the Point groups in three dimensions#The seven remaining point groups, either tetrahedral Td, octahedral Oh or icosahedral Ih. Truncated forms Regular polyhedra and their truncated forms Column A lists all the regular polyhedra, column B list their truncated forms. Regular polyhedra all have vertex figures pr: p.p.p etc. and Wythoff symbol p|q r. The truncated forms have vertex figure q.q.r (where q=2p and r) and Wythoff p q|r. In addition there are three quasi-truncated forms. These also class as truncated-regular polyhedra. Truncated forms of quasi-regular polyhedra Column A lists some quasi-regular polyhedra, column B lists normal truncated forms, column C shows quasi-truncated forms, column D shows a different method of truncation. These truncated forms all have a vertex figure p.q.r and a Wythoff symbol p q r|. Polyhedra sharing edges and vertices Regular These are all mentioned elsewhere, but this table shows some relations. They are all regular apart from the tetrahemihexahedron which is versi-regular. Quasi-regular and versi-regular Rectangular vertex figures, or crossed rectangles first column are quasi-regular second and third columns are hemihedra with faces passing through the origin, called versi-regular by some authors. Ditrigonal regular and versi-regular Ditrigonal (that is di(2) -tri(3)-ogonal) vertex figures are the 3-fold analog of a rectangle. These are all quasi-regular as all edges are isomorphic. The compound of 5-cubes shares the same set of edges and vertices. The cross forms have a non-orientable vertex figure so the "-" notation has not been used and the "*" faces pass near rather than through the origin. versi-quasi-regular and quasi-quasi-regular Group III: trapezoid or crossed trapezoid vertex figures. The first column include the convex rhombic polyhedra, created by inserting two squares into the vertex figures of the Cuboctahedron and Icosidodecahedron. References Uniform polyhedra