It is known that the class of regular-faced polyhedra in $E^3$ with conditional edges, which generalizes the class of Johnson polyhedra, contains, in particular, polyhedra with symmetric rhombic face stars of some vertices; however, the rhombuses of the vertex stars can only have an acute angle of 60 degrees. In this paper, we consider a class of polyhedra (the so-called $RR$-polyhedra) that, in addition to regular faces, contain rhombic face stars with angles not necessarily equal to 60 degrees. Previously, the author found twenty-four RR-polyhedra with regular faces of the same type and proved the completeness of the list of such polyhedra. A complete list of forty-six composite $RR$-polyhedra with regular faces of various types, lacking conditional edges and conditional vertices, was also found. In this paper, the author continues the study of $RR$-polyhedra: a complete list of composite RR-polyhedra with conditional edges is found. Since the $RR$-polyhedron also has irregular faces (rhombuses), a different definition of a composite $RR$-polyhedron (compared to the definition of a composite regular-faced polyhedron) is used in this work.
Keywords: RR polyhedron, a compound polyhedron with conditional edges, a symmetric rhombic vertex
Received November 5, 2025
Revised December 4, 2025
Accepted December, 12 2025
Vladimir Ivanovich Subbotin, Cand. Sci. (Phys.-Math.), Platov South Russian State Polytechnic University (NPI), Novocherkassk, 346400 Russia, e-mail: geometry@mail.ru
REFERENCES
1. Coxeter H.S.M. Regular polytopes. NY, Dover Publ., 1973, 321 p. ISBN-10: 0486614808 .
2. Rajwade A.R. Convex polyhedra with regularity conditions and Hilbert’s third problem. Gurgaon, Hindustan Book Agency, 2001, 128 p. https://doi.org/10.1007/978-93-86279-06-4
3. Cromwell P.R. Polyhedra. Cambridge, Cambridge Univ. Press, 1999, 476 p. ISBN-13: 9780521664059 .
4. Deza М., Grishukhin V.P., Shtogrin M.I. Scale-isometric polytopal graphs in hypercubes and cubic lattices: polytopes in hypercubes and Zn. London, Imperial College Press, 2004, 188 p. https://doi.org/10.1142/p308 . Translated to Russian under the title Izometricheskiye poliedral’nyye podgrafy v giperkubakh i kubicheskikh reshetkakh, Moscow, MTSNMO, 2008, 192 p.
5. Gilson B.R. The Johnson solids and their duals: a comprehensive survey. CreateSpace Independent Publ. Platform. 2014, 446 p. ISBN-13: 9781500932459 .
6. Piette B.M.A.G, Kowalczyk A, Heddle J.G. Characterization of near-miss connectivity-invariant homogeneous convex polyhedral cages. Proc. Royal Soc. A: Math, Phys. and Engin. Sci., 2022, vol. 478, no. 2260, art. no. 20210679. https://doi.org/10.1098/rspa.2021.0679
7. Jonson N.W. Convex polyhedra with regular faces. Canad. J. Math., 1966, vol. 18, no. 1, pp. 169–200. https://doi.org/10.4153/CJM-1966-021-8
8. Zalgaller V.A. Convex polyhedra with regular faces. Seminars in mathematics V. A. Steklov Mathematical Institute, vol. 2. New York, 1969, 95 p. Original Russian text published in Zalgaller V. A. Vypuklyye mnogogranniki s pravil’nymi granyami. Zapiski nauchnykh seminarov Leningradskogo otdeleniya Mat. In-ta V. A. Steklova AN SSSR “LOMI”, tom 2. Moscow, Leningrad, Nauka Publ., 1967, 220 p.
9. Tupelo-Schneck R. Convex regular-faced polyhedra with conditional edges [Elect. res.]. 2025. Available on: http://tupelo-schneck.org/polyhedra
10. Tupelo-Schneck R. Regular-faced polyhedra [Elect. res.]. 2025. Available on: https://tupelo-schneck.org/polyhedra/background.html
11. Subbotin V.I. On two classes of polyhedra with rhombic vertices. J. Math. Sci., 2020, vol. 251, pp. 531–538. https://doi.org/10.1007/s10958-020-05114-3
12. Subbotin V.I. On the composite RR-polyhedra of the second type. Sib. Math. J., 2023, vol. 64, no. 2, pp. 500–506. https://doi.org/10.1134/S0037446623020209
Cite this article as: Subbotin V.I. Composite RR-polytopes with conditional edges. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 263–272.
