This article is about the 3-dimensional shape. For cubes in any cube steak in instant pot, see Hypercube. 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 a 3-zonohedron. 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. The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Not to be confused with Squircle. The cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection.

This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. A cube can also be considered the limiting case of a 3D superellipsoid as all three exponents approach infinity. The cube has three uniform colorings, named by the unique 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 solid, with all the six sides being different colors. The 11 nets of the cube. These familiar six-sided dice are cube-shaped. To color the cube so that no two adjacent faces have the same color, one would need at least three colors. The cube is the cell of the only regular tiling of three-dimensional Euclidean space. The cube can be cut into six identical square pyramids. They also appear in Judaism as Teffilin and New Jerusalem in the New Testament is also described as being a Cube.

The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. Euclidean space and a tesseract is the order-4 hypercube. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions. The dual of a cube is an octahedron, seen here with vertices at the center of the cube’s square faces. The hemicube is the 2-to-1 quotient of the cube. The quotient of the cube by the antipodal map yields a projective polyhedron, the hemicube.