New results for the Mondrian art problem

TL;DR

This paper investigates optimal ways to dissect squares into rectangles with minimal area difference, providing bounds, constructions, and density results related to perfect and near-perfect partitions.

Contribution

It introduces new bounds on the area difference, constructs partitions with diminishing relative difference, and analyzes the distribution of side lengths lacking perfect partitions.

Findings

Bounds on the difference $d(n)$ in terms of the number of rectangles.

Existence of partitions where $d(n)/n^2$ tends to zero as $n$ grows.

Asymptotic density of side lengths without perfect partitions is $rac{( ext{log}( ext{log}(x)))^2}{2 ext{log} x}$.

Abstract

The Mondrian problem consists of dissecting a square of side length $n\in \NN$ into non-congruent rectangles with natural length sides such that the difference $d(n)$ between the largest and the smallest areas of the rectangles partitioning the square is minimum. In this paper, we compute some bounds on $d(n)$ in terms of the number of rectangles of the square partition. These bounds provide us optimal partitions for some values of $n \in \NN$. We provide a sequence of square partitions such that $d(n)/n^2$ tends to zero for $n$ large enough. For the case of `perfect' partitions, that is, with $d(n)=0$, we show that, for any fixed powers $s_1,\ldots, s_m$, a square with side length $n=p_1^{s_1}\cdots p_m^{s_m}$, can have a perfect Mondrian partition only if $p_1$ satisfies a given lower bound. Moreover, if $n(x)$ is the number of side lengths $x$ (with $n\le x$) of squares not having a perfect partition, we prove that its `density' $\frac{n(x)}{x}$ is asymptotic to $\frac{(\log(\log(x))^2}{2\log x}$, which improves previous results.