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.

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.

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.

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.

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.

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.

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.

Octahedra having triangular faces other than equilateral occur as crystal forms. See **Polyhedron** and Crystallography.

**POLYHEDRON** (Gr.