Inductive constructions for frameworks on a twodimensional fixed torus
Abstract
An infinite periodic framework in the plane can be represented as a framework on a torus, using a $\mathbb Z^2$labelled gain graph. We find necessary and sufficient conditions for the generic minimal rigidity of frameworks on the twodimensional fixed torus $\mathcal T_0^2$. It is also shown that every minimally rigid periodic orbit framework on $\mathcal T_0^2$ can be constructed from smaller frameworks through a series of inductive constructions. These are fixed torus adapted versions of the results of Laman and Henneberg respectively for finite frameworks in the plane. The proofs involve the development of inductive constructions for $\mathbb Z^2$labelled graphs.
 Publication:

arXiv eprints
 Pub Date:
 March 2012
 arXiv:
 arXiv:1203.6561
 Bibcode:
 2012arXiv1203.6561R
 Keywords:

 Mathematics  Metric Geometry;
 52C25
 EPrint:
 41 pages, 17 figures