The non-Platonic and non-Archimedean noncomposite polyhedra (Q844552)

If a convex polyhedron with regular faces can be divided by some plane into two polyhedra with regular faces, then it is said to be composite; otherwise, the polyhedron is noncomposite. Noncomposite Platonic polyhedra are the dodecahedron, the tetrahedron, and the cube. The octahedron is composed of two pyramids, and the icosahedron consists of at least three noncomposite polyhedra. It is proved in [\textit{V. A. Zalgaller}, Convex polyhedra with regular faces. Semin. in Mathematics, V.A. Steklov Math. Inst., Leningrad 2 (1969; Zbl 0177.24802)] that except for infinite series of prisms and antiprisms, there exist only 28 noncomposite polyhedra. In the paper the author indicates the exact coordinates of the vertices of noncomposite polyhedra that are neither Platonic, nor Archimedean, nor their parts cut by no more than three planes. Such a description allows to obtain a short proof of the existence of each of the eight such polyhedra and to obtain other applications.