorings and symmetry
caption_attribution_description: 3D point symmetry subgroup hierarchy
context_page_description: In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
  The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
  The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
  The cube is dual to the octahedron. It has cubical or octahedral symmetry.
  The cube is the only convex polyhedron whose faces are all squares.
context_section_description: The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.
  The cube has four 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 prismatic subsets D2d has the same coloring as the previous one and D₂ₕ has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.
caption: Octahedral symmetry tree
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