An Algebraic Approach to the Arithmetic of Square Tilings
It is a classical result dating back to Dehn (1903) that squares tiling a rectangle must have rational side lengths. However, bounding the arithmetic complexity of these tilings, specifically the growth of their denominators, is an active area of research. Recent contributions have tackled these arithmetic constraints through the complexity of planar multigraphs and Diophantine approximations. In this paper, we offer an algebraic perspective of this problem by encoding the tiling's geometry into a block matrix. This approach allows us to reduce geometric constraints to a finite set of additive relations. By exploiting the total unimodularity of interval matrices (Ghouila-Houri, 1962), we provide a self-contained proof that the least common multiple of the denominators is bounded by $2^n$ in two dimensions, where $n$ is the order of the tiling. Furthermore, because we can naturally generalizes this matrix to represent $d$-dimensional hypercube tilings, yielding a bound of $d^n$. We demonstrate the utility of this framework by providing a brief proof of Richard Kenyon's 1996 theorem regarding the minimum number of squares required to tile an integer rectangle.