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Compact polyhedra of cubic point symmetry Oh, exhibit surfaces of planarsections (facets) characterized by normal vector families {abc} with up to 48members each, compatible with Oh symmetry. We focus first on polyhedra confinedby facets with normal vectors of one family, {100}, {110}, and {111},separately. This yields generic polyhedra which serve for the definition ofgeneral compact polyhedra as intersections of the three generic species. Theirstructural properties, such as shape, size, volume, and surface facets, arefound to be described by only three polyhedral structure parameters A, B, C. Inaddition, we examine compact polyhedra exhibiting facets defined by normalvectors of only one general {abc} family resulting in up to 48 facets. Thesepolyhedra can be described by four polyhedral structure parameters Q, a, b, c.Geometrical properties of all polyhedra are discussed in analytical detail andsubstantiated by visualization of characteristic examples. Correspondingrelationships can also be used to classify shapes and estimate sizes of realcompact metal nanoparticles observed e.g. in electron microscopy experiments.

Compact Polyhedra of Cubic Symmetry: Geometrical Analysis and Classification