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Russian Mathematical Surveys, 2011, Volume 66, Issue 3, Pages 445–505
DOI: https://doi.org/10.1070/RM2011v066n03ABEH004748 (Mi rm9396) 		 
This article is cited in 24 scientific papers (total in 24 papers)

Algebraic methods for solution of polyhedra

I. Kh. Sabitov

M. V. Lomonosov Moscow State University
English version PDF (955 kB) Citations (24)
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DOI: https://doi.org/10.1070/RM2011v066n03ABEH004748
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.
Received: 08.07.2010
Russian version:
Uspekhi Matematicheskikh Nauk, 2011, Volume 66, Issue 3(399), Pages 3–66
DOI: https://doi.org/10.4213/rm9396
Bibliographic databases:
Document Type: Article
UDC: 514.772.35
MSC: Primary 51M20, 52C25; Secondary 51M10, 52B11
Language: English
Original paper language: Russian
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
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  • https://www.mathnet.ru/eng/rm/v66/i3/p3
    • This publication is cited in the following 24 articles:
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    References:121
    First page:87
     
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