Flexible polyhedra: Constructions, Volume, Scissors Congruence

Flexible polyhedra are polyhedral surfaces with rigid faces and hinges at edges that admit non-trivial deformations, that is, deformations not induced by ambient isometries of the space. Main steps in theory of flexible polyhedra are: Bricard’s construction of self-intersecting flexible octahedra (1897), Connelly’s construction of flexors, i.e., non-self-intersecting flexible polyhedra (1977), and Sabitov’s proof of the Bellows conjecture claiming that the volume of any flexible polyhedron remains constant during the flexion (1996). In my talk I will give a survey of these classical results and ideas behind them, as well as of several more recent results by the speaker, including the proof of Strong Bellows conjecture claiming that any flexible polyhedron in Euclidean three-space remains scissors congruent to itself during the flexion (joint with Leonid Ignashchenko, 2017)