Truncated cuboctahedron

In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.The name truncated cuboctahedron, given originally by Johannes Kepler, is misleading. An actual truncation of a cuboctahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid. In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicuboctahedron. The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all permutations of: The area A and the volume V of the truncated cuboctahedron of edge length a are: The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons and 8 triangular cupolas below the hexagons. A dissected truncated cuboctahedron can create a genus 5, 7 or 11 Stewart toroid by removing the central rhombicuboctahedron and either the square cupolas, the triangular cupolas or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupolas creates a genus 3 torus, which (if they are chosen appropriately) has tetrahedral symmetry. There is only one uniform coloring of the faces of this polyhedron, one color for each face type. A 2-uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons. The truncated cuboctahedron has two special orthogonal projections in the A2 and B2 Coxeter planes with and projective symmetry, and numerous symmetries can be constructed from various projected planes relative to the polyhedron elements.