Bending Fields, Rigidity, and the Bellows Conjecture
نویسنده
چکیده
Aleksandrov and Pogorelov used bending fields (the velocity fields of continuous isometric deformations) to show infinitesimal rigidity of some classes of surfaces ([2] and [12]). Bending fields (and the rigidity matrix) were also successfully employed in the study of infinitesimal rigidity of polyhedra and tensegrity frameworks (see [7]). We will use these tools to re-derive a previously known result that almost all spherical polyhedra are rigid, and to finding the rate of change in volume of a flexible polyhedron during a flex. The latter leads us to two claims whose validity would imply the bellows conjecture. The first involves the bending vectors at the polyhedron’s vertices, and is very elementary (see claim 4.3). The second involves a linear system whose matrix is the rigidity matrix (see claim 4.7). This seems to be a rather natural approach to proving the bellows conjecture that the volume of a polyhedron is constant under flexing.
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