论文标题
矩形分解成相等区域的非统一矩形
Decompositions of a rectangle into non-congruent rectangles of equal area
论文作者
论文摘要
在本文中，我们处理一个简单的几何问题：是否可以将矩形划分为$ k $相等区域的$ k $非统一矩形？这个问题是由所谓的“蒙德里亚艺术问题”引起的，该问题提出了一个类似的问题，该问题是针对整数方面的矩形。在这里，我们通过允许真实边的矩形来概括蒙德里安问题。在这种情况下，我们表明，矩形具有“完美的蒙德里安分区”（即具有非统一矩形）的最低价值为七个。此外，我们证明了这样的分区是唯一的（直到对称性），并且完全存在两个适当的完美蒙德里亚分区，用于$ k = 8 $。最后，我们还证明，任何广场都有$ k \ ge 7 $的完美蒙德里安分解。
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 $k\ge 7$.
