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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 3(399), Pages 3–66 (Mi umn9396)  

This article is cited in 17 scientific papers (total in 17 papers)

Algebraic methods for solution of polyhedra

I. Kh. Sabitov

M. V. Lomonosov Moscow State University

Abstract: By analogy with the solution of triangles, the solution of polyhedra means a theory and methods for calculating some geometric parameters of polyhedra in terms of other parameters of them. The main content of this paper is a survey of results on calculating the volumes of polyhedra in terms of their metrics and combinatorial structures. It turns out that a far-reaching generalization of Heron's formula for the area of a triangle to the volumes of polyhedra is possible, and it underlies the proof of the conjecture that the volume of a deformed flexible polyhedron remains constant.
Bibliography: 110 titles.

Keywords: polyhedra, combinatorial structure, metric, volume, bending, bellows conjecture, volume polynomials, generalization of Heron's formula.


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English version:
Russian Mathematical Surveys, 2011, 66:3, 445–505

Bibliographic databases:

UDC: 514.772.35
MSC: Primary 51M20, 52C25; Secondary 51M10, 52B11
Received: 08.07.2010

Citation: I. Kh. Sabitov, “Algebraic methods for solution of polyhedra”, Uspekhi Mat. Nauk, 66:3(399) (2011), 3–66; Russian Math. Surveys, 66:3 (2011), 445–505

Citation in format AMSBIB
\by I.~Kh.~Sabitov
\paper Algebraic methods for solution of polyhedra
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 3(399)
\pages 3--66
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 3
\pages 445--505

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    This publication is cited in the following articles:
    1. Alexandrov V., “Necessary Conditions For the Extendibility of a First-Order Flex of a Polyhedron to Its Flex”, Beitr. Algebr. Geom.  crossref  isi
    2. Sabitov I.Kh., Stepanov D.A., “Elimination of Parasitic Solutions in the Theory of Flexible Polyhedra”, Beitr. Algebr. Geom.  crossref  isi
    3. Oscar Miguel Rivera-Borroto, José Manuel García-de la Vega, Yoandy Hernández-Díaz, “Theoretical advances on coefficients of relational agreement: application to cheminformatics ask-way biomolecular similarity measures”, J. Chemometrics, 27:11 (2013), 420–430  crossref  isi  scopus
    4. I. Kh. Sabitov, “Giperbolicheskii tetraedr: vychislenie ob'ema s primeneniem k dokazatelstvu formuly Shlefli”, Model. i analiz inform. sistem, 20:6 (2013), 149–161  mathnet
    5. A. A. Gaifullin, “polynomials for volumes of polyhedra in four dimensions”, Adv. Math., 252 (2014), 586–611  crossref  mathscinet  zmath  isi  elib  scopus
    6. Abrosimov N.V., Makai Jr. E., Mednykh A.D., Nikonorov Yu.G., Rote G., “The Infimum of the Volumes of Convex Polytopes of Any Given Facet Areas Is 0”, Stud. Sci. Math. Hung., 51:4 (2014), 466–519  crossref  mathscinet  zmath  isi  elib  scopus
    7. Alexandrov V., “Continuous Deformations of Polyhedra That Do Not Alter the Dihedral Angles”, Geod. Dedic., 170:1 (2014), 335–345  crossref  mathscinet  zmath  isi  scopus
    8. R. E. Schwartz, “Lengthening a tetrahedron”, Geom. Dedicata, 174:1 (2015), 121–144  crossref  mathscinet  zmath  isi  elib  scopus
    9. M. I. Shtogrin, “On flexible polyhedral surfaces”, Proc. Steklov Inst. Math., 288 (2015), 153–164  mathnet  crossref  crossref  isi  elib
    10. A. A. Gaifullin, “Embedded flexible spherical cross-polytopes with nonconstant volumes”, Proc. Steklov Inst. Math., 288 (2015), 56–80  mathnet  crossref  crossref  isi  elib
    11. V. A. Alexandrov, “The set of nondegenerate flexible polyhedra of a prescribed combinatorial structure is not always algebraic”, Siberian Math. J., 56:4 (2015), 569–574  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. 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
    13. 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
    14. 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
    15. Alexandrov V., “How Many Times Can the Volume of a Convex Polyhedron Be Increased By Isometric Deformations?”, Beitr. Algebr. Geom., 58:3 (2017), 549–554  crossref  mathscinet  zmath  isi
    16. 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
    17. Aomoto K., Machida Y., “Generalization of Schlafli Formula to the Volume of a Spherically Faced Simplex”, J. Math. Soc. Jpn., 72:1 (2020), 213–249  crossref  isi
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