Deformations of period lattices of flexible polyhedral surfaces

TL;DR

This paper investigates the deformation properties of two-periodic flexible polyhedral surfaces in three-dimensional space, showing that their period lattice cannot vary with two degrees of freedom during flexion.

Contribution

It proves that the period lattice of a flexible two-periodic surface homeomorphic to a plane has at most one degree of freedom, advancing understanding of flexible polyhedral structures.

Findings

The period lattice cannot have two degrees of freedom during flexion.

Flexible two-periodic surfaces are constrained in their lattice deformations.

The result extends the theory of flexible polyhedra to periodic 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.