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.
\Bibitem{Sab98}
\by I.~Kh.~Sabitov
\paper A generalized Heron--Tartaglia formula and some of its consequences
\jour Sb. Math.
\yr 1998
\vol 189
\issue 10
\pages 1533--1561
\mathnet{http://mi.mathnet.ru/eng/sm354}
\crossref{https://doi.org/10.1070/sm1998v189n10ABEH000354}
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This publication is cited in the following 37 articles:
S. N. Mikhalev, “A metric description of flexible octahedra”, Sb. Math., 214:7 (2023), 952–981
Nikolay Abrosimov, Alexander Kolpakov, Alexander Mednykh, “Euclidean volumes of hyperbolic knots”, Proc. Amer. Math. Soc., 2023
Sabitov I.Kh., Stepanov D.A., “Elimination of Parasitic Solutions in the Theory of Flexible Polyhedra”, Beitr. Algebr. Geom., 62:3 (2021), 687–703
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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
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I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175
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Gaifullin A.A., “Sabitov Polynomials for Volumes of Polyhedra in Four Dimensions”, Adv. Math., 252 (2014), 586–611
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Nikolay Abrosimov, Alexander Mednykh, Fields Institute Communications, 70, Rigidity and Symmetry, 2014, 1
A. D. Mednykh, “Brahmagupta formula for cyclic quadrilaterals in the hyperbolic plane”, Sib. elektron. matem. izv., 9 (2012), 247–255
I. Kh. Sabitov, “Algebraic methods for solution of polyhedra”, Russian Math. Surveys, 66:3 (2011), 445–505
P. Bibikov, “On algebraic dependencies in polygons and polyhedra”, Lobachevskii J Math, 32:2 (2011), 146
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