Tiling with Squares and Square-tileable Surfaces

We introduce thèsquare tiling group' and use it to give necessary conditions on a planar polygon to be tileable with squares. We deene square tilings on Riemann surfaces, and compute the Euclidean structures on a torus which are square-tileable. We give necessary conditions on surfaces of higher genus to be tileable. A higher-dimensional version of the square-tiling group yields necessary conditions on an n-dimensional rectilinear polyhedron to be tileable by boxes, each having k rational relations among its side lengths. We deene the rectilinear scissors congruence problem for R n related to box tilings and construct a complete invariant.