Generalization of Square Tiling Properties via Linear Algebra

TL;DR

This paper establishes an explicit upper bound on the denominators of rational side lengths in square tilings of rectangles, using an algebraic approach and total unimodularity theory.

Contribution

It introduces a purely algebraic framework for analyzing tilings, providing a bound on denominators and a concise proof of a classical tiling theorem.

Findings

Bound on denominators is 2^n, where n is the tiling order.

Algebraic encoding simplifies geometric constraints to additive relations.

Provides a short proof of Kenyon's theorem on minimum squares in a rectangle.

Abstract

While it is a classical result dating back to Dehn (1903) that squares composing a perfect rectangle must have rational side lengths, the arithmetic complexity of these tilings, specifically the growth of the denominators of these rational sizes, has remained largely unexplored. This paper addresses this gap by providing an explicit upper bound on these denominators. Departing from traditional electrical network analogies, we introduce a purely algebraic framework where the tiling's incidence geometry is encoded into a block matrix. This approach allows us to reduce geometric constraints to a finite set of additive relations and to prove, through the theory of total unimodularity of interval matrices (Ghouila-Houri, 1962), that the least common multiple of the denominators is bounded by $2^n$, where $n$ is the order of the tiling. We demonstrate the power of this result by providing a remarkably brief and self-contained proof of Richard Kenyon's 1996 theorem regarding the minimum number of squares required to tile a rectangle of integer dimensions.