  This article shines new light on the classical problem of tiling rectangles
with squares efficiently with a novel method. With a twist on the traditional
approach of resistor networks, we provide new and improved results on the
matter using the theory of Diophantine Approximation, hence overcoming
long-established difficulties, such as generalizations to higher-dimensional
analogues. The universality of the method is demonstrated through its
applications to different tiling problems. These include tiling rectangles with
other rectangles, with their respective higher-dimensional counterparts, as
well as tiling equilateral triangles, parallelograms, and trapezoids with
equilateral triangles.
