Decompositions of a rectangle into non-congruent rectangles of equal area

TL;DR

This paper investigates the geometric problem of partitioning rectangles into non-congruent rectangles of equal area, establishing minimal numbers, uniqueness, and existence conditions for such dissections.

Contribution

It generalizes the Mondrian art problem to real-sided rectangles, proving minimal k, uniqueness of partitions, and existence for squares.

Findings

Minimum k for perfect partition is 7.

Unique partition exists for k=7.

Two proper partitions exist for k=8.

Abstract

In this paper, we deal with a simple geometric problem: Is it possible to partition a rectangle into $k$ non-congruent rectangles of equal area? This problem is motivated by the so-called `Mondrian art problem' that asks a similar question for dissections with rectangles of integer sides. Here, we generalize the Mondrian problem by allowing rectangles of real sides. In this case, we show that the minimum value of $k$ for a rectangle to have a `perfect Mondrian partition' (that is, with non-congruent equal-area rectangles) is seven. Moreover, we prove that such a partition is unique (up to symmetries) and that there exist exactly two proper perfect Mondrian partitions for $k=8$. Finally, we also prove that any square has a perfect Mondrian decomposition for $k\ge 7$.