Simple equations giving shapes of various convex polyhedra: the regular polyhedra and polyhedra composed of crystallographically low-index planes

Simple equations are derived that give the shapes of various convex polyhedra. The five regular polyhedra, called Platonic solids ( the tetrahedron, hexahedron or cube, octahedron, dodecahedron and icosahedron), and polyhedra composed of crystallographically low-index planes are treated. The equations also give shapes that are nearly polyhedral with round edges, or intermediate shapes between a sphere and convex polyhedra, and which are observed as the shapes of small particles or precipitates in materials.