Jucovič, E.; Ševec, S.; Trenkler, M. Constructions of triangular and quadrangular polyhedra of inscribable type. (English) Zbl 1023.52001 Math. Slovaca 47, No. 3, 313-317 (1997). Summary: A polyhedron \(P\) is of inscribable type if there exists a polyhedron \(P^\ast \) combinatorially isomorphic to \(P\) and a sphere \(\gamma \) such that all the vertices of \(P^\ast \) belong to \(\gamma \). There has been some effort to characterize both the inscribable and non-inscribable types of polyhedra in the literature. In the present paper two sufficient conditions for a polyhedron to be of inscribable type are given. The conditions are, in fact, constructions of infinite families of triangular and quadrangular polyhedra of inscribable type. The paper ends with the interesting question whether there are infinitely many pentagonal polyhedra of inscribable type or the dodecahedron is the only one. MSC: 52B10 Three-dimensional polytopes Keywords:polyhedron; inscribable type × Cite Format Result Cite Review PDF Full Text: EuDML References: [1] DILLENCOURT M. B-SMITH W. D.: A linear time algorithm for testing the inscribability of trivalent polyhedra. Internat. J. Comput. Geom. Appl. 5 (1995), 21-36. · Zbl 0818.68079 · doi:10.1142/S0218195995000039 [2] DILLENCOURT M. B.-SMITH W. D.: A simple method for resolving degeneracies in Delaunay triangulations. Automata, Languages, and Programming: Proc. 20th Internal. Coll., Lund Sweden ICALP, Lund, Sweden, July 1993. Lecture Notes in Comput. Sci. 700, Springer, Berlin-New York, 1993, pp. 177-188. [3] GRÜNBAUM B.: On Steinitz’s theorem about non-inscribable polyhedra. Nederl. Akad. Wetensch. Proc. Ser. A 66 (1963), 452-455. · Zbl 0115.15005 [4] GRÜNBAUM B.-SHEPHARD G. C.: Some problems on polyhedra. J. Geom. 29 (1987), 182-190. · Zbl 0618.52004 · doi:10.1007/BF01225208 [5] HARARY F.: Graph Theory. Addison Wesley, Reading, 1969. · Zbl 0196.27202 [6] HODGSON C. D.-RIVIN I.-SMITH W. D.: A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bull. Amer. Math. Soc. (N.S.) 27 (1992), 246-251. · Zbl 0759.52010 · doi:10.1090/S0273-0979-1992-00303-8 [7] JUCOVIČ E.-ŠEVEC S.: Note on inscribability of quadrangular polyhedra with restricted number of edge-types. J. Geom. 42 (1991), 126-131. · Zbl 0742.52011 · doi:10.1007/BF01231872 [8] STEINER J.: Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander. Reimer, Berlin, 1832; Jacob Steiner’s Collected Works, Vol. 1, Berlin, 1881. · Zbl 0015.36602 [9] STEINITZ E.: Uber isoperimetrische Probleme bei konvexen Polyedern I; II. J. Reine Angew. Math. 158; 159 (1927; 1929), 129-153; 133-143. · JFM 53.0480.02 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.