New results for the Mondrian art problem

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.