A simpler construction of volume polynomials for a polyhedron
type: Item
instance of: scholarly article
title: A simpler construction of volume polynomials for a polyhedron
zbMATH Open document ID: 1043.52007
author: Seiya Negami
author: Serge Lawrencenko
author: I. Kh Sabitov
published in: Beiträge zur Algebra und Geometrie
publication date: 2002-08-20T00:00:00Z
full work available at URL: https://eudml.org/doc/226197
full work available at URL: 20.html
review text: This article builds on earlier work to give a simple constructive proof of the existence of so-called ``volume polynomials'' for (triangulated) spherical polyhedra. For a polyhedron with a given combinatorial structure, a volume polynomial is a polynomial in \(V^2\) whose coefficients are fixed functions of the edge-lengths of the polyhedron, which has as a root the volume \(V\) of the polyhedron. An example is Franscesca's formula \[ \begin{multlined} 144V^2=-abc-ab'c'-a'bc'-a'b'c+(a+a'+b+b'-c-c')\\ cc'+(a+a'-b-b'+c+c')bb'+(-a- a'+b+b'+c+c')aa'\end{multlined} \] for the volume of the tetrahedron where \((a,a')\), \((b,b')\) and \((c,c')\) are pairs of squares of lengths of opposite edges. It was shown by \textit{I. Kh. Sabitov} [Fundam. Prikl. Mat. 2, 1235--1246 (1996; Zbl 0904.52002), Discrete Comput. Geom. 20, 405--425 (1998; Zbl 0922.52006)], that volume polynomials exist for all spherical triangulations. In the article under review, a simplification of the earlier proof is given, by proving and then using the fact that every triangulation of the 2-sphere contains a vertex which is not incident with any non-facial 3-cycle of edges. The final section of the article examines the case of triangulated toroids and shows under what circumstances they may be cut to yield triangulated spheres with properties the earlier sections show to be important in the search for low-degree volume polynomials.
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 51M25
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57Q15
zbMATH DE Number: 1784607
zbMATH Keywords: volume polynomial
zbMATH Keywords: triangulation
zbMATH Keywords: 2-sphere
zbMATH Keywords: torus
reviewed by: Michael I. Hartley
MaRDI profile type: Publication
Recommended article: Q5310600
Recommended article: Volume polynomial for polyhedra of hexaedral type
Recommended article: Some applications of a formula for the volume of a polyhedron
Recommended article: A generalized Heron-Tartaglia formula and some of its consequences
Recommended article: Q4396594
Recommended article: Some applications of the formula for the volume of an octahedron
Recommended article: Q3365083
Recommended article: Generalization of Sabitov's theorem to polyhedra of arbitrary dimensions
Recommended article: A canonical polynomial for the volume of a polyhedron
Recommended article: SPHERICAL SIMPLICES AND THEIR POLARS
description: scientific article; zbMATH DE number 1784607