The non-Platonic and non-Archimedean noncomposite polyhedra

If a convex polyhedron with regular faces cannot be divided by any plane into two polyhedra with regular faces, then it is said to be noncomposite. We indicate the exact coordinates of the vertices of noncomposite polyhedra that are neither regular (Platonic), nor semiregular (Archimedean), nor their parts cut by no more than three planes. Such a description allows one to obtain a short proof of the existence of each of the eight such polyhedra (denoted by $M_8$, $M_{20}$–$M_{25}$, $M_{28}$) and to obtain other applications.