We find all convex compounds of the Zalgaller polyhedra \(M_3\), \(M_8\), \(M_{20}\). Besides the eight such compounds
that appear in the classification theorem on convex regular-faced polyhedra (A. M. Gurin, V. A. Zalgaller, A. V. Timofeenko, 2008–2011, see, e. g., [1]), we obtain four more polyhedra some of whose faces are composed of
regular polygons in such a way that some of the polygons' vertices are in the interior of a polyhedron's edge. There exists no theorem so far that describes all convex polyhedra with such vertices. A part of such a theorem is the following
assertion:
Theorem.A convex polyhedron is composed of the bodies \(M_3\), \(M_8\), \(M_{20}\) if and only if it is one of the following compounds:
\[S_1 = M_3+M_3, S_2 = M_3+M_8, S_3 = M_3+M_{20};\]
\[S_4 = S_2+M_3, S_5 = S_2+M_3', S_6 = S_2+M_3'', S_7 = S_3+M_3;\]
\[S_8 = S_4+M_3, S_9 = M_3+S_7;\]
\[S_{10} = M_3+S_6.\]
The primes in \(M_3'\) and \(M_3''\) mean that the pyramid \(M_3\) is attached in \(S_5\) and \(S_6\) to other
faces than in \(S_4\).
References
A. V. Timofeenko, On the list of regular-faced polyhedra, Modern Problems in Mathematics and Mechanics, V. VI. Mathematics, Issue 3, to the 100th Anniversary of N. V. Efimov., MSU, Moscow (2011), 155–170. (Russian)
