金碧辉煌的圣殿（5. perfectly squared）The Golden Temple (5) 

本篇可以称为“外一篇”，因为它不涉及欧几里得几何的内容，跟公理、定理无关。但它的确是关于几何图形的，内容也相当神奇且美观。所以它跟Morley’s trisector theorem 和John’s theorem一样，在我家的墙壁上占有一席之地。

This article can be called an "extra piece" because it does not involve the content of Euclidean geometry and is unrelated to axioms and theorems. However, it is indeed about geometric shapes, and its content is quite magical and beautiful. So, like Morley's trisector theorem and John's theorem, it has a place on the walls of my house.

整整一百年前的1923年，在波兰的University of Lwoów，有两个学习数学的学生Zbigniew Moroń和Wladyslaw Orlicz。他们在课上听教授说起完美矩形（perfect rectangle，或称squared rectangle)的问题。这个问题早就有人提出来，但无人能解决。

Exactly one hundred years ago, in 1923, at the University of Lwoów in Poland, two mathematics students, Zbigniew Moroń and Wladyslaw Orlicz, heard their professor mention the problem of perfect rectangles (or squared rectangles) in class. This problem had been posed before, but no one could solve it.

问题是这样的：是否存在整数边长m x n的矩形，它能被分割成多个不同大小整数边长的正方形？在这个问题中，“不同大小整数边长”是关键。假如有一个2 x 3的矩形，我们当然能把它分成六个1 x 1的正方形，一点意思都没有。

The problem is as follows: Does there exist a rectangular shape with integer side lengths m x n that can be divided into multiple squares of different sizes with integer side lengths? In this problem, "different sizes of integer side lengths" are crucial. If there is a 2 x 3 rectangle, for example, we can certainly divide it into six 1 x 1 squares, but that is not interesting at all.

两个年轻人兴致勃勃地开始研究这个问题，但过了一段时间，他们就得出结论，这恐怕和许多表面上看似简单的数论问题一样，其实是极为困难的。Orlicz放弃了，他后来成了大学教授；Moron（唉，这个词在英文里的意思太不好了！）继续坚持，他一辈子当中学数学老师。在苦苦探索了两年（不算太长）后，他就有了突破。1925年，他用波兰语发表了论文《论将矩形分解为正方形》。而且，他一下子发现了两个这样的长方形，如下图所示：

The two young men enthusiastically began to study this problem, but after a while, they concluded that it was probably as difficult as many seemingly simple number theory problems. Orlicz gave up, later becoming a university professor; Moroń (oh, the English meaning of this word is so bad!) persisted. He is a lifelong middle school math teacher. After two years of arduous exploration (not too long), he had a breakthrough. In 1925, he published a paper in Polish titled "On the Decomposition of Rectangles into Squares." Moreover, he discovered two rectangles at once, as shown in the figure below:

矩形A 边长为 65x47，是10阶（order，即组成它的正方形的数目）。这个矩形就是在我家墙上挂的那个；矩形B 边长为33x32，它很像正方形，但不是。它只有9阶。Reichert 和 Toepkin 在1940 年证明，9阶是完美矩形的最小阶数。少于9个正方形是不可能拼成一个矩形的。也就是说，Moron在一开始发现的那个33 x 32的矩形，已经是最简单的完美矩形。

Rectangle A has side lengths of 65x47 with 10 orders (i.e., the number of squares that make it up). This rectangle is the one hanging on the wall in my house; Rectangle B has side lengths of 33x32, resembling a square but not one. It has only 9 orders. Reichert and Toepkin proved in 1940 that 9 orders are the minimum order for a perfect rectangle. It is impossible to compose a rectangle with fewer than 9 squares. In other words, the 33 x 32 rectangle that Moroń initially discovered is already the simplest perfect rectangle.

在得到一个完美矩形以后，你可以在任何一边添加相同边长的正方形，进而无限扩大矩形。通过交替边继续这个过程，加入的正方形边长可以类似于斐波那契数列（Fibonacci sequence），这是很有意思的现象。

After obtaining a perfect rectangle, you can add squares of the same side length on any side, thus infinitely expanding the rectangle. By alternating sides, the side lengths of the added squares can resemble the Fibonacci sequence, which is very interesting.

随后的几年，越来越多的完美矩形被发现了（不是以上面那种以无限交替重复方式得到）。仅一位日本老兄安倍路生（Michio Abe）就发现了600多个。于是有人就问，是否存在完美正方形（perfect square，或squared square）呢，即是否存在整数边长的正方形，它能被分割成多个不同大小整数边长的正方形呢？人们寻找了一段时间，没有答案。于是有人怀疑这样的正方形是不存在的。

In the following years, more and more perfect rectangles were discovered (not obtained by infinite alternating repetition as described above). Only one Japanese, Michio Abe, discovered more than 600. So someone asked, does a perfect square exist? That is, does a square with integer side lengths exist, which can be divided into squares of different sizes with integer side lengths? People searched for a while with no answer. Some people doubted the existence of such a square.

在寻找的过程当中，有人提出了一个思路：如果存在任何一个 a x b 的矩形，它不仅是完美矩形，而且可以有两种方式拼成，那么边为 (a+b) x (a+b) 的正方形就一定是完美正方形。

In the process of searching, people proposed an idea: If there is any a x b rectangle that is not only a perfect rectangle but can also be composed in two ways, then a square with side length (a+b) x (a+b) must be a perfect square.

照着这个路子，德国人Roland Sprague去构架和寻找。终于，他在1939年发现，以1885 x 2320 为边长的矩形可以用两种（上图的X 和 X’）方式构成为完美矩形，这样，一个边长为4205（=1885 + 2320）的完美正方形就找到了，它由55个小正方形组成（图Y）。随后，其他人通过类似的方法，又得到了一些更简单的完美正方形。

Following this path, the German Roland Sprague constructed and searched. Finally, in 1939, he found that a rectangle with side lengths of 1885 x 2320 could be composed in two ways (shown in the figure as X and X'). Thus, a perfect square with side length 4205 (=1885 + 2320) with 55 orders was found (figure Y). Subsequently, others, using similar methods, found simpler perfect squares.

到了1962年， 荷兰数学家Arie Duijvestijn证明，不存在低于 21 阶的完美正方形。然而这个21阶的完美正方形在哪里呢？又过了16年，到了1978年，这个唯一的阶数最小的正方形，被Duijvestijn 找到了。这是一个 112 × 112 的完美正方形（图A）。此外，人们还找到了三个边长更短（均为110），阶数为22 和23的完美正方形。其中两个还是由Duijvestijn贡献的（即图B和D）。

By 1962, Dutch mathematician Arie Duijvestijn proved that there is no perfect square below 21orders. However, where is this 21-order perfect square? Sixteen years later, in 1978, this unique and smallest-order square was found by Duijvestijn. It is a 112 x 112 perfect square (figure A). In addition, people found three squares with shorter side lengths (all 110), with 22 and 23 orders. Two of them were also contributed by Duijvestijn (namely figures B and D).

有人可能会进一步想，如果扩展到3维空间会怎么样呢？一个边长为整数的长方体或正方体，是否可以被边长为整数的不同的大小的正方体完全填充呢？其实，你只要发挥一点儿空间想象力，这个问题的答案出奇地明确！

Some may further wonder, what if we extend this to three dimensions? Is it possible a rectangular prism or cube with integer side lengths is completely filled with cubes of different sizes with integer side lengths? In fact, if you use a little spatial imagination, the answer to this question is surprisingly clear!

写到这里，这个几何系列文章就告一段落了。日前，我突然想到，其实我自己也有一个手工几何作品呢，我都几乎忘记了。

这是一个木制的jigsaw puzzle，它是整整20年前，我为儿子周岁做的生日礼物。经过了这么多年，虽然有些磨损，但色彩还是那样的鲜亮。我不知道这个礼物对孩子后来到底产生了什么影响。但我精心设计、做出这样一个东西，也仿佛是给自己的一份礼物。在后来的岁月里，我陪他走过了迥然不同于一般孩子的、跌宕起伏的人生道路。最终，他站立起来，走出家门，以他的方式去独立面对这个世界了。今天，看着这个jigsaw puzzle，想到在3000英里外的那个沉默而忙碌的孩子，我心中百感交集……

【Edited from ChatGPT translation.】