Deformations of period lattices of flexible polyhedral surfaces

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 the 3-space 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.

12 pages, 5 figures

References in corpus (3)

• Generic combinatorial rigidity of periodic frameworks
• Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions
• Sabitov polynomials for volumes of polyhedra in four dimensions

Cited by in corpus (1)

• Domes over curves