A cube packs six equal squares into twelve equal edges
The cube is a polyhedron with eight vertices, twelve equal edges and six square faces, a special case of the cuboid, parallelepiped and rhombohedron. Plato's Timaeus paired it with the element earth, and a cube of unit edge is still the standard unit against which volume is measured.
Three squares meet at every corner, each perpendicular to the others, so every dihedral angle is a right angle and the cube counts as an orthogonal polyhedron. Its faces, edges and vertices are all interchangeable under its symmetries, which makes it one of the regular solids. It is also elementary: no plane slicing only along its edges can split it into smaller convex pieces with regular faces. Like every convex polyhedron it has Euler characteristic 2, and it is the three-dimensional member of the hypercube family that runs from the square to the four-dimensional tesseract.
Its symmetry group has 48 elements. Nine mirror planes divide it, three through edge midpoints and six along diagonals, and thirteen rotation axes pass through it: three through opposite face centres with quarter-turn symmetry, six through opposite edge midpoints with half-turn symmetry, and four through opposite corners with one-third-turn symmetry. Its dual, formed by polar reciprocation, is the regular octahedron, which shares the same octahedral symmetry. Because volume equals edge length raised to the third power, cubing a number came to mean exactly that.
Doubling the cube, building one with twice the volume using only compass and straightedge, occupied geometers for centuries until Pierre Wantzel showed in 1837 that the cube root of 2 cannot be constructed that way; Messer managed it by folding paper in 1986. Cubes stack face to face into a honeycomb that fills space with no gaps, and gluing them together gives polycubes. A path drawn straight across its faces can close on itself in three ways, one of them a regular hexagon.
In everyday life the cube is the usual shape of dice and the basis of puzzles such as the Rubik's Cube, the Skewb and the Soma cube, as well as the blocks of Minecraft. M. C. Escher explored the impossible cube and the Necker cube, Alberti's 1450 architecture treatise used it, and the Dutch cube houses stand tilted on a corner.
Source: Cube
