The s, p, d and f atomic orbitals are classified by the numbers and orientations of their major lobes, which can be conveniently depicted as an orbital graph. Coordination polyhedra are formed from these s, p, d and f orbitals by adding or subtracting orbitals from the spherical four-orbital sp 3 and nine-orbital sp 3 d 5 manifolds. The five-coordinate trigonal bipyramid and square pyramid arise from adding the z 2 and x 2− y 2 orbitals, respectively, to a spherical sp 3 manifold. Six-coordinate polyhedra are formed by adding pairs of d orbitals to the spherical sp 3 manifold with the pairs ( x 2− y 2, z 2), ( xz,z 2), ( xy,x 2− y 2) and ( xy,xz), giving the octahedron, bicapped tetrahedron, pentagonal pyramid and trigonal prism, respectively. Similarly, the eight-coordinate square antiprism and bisdisphenoid ( D 2 d dodecahedron) arise from subtracting the z 2 and x 2− y 2 orbitals, respectively, from a spherial nine-orbital sp 3 d 5 manifold. The seven-coordinate pentagonal bipyramid, capped octahedron and capped trigonal prisms can arise from subtractions of various d orbital pairs from the spherical nine-orbital sp 3 d 5 manifold. The stereochemical non-rigidity of many five-coordinate ML 5 complexes and eight-coordinate ML 8 complexes can arise from continuous transformations of the d orbital added to the sp 3 manifold or subtracted from the sp 3 d 5 manifold, respectively, from the shape of a z 2 orbital to that of an z 2− y 2 orbital. The seven-vertex hexagonal bipyramid and eight-vertex cube, hexagonal bipyramid and D 3 h 3,3-bicapped trigonal prism cannot be formed using hybrids of only s, p and d orbitals, but require an additional f orbital, which is an xyz orbital with eight major lobes in case of the cube, an x( x 2−3 y 2) orbital with six major lobes in the cases of the hexagonal pyramid and bipyramid with six co-planar atoms, and a z 3 orbital in the case of the 3,3-bicapped trigonal prism. The highly symmetrical 12-coordinate icosahedron and cuboctahedron can arise by addition of a triply degenerate set of cubic f ε orbitals, namely the [ x( z 2− y 2), y( z 2− x 2), z( x 2− y 2)] set, to the spherical nine-orbital sp 3 d 5 manifold.