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Mat. Sb., 1998, Volume 189, Number 10, Pages 105–134 (Mi msb354)  

This article is cited in 30 scientific papers (total in 30 papers)

A generalized Heron–Tartaglia formula and some of its consequences

I. Kh. Sabitov

M. V. Lomonosov Moscow State University

Abstract: The well-known formula for finding the area of a triangle in terms of its sides is generalized to volumes of polyhedra in the following way. It is proved that for a polyhedron (with triangular faces) with a given combinatorial structure $K$ and with a given collection $(l)$ of edge lengths there is a polynomial such that the volume of the polyhedron is a root of it, and the coefficients of the polynomial depend only on $K$ and $(l)$ and not on the concrete configuration of the polyhedron itself. A number of problems in the metric theory of polyhedra are solved as a consequence.

DOI: https://doi.org/10.4213/sm354

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English version:
Sbornik: Mathematics, 1998, 189:10, 1533–1561

Bibliographic databases:

UDC: 513.7
MSC: Primary 52C25; Secondary 51M25, 52B10, 52B45
Received: 07.05.1998

Citation: I. Kh. Sabitov, “A generalized Heron–Tartaglia formula and some of its consequences”, Mat. Sb., 189:10 (1998), 105–134; Sb. Math., 189:10 (1998), 1533–1561

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Alexandrov, V, “Implicit function theorem for systems of polynomial equations with vanishing Jacobian and its application to flexible polyhedra and frameworks”, Monatshefte fur Mathematik, 132:4 (2001), 269  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Sabitov, IK, “The solution of polyhedra”, Doklady Mathematics, 63:2 (2001), 170  mathscinet  zmath  isi  elib
    3. S. N. Mikhalev, “Izometricheskie realizatsii oktaedrov Brikara 1-go i 2-go tipa s izvestnymi znacheniyami ob'ema”, Fundament. i prikl. matem., 8:3 (2002), 755–768  mathnet  mathscinet  zmath
    4. I. Kh. Sabitov, “Algorithmic solution of the problem of isometric realization for two-dimensional polyhedral metrics”, Izv. Math., 66:2 (2002), 377–391  mathnet  crossref  crossref  mathscinet  zmath  elib
    5. Siberian Math. J., 43:4 (2002), 661–673  mathnet  crossref  mathscinet  zmath  isi  scopus
    6. V. V. Varfolomeev, “Inscribed polygons and Heron polynomials”, Sb. Math., 194:3 (2003), 311–331  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Alexandrov V., “Flexible polyhedra in Minkowski 3-space”, Manuscripta Math., 111:3 (2003), 341–356  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. R. V. Galiulin, S. N. Mikhalev, I. Kh. Sabitov, “Some Applications of the Formula for the Volume of an Octahedron”, Math. Notes, 76:1 (2004), 25–40  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. Schlenker, JM, “The bellows conjecture”, Asterisque, 2004, no. 294, 77  mathscinet  zmath  isi
    10. Sabitov, I, “Solution of polyhedra”, Bulletin of the Brazilian Mathematical Society, 35:2 (2004), 199  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    11. V. V. Varfolomeev, “Galois groups of the Heron–Sabitov polynomials for inscribed pentagons”, Sb. Math., 195:2 (2004), 149–162  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. Fedorchuk, M, “Rigidity and polynomial invariants of convex polytopes”, Duke Mathematical Journal, 129:2 (2005), 371  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    13. Pak, I, “The area of cyclic polygons: Recent progress on Robbins' conjectures”, Advances in Applied Mathematics, 34:4 (2005), 690  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    14. S. N. Mikhalev, “A method for solving the problem of isometric realization of developments”, J. Math. Sci., 149:1 (2008), 971–995  mathnet  crossref  mathscinet  zmath  elib
    15. A. V. Timofeenko, “The non-Platonic and non-Archimedean noncomposite polyhedra”, J. Math. Sci., 162:5 (2009), 710–729  mathnet  crossref  mathscinet  zmath  elib  elib
    16. V. A. Alexandrov, “On the total mean curvature of a nonrigid surface”, Siberian Math. J., 50:5 (2009), 757–759  mathnet  crossref  mathscinet  isi  elib
    17. I. Kh. Sabitov, “Algebraic methods for solution of polyhedra”, Russian Math. Surveys, 66:3 (2011), 445–505  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. A. D. Mednykh, “Brahmagupta formula for cyclic quadrilaterals in the hyperbolic plane”, Sib. elektron. matem. izv., 9 (2012), 247–255  mathnet
    19. Gaifullin A.A., “Sabitov Polynomials for Volumes of Polyhedra in Four Dimensions”, Adv. Math., 252 (2014), 586–611  crossref  mathscinet  zmath  isi  scopus  scopus
    20. A. A. Gaifullin, “Flexible cross-polytopes in spaces of constant curvature”, Proc. Steklov Inst. Math., 286 (2014), 77–113  mathnet  crossref  crossref  isi  elib  elib
    21. A. A. Gaifullin, “Embedded flexible spherical cross-polytopes with nonconstant volumes”, Proc. Steklov Inst. Math., 288 (2015), 56–80  mathnet  crossref  crossref  isi  elib
    22. A. A. Gaifullin, “The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces”, Sb. Math., 206:11 (2015), 1564–1609  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    23. Kharazishvili A., “on Inscribed and Circumscribed Convex Polyhedra”, Proc. A Razmadze Math. Inst., 167 (2015), 123–129  mathscinet  zmath  isi
    24. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
    25. Chen Ch., Lin J., Liao M., Li G., Huang G., “Learning To Detect Salient Curves of Cartoon Images Based on Composition Rules”, 2016 11Th International Conference on Computer Science & Education (Iccse), International Conference on Computer Science & Education, IEEE, 2016, 808–813  isi
    26. D. I. Sabitov, I. Kh. Sabitov, “Kanonicheskie mnogochleny ob'ema dlya mnogogrannikov kombinatornogo tipa geksaedra”, Sib. elektron. matem. izv., 14 (2017), 1078–1087  mathnet  crossref
    27. Alexander A. Gaifullin, “The bellows conjecture for small flexible polyhedra in non-Euclidean spaces”, Mosc. Math. J., 17:2 (2017), 269–290  mathnet  mathscinet
    28. Aminov Yu.A., “Polyhedrons At the Nuclear Structure”, Probl. At. Sci. Technol., 2017, no. 3, 21–25  isi
    29. Alexander A. Gaifullin, Leonid S. Ignashchenko, “Dehn invariant and scissors congruence of flexible polyhedra”, Proc. Steklov Inst. Math., 302 (2018), 130–145  mathnet  crossref  crossref  isi  elib
    30. D. I. Sabitov, I. Kh. Sabitov, “Mnogochleny ob'ema dlya mnogogrannikov kombinatornogo tipa $n$-grannykh prizm v sluchayakh $n=5,6,7$”, Sib. elektron. matem. izv., 16 (2019), 439–448  mathnet  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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