  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
two-dimensional 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.
