Sufficient conditions for the extendibility of an \(n\)-th order flex of polyhedra
type: Item
instance of: scholarly article
title: Sufficient conditions for the extendibility of an \(n\)-th order flex of polyhedra
zbMATH Open document ID: 0916.52005
author: Victor Alexandrov
published in: Beiträge zur Algebra und Geometrie
publication date: 1999-07-07T00:00:00Z
full work available at URL: https://eudml.org/doc/230087
full work available at URL: no.2
review text: A closed polyhedron \(Q\) in the Euclidean 3-space is called flexible if there exists an analytic (with respect to a parameter \(t\in \left[ 0,1\right])\) family of polyhedra \(Q_{t}\), such that i) \(Q\) coincides with \(Q_{0}\); ii) for each \(t\in \left[ 0,1\right] \), the polyhedron Q\(_{t}\) is isometric to the polyhedron \(Q_{0}\) in the intrinsic metrics; iii) the polyhedron Q\(_{1}\) cannot be obtained from Q\(_{0}\) by a rigid motion. The family of polyhedra Q\(_{t}\) \ is called an authentic flex. Let \(Q\) be a closed flexible polyhedra and let us suppose all its faces are triangular. Let us suppose \(Q\) has \(v\) vertices \((x_{i},y_{i},z_{i})\) \(1\leq i\leq v\), and \(e\) edges. If vertices \((x_{i},y_{i},z_{i})\) and \((x_{j},y_{j},z_{j})\) are joined by an edge of \(Q\) then a number \(e(i,j)\), \(1\leq e(i,j)\leq e\) is assigned to this edge. Let \(B:R^{3v}\times R^{3v}\rightarrow R^{e}\) be a bilinear form, such that \(B\) maps the vectors \(X=(x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},\dots ,x_{v},y_{v},z_{v})\) and \(U=(u_{1},v_{1},w_{1},u_{2},v_{2},w_{2},\dots ,u_{v},v_{v},w_{v})\) into a vector whose \(e(i,j)\)-th component is equal to \((x_{i}-x_{j})(u_{i}-u_{j})+(y_{i}-y_{j})(v_{i}-v_{j})+(z_{i}-z_{j})(w_{i}- w_{j})\). If for \(n\geq 1\) all vertices \(i\) and \(j\) are joined by an edge we can write \(\sum\limits_{m=0}^{n}B(X_{m},X_{n-m})=0\), where \(X_{k}=(x_{1,k},y_{1,k},z_{1,k},x_{2,k},y_{2,k},z_{2,k},\dots ,x_{v,k},y_{v,= k},z_{v ,k})\). The vector \(X_{0}+tX_{1}+\dots +t^{k}X_{k}\) is called the \(k\)-th order infinitesimal flex of the polyhedron \(X_0\). Making use of this bilinear map the author reformulates the problem of finding out if a polyhedron is flexible to an algebraic problem, and proves some linear algebra statements that will help to prove the main result: give some sufficient conditions under which an infinitesimal flex of a polyhedron can be extended to an authentic flex. He also gives computer results that prove that a set …
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 1208011
zbMATH Keywords: flex polyhedra
zbMATH Keywords: bilinear game
zbMATH Keywords: Bricard octahedron
zbMATH Keywords: smooth compact surfaces
reviewed by: Ana Pereira do Vale
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1208011