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Sufficient conditions for the extendibility of an \(n\)-th order flex of polyhedra. (English) Zbl 0916.52005
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 of these conditions is fulfilled for the Bricard octahedra. From the historical point of view flexible polyhedra have become connected to the attempts of solution to the open problem to prove that a smooth compact surface in the Euclidean 3-space is rigid. For further details see I. Ivanova-Karatopraklieva and I. Kh. Sabitov [J. Math. Sci., New York 70, No. 2, 1685-1716 (1994); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 23, 131-184 (1991; Zbl 0835.53003)] and I. Kh. Sabitov [Encycl. Math. Sci. 48, 179-250 (1992); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 196-270 (1989; Zbl 0781.53008)].

52B70 Polyhedral manifolds
53A05 Surfaces in Euclidean and related spaces
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