Cube this problem, and in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number. 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 O has all the faces the same color. The dihedral symmetry D comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D is also a prismatic symmetry, with
Cube only member of that group that is a zonohedron (every face has point symmetry). The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces). The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube (or "n"-dimensional cube or simply "n"-cube) is the analogue of the cube in "n"-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is also called