In the 1890's H. Minkowski identified the 2 by 2 matrices whose image of the unit square tiles the plane under integer translations. He conjectured a condition for all dimensions. This was proved to be true in 1941. These matrices define maps of the torus which are continuous on all but a small set. The discontinuities lead to unexpected behavior in the dynamics. I will describe the matrices and look at some examples.
The simplest case displays remarkable complexity and I will show some pictures which demonstrate this.