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