Tiling of rectangles with squares via Diophantine approximation

TL;DR

This paper introduces a novel Diophantine approximation-based method for efficiently tiling various polygons with squares and other shapes, overcoming classical difficulties and extending to higher dimensions.

Contribution

It presents a new universal approach to tiling problems using Diophantine approximation, applicable to multiple geometric shapes and higher-dimensional analogues.

Findings

Improved results on tiling rectangles with squares.

Extension of tiling methods to higher-dimensional shapes.

Application to tiling triangles, parallelograms, and trapezoids.

Abstract

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.