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Discrete Comput. Geom., 2014, Volume 51, Issue 3, Pages 650–665 (Mi dcg2)  

Deformations of period lattices of flexible polyhedral surfaces

A. A. Gaifullinabcd, S. A. Gaifullince

a Demidov Yaroslavl State University, Yaroslavl, Russia
b Steklov Mathematical Institute, Moscow, Russia
c Lomonosov Moscow State University, Moscow, Russia
d Kharkevich Institute for Information Transmission Problems, Moscow, Russia
e Higher School of Economics, Moscow, Russia

Abstract: At the end of the 19th century Bricard discovered the phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit nontrivial 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 $\mathbb{R}^3$, doubly periodic with respect to translations by two non-collinear vectors, that can vary continuously during the flexion. The main result is that the period lattice of a flexible doubly periodic surface that is homeomorphic to the plane cannot have two degrees of freedom.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-31444
13-01-12469
12-01-31342
12-01-00704
Ministry of Education and Science of the Russian Federation MD-4458.2012.1
11.G34.31.0053
8214
Russian Academy of Sciences - Federal Agency for Scientific Organizations
Dynasty Foundation
The first author was partially supported by the Russian Foundation for Basic Research (projects 12-01-31444 and 13-01-12469), by a grant of the President of the Russian Federation (project MD-4458.2012.1), by a grant of the Government of the Russian Federation (project 11.G34.31.0053), by a program of the Branch of Mathematical Sciences of the Russian Academy of Sciences, and by a grant from Dmitry Zimin’s “Dynasty” foundation. The second author was partially supported by the Russian Foundation for Basic Research (projects 12-01-31342 and 12-01-00704) and by the Ministry of Education and Science of the Russian Federation (project 8214).


DOI: https://doi.org/10.1007/s00454-014-9575-8


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Received: 02.06.2013
Revised: 13.01.2014
Accepted:18.01.2014
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