  We introduce a family of polytope exchange transformations (PETs) acting on
parallelotopes in $\R^{2n}$ for $n=1,2,3...$. These PETs are constructed using
a pair of lattices in $\R^{2n}$. The moduli space of these PETs is $GL_n(\R)$.
We study the case n=1 in detail. In this case, we show that the 2-dimensional
family is completely renormalizable and that the $(2,4,\infty)$ hyperbolic
reflection triangle group acts (by linear fractional transformations) as the
renormalization group on the moduli space. These results have a number of
geometric corollaries for the system. Most of the paper is traditional
mathematics, but some part of the paper relies on a rigorous computer-assisted
proof involving integer calculations.
