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Like other cuboids, every face of a cube has four vertices, each of which connects with three congruent lines. These edges form square faces, making the dihedral angle of a cube between every two adjacent squares being the interior angle of a square, 90°. Hence, the cube has six faces, twelve edges, and eight vertices.
This process is known as rectification, making the cuboctahedron being named the rectified cube and rectified octahedron. [3] An alternative construction is by cutting of all of the vertices, known as truncation. can be started from a regular tetrahedron, cutting off the vertices and beveling the edges
The chamfered cube is constructed as a chamfer of a cube: the squares are reduced in size and new faces, hexagons, are added in place of all the original edges. The cC is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 6 congruent (and regular) squares, and 12 congruent flattened hexagons.
Faces are reduced to half as many sides, and square faces degenerate into edges. For example, the tetrahedron is an alternated cube, h{4,3}. Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices.
The cuboctahedron can flex this way even if its edges (but not its faces) are rigid. The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity. Consequently, its vertices can be repositioned by folding (changing the dihedral angle ...
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges.
A rectified cube is a cuboctahedron – edges reduced to vertices, and vertices expanded into new faces A birectified cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices. A rectified cubic honeycomb – edges reduced to vertices, and vertices expanded into new cells.
The snub cube is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [6] It is chiral, meaning there are two distinct forms whenever being mirrored. Therefore, the snub cube has the rotational octahedral symmetry.