This is one of the Platonic solids, and is treated in the article Polyhedron, as is also the derived Archimedean solid named the "truncated tetrahedron"; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system.
More briefly, the figure may be defined as a polyhedron with two parallel faces containing all the vertices.
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.
As regards the former figure this is evident at once; viz, the polyhedron consists of two pyramids with vertices represented by 0, 0, and a common base whose perimeter is represented by the forcepolygon (only one of these is shown in fig.
The points thus obtained are evidently the vertices of a polyhedron with plane faces.
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.
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.