《金碧輝煌的聖殿（5. perfectly squared）》The Golden Temple (5.)

金碧輝煌的聖殿（5. perfectly squared）The Golden Temple (5) 

 

本篇可以稱為“外一篇”，因為它不涉及歐幾裏得幾何的內容，跟公理、定理無關。但它的確是關於幾何圖形的，內容也相當神奇且美觀。所以它跟Morley’s trisector theorem 和John’s theorem一樣，在我家的牆壁上占有一席之地。

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，在波蘭的University of Lwoów，有兩個學習數學的學生Zbigniew Moroń和Wladyslaw Orlicz。他們在課上聽教授說起完美矩形（perfect rectangle，或稱squared rectangle)的問題。這個問題早就有人提出來，但無人能解決。

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.

問題是這樣的：是否存在整數邊長m x n的矩形，它能被分割成多個不同大小整數邊長的正方形？在這個問題中，“不同大小整數邊長”是關鍵。假如有一個2 x 3的矩形，我們當然能把它分成六個1 x 1的正方形，一點意思都沒有。

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.

兩個年輕人興致勃勃地開始研究這個問題，但過了一段時間，他們就得出結論，這恐怕和許多表麵上看似簡單的數論問題一樣，其實是極為困難的。Orlicz放棄了，他後來成了大學教授；Moron（唉，這個詞在英文裏的意思太不好了！）繼續堅持，他一輩子當中學數學老師。在苦苦探索了兩年（不算太長）後，他就有了突破。1925年，他用波蘭語發表了論文《論將矩形分解為正方形》。而且，他一下子發現了兩個這樣的長方形，如下圖所示：

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 邊長為 65x47，是10階（order，即組成它的正方形的數目）。這個矩形就是在我家牆上掛的那個；矩形B 邊長為33x32，它很像正方形，但不是。它隻有9階。Reichert 和 Toepkin 在1940 年證明，9階是完美矩形的最小階數。少於9個正方形是不可能拚成一個矩形的。也就是說，Moron在一開始發現的那個33 x 32的矩形，已經是最簡單的完美矩形。

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.

在得到一個完美矩形以後，你可以在任何一邊添加相同邊長的正方形，進而無限擴大矩形。通過交替邊繼續這個過程，加入的正方形邊長可以類似於斐波那契數列（Fibonacci sequence），這是很有意思的現象。

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.

隨後的幾年，越來越多的完美矩形被發現了（不是以上麵那種以無限交替重複方式得到）。僅一位日本老兄安倍路生（Michio Abe）就發現了600多個。於是有人就問，是否存在完美正方形（perfect square，或squared square）呢，即是否存在整數邊長的正方形，它能被分割成多個不同大小整數邊長的正方形呢？人們尋找了一段時間，沒有答案。於是有人懷疑這樣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.

在尋找的過程當中，有人提出了一個思路：如果存在任何一個 a x b 的矩形，它不僅是完美矩形，而且可以有兩種方式拚成，那麽邊為 (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.

照著這個路子，德國人Roland Sprague去構架和尋找。終於，他在1939年發現，以1885 x 2320 為邊長的矩形可以用兩種（上圖的X 和 X’）方式構成為完美矩形，這樣，一個邊長為4205（=1885 + 2320）的完美正方形就找到了，它由55個小正方形組成（圖Y）。隨後，其他人通過類似的方法，又得到了一些更簡單的完美正方形。

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， 荷蘭數學家Arie Duijvestijn證明，不存在低於 21 階的完美正方形。然而這個21階的完美正方形在哪裏呢？又過了16年，到了1978年，這個唯一的階數最小的正方形，被Duijvestijn 找到了。這是一個 112 × 112 的完美正方形（圖A）。此外，人們還找到了三個邊長更短（均為110），階數為22 和23的完美正方形。其中兩個還是由Duijvestijn貢獻的（即圖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).

有人可能會進一步想，如果擴展到3維空間會怎麽樣呢？一個邊長為整數的長方體或正方體，是否可以被邊長為整數的不同的大小的正方體完全填充呢？其實，你隻要發揮一點兒空間想象力，這個問題的答案出奇地明確！

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!

寫到這裏，這個幾何係列文章就告一段落了。日前，我突然想到，其實我自己也有一個手工幾何作品呢，我都幾乎忘記了。

這是一個木製的jigsaw puzzle，它是整整20年前，我為兒子周歲做的生日禮物。經過了這麽多年，雖然有些磨損，但色彩還是那樣的鮮亮。我不知道這個禮物對孩子後來到底產生了什麽影響。但我精心設計、做出這樣一個東西，也仿佛是給自己的一份禮物。在後來的歲月裏，我陪他走過了迥然不同於一般孩子的、跌宕起伏的人生道路。最終，他站立起來，走出家門，以他的方式去獨立麵對這個世界了。今天，看著這個jigsaw puzzle，想到在3000英裏外的那個沉默而忙碌的孩子，我心中百感交集……

 

【Edited from ChatGPT translation.】