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On Polyhedra with Rhombic Vertices and Regular Faces

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Abstract

In this paper, we consider the class of closed convex polyhedra with regular faces in E3 for which the stars of some vertices are symmetric and consist of equal and identically located rhombuses (RR-polyhedra). We obtain a complete classification of RR-polyhedra with two acute-angled rhombic vertices whose stars are separated by a belt of regular faces of the same type. The proof is based on a result on the existence of two polyhedra of this class obtained by the author earlier.

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References

  1. H. S. M. Coxeter, Regular Polytopes, Dover, New York (1973).

    MATH  Google Scholar 

  2. H. S. M. Coxeter, “Regular and semi-regular polytopes, III,” Math. Z., 200, No. 21, 3–45 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  3. P. R. Cromwell, Polyhedra, Cambridge Univ. Press, Cambridge (1999).

    MATH  Google Scholar 

  4. M. Deza, V. P. Grishukhin, and A. I. Shtogrin, Isometric Polyhedral Subgraphs in Hypercubes and Cubic Lattices [in Russian], MCCME, Moscow (2007).

    MATH  Google Scholar 

  5. S. L. Farris, “Completely classifying all vertex-transitive and edge-transitive polyhedra,” Geom. Dedic., 26, No. 1, 111–124 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Grunbaum, “Regular polyhedra—old and new,” Aequat. Math., 16, No. 1-2, 1–20 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  7. N. W. Johnson, “Convex polyhedra with regular faces,” Can. J. Math., 18, No. 1, 169–200 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  8. V. I. Subbotin, “On two classes of polyhedra with rhombic vertices,” Zap. Nauch. Semin. POMI, 476, 153–164 (2018).

    Google Scholar 

  9. V. I. Subbotin, “On a one class of strongly symmetric polyhedra,” Chebyshev. Sb., No. 4, 132–140 (2016).

  10. V. A. Zalgaller, “Convex polyhedra with regular faces,” Zap. Nauch. Semin. LOMI, 2, 1–220 (1967).

    MathSciNet  Google Scholar 

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Correspondence to V. I. Subbotin.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 181, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor V. T. Bazylev. Moscow, April 22-25, 2019. Part 3, 2020.

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Subbotin, V.I. On Polyhedra with Rhombic Vertices and Regular Faces. J Math Sci 276, 802–806 (2023). https://doi.org/10.1007/s10958-023-06803-5

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  • DOI: https://doi.org/10.1007/s10958-023-06803-5

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