More briefly, the figure may be defined as a polyhedron with two parallel faces containing all the vertices.
The points thus obtained are evidently the vertices of a polyhedron with plane faces.
The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article Crystallography for its significance as a crystal form.
If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former.
A polyhedron (A) is said to be the summital or facial holohedron of another (B) when the faces or vertices of A correspond to the edges of B, and the vertices or faces of A correspond to the vertices and faces together of B.
If the figure be entirely to one side of any face the polyhedron is said to be " convex, " and it is obvious that the faces enwrap the centre once; if, on the other hand, the figure is to both sides of every face it is said to be concave, " and the centre is multiply enwrapped by the faces.
A polyhedron is said to be the hemihedral form of another polyhedron when its faces correspond to the alternate faces of the latter or holohedral form; consequently a hemihedral form has half the number of faces of the holohedral form.
The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which one polyhedron may be derived from another.
Take the pole of each face of such a polyhedron with respect to a paraboloid of revolution, these poles will be the vertices of a second polyhedron whose edges are the conjugate lines of those of the former.
Octahedra having triangular faces other than equilateral occur as crystal forms. See Polyhedron and Crystallography.