Deformations of Period Lattices of Flexible Polyhedral Surfaces

Abstract

Abstract. In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of Sabitov asserting that the volume of a flexible polyhedron is constant during the flexion. In this paper we study flexible polyhedral surfaces in R 3 two-periodic with respect to translations by two non-colinear vectors that can vary continuously during the flexion. The main result is that the period lattice of a flexible two-periodic surface homeomorphic to a plane cannot have two degrees of freedom.

Domes over curves

“…Theorem 4.3 (Gaifullin-Gaifullin [12]). Let K be a simplicial pure 2-dimensional complex homeomorphic to R 2 with a free action of the group Z ⊕ Z.…”

“…Remark 4.4. In fact, the result in [12] is more general, as the authors consider all polygonal doubly periodic surfaces homeomorphic to the plane with arbitrary sets of side lengths. In this setting, the coefficients of polynomials p and q are obtained from the ideal generated by squares of all side lengths of polygons in the polygonal surface Proof of Proposition 4.1.…”

“…In Section 4, we prove Theorem 1.2 by extending the results of Gaifullin brothers [12]. We assume that the reader is familiar with the theory of places, see e.g.…”

“…In the Appendix A, we include a negative solution of the question in [12] on the dimension of the flexes of doubly periodic surfaces. This counterexample arose upon careful inspection of our proof of Theorem 1.2, and is of independent interest (cf.…”

Volume of a simplex as a multivalued algebraic function of the areas of its two-faces

“…Places of fields were first used in the metric theory of polyhedra by Connelly, Sabitov, and Walz [2] to obtain another proof of Sabitov's theorem on the volume. Then they were used by the author [3], [4] to generalize Sabitov's theorem to higher dimensions, and by S. A. Gaifullin and the author [5] to study flexible periodic polyhedral surfaces. In our situation the language of valuations turns out to be more convenient than the language of places.…”