《金碧輝煌的聖殿 （2. Morley’s trisector theorem）》The Glorious Temple of

金碧輝煌的聖殿 （2. Morley’s trisector theorem）

The Glorious Temple of Geometry 2. Morley’s trisector theorem

在我家的牆上，掛有兩類圖畫，第一類是我拍攝的自然、風景照片，第二類並非什麽李杜詩詞之類，而是幾幅相當優美（我以為）的幾何圖案。這幾個幾何定理不僅優美，而且是最近一百年左右“新近”被發現和證明的。比如下麵鏡框中的圖案叫“Morley’s trisector theorem”，它被無數人忽視了2000多年，直到1899年才被數學家Frank Morley 揭示。我們後麵細談。

On the walls of my home, there are two types of paintings. The first type consists of nature and landscape photographs that I've taken. The second type, rather than featuring poems or verses like those of Li Bai and Du Fu, comprises several quite beautiful (in my opinion) geometric patterns. These geometric theorems are not only aesthetically pleasing but have also been discovered and proven in the past hundred years or so. For example, the pattern in the frame below is called "Morley's trisector theorem." It remained overlooked by countless individuals for over 2000 years until mathematician Frank Morley revealed it in 1899. We'll delve into this further.

自從《幾何原本》問世以來，古希臘數學家的聰明才智展現在世人的麵前。隨後，世界各民族、各文化的數學家又不斷給這座大廈添磚加瓦。可以說，現在，歐式幾何已經非常全麵、完備了。一些困擾人們千年的難解問題也得到了明確的答案。我們可以用“尺規作圖”作為例子來說明一下：

Ever since the publication of "The Elements" in which the brilliance of ancient Greek mathematicians have showcased to the world, mathematicians from various nations and cultures have continued to contribute to this edifice. It can be said that Euclidean geometry is now very comprehensive and complete. Some millennium-old perplexing problems have also received clear answers. We can take the "compass-and-straightedge construction" as an example:

尺規作圖（Compass-and-straightedge construction或 ruler-and-compass construction）是起源於古希臘、與歐式幾何密切相關的作圖法。該法使用圓規（無角度，但可無限寬）和直尺（無刻度，但可無限長），且隻準許使用有限次，來解決幾何作圖問題。千百年來，人們使用尺規作圖的原則，實現了各種簡單或複雜的操作，比如下圖中A、B、C分別是用尺規做線段的垂直平分線，角的平分線和正六邊形，這些都非常簡單。而圖D是用尺規做正17邊形，極其複雜，多達51步。但這個難得不可想象的做圖，被德國著名數學家高斯在他大學二年級的時候攻克了。

Compass-and-straightedge construction is a graphing method originating from ancient Greece closely related to Euclidean geometry. This method uses a compass (angle-free but can be infinitely wide) and a straightedge (without markings but can be infinitely long) and allows only a finite number of uses to solve geometric construction problems. For centuries, people have employed the principles of compass-and-straightedge construction to achieve various simple or complex operations. For instance, in the figure below, A, B, and C are constructed using compass and straightedge to represent the perpendicular bisector, angle trisector, and hexagon, respectively. These are relatively simple constructions. However, figure D, representing a regular 17-gon, is extremely complex, requiring as many as 51 steps. This seemingly unimaginable construction was conquered by the renowned German mathematician Gauss during his second year at university.

然而，有些看似簡單的問題，人們無論怎樣努力，也無法用尺規作圖來解決。比如用尺規作圖法做正七邊形，以及著名的尺規作圖的“古希臘三大難題”——

** 化圓為方問題: 求一個正方形的邊長，使其麵積與一已知圓的相等 【Squaring a circle (constructing a square with the same area as a given circle)】

** 三等分角問題: 求一角，使其角度是一已知角度的三分之一 【Trisecting an angle (dividing a given angle into three equal angles)】

** 倍立方問題: 求一立方體的棱長，使其體積是一已知立方體的二倍【Doubling a cube (constructing a cube with twice the volume of a given cube)】

 

Nevertheless, despite considerable effort, some seemingly simple problems cannot be solved using compass-and-straightedge construction. Examples include constructing a regular heptagon and the famous "Three Classical Problems " in compass-and-straightedge construction:

Squaring a circle: Constructing a square with an area equal to that of a given circle.

Trisecting an angle: Dividing a given angle into three equal angles.

Doubling a cube: Constructing a cube with twice the volume of a given cube.

這些問題，無論人們怎樣努力，總是無法解決，又無法在歐氏幾何的範圍內證偽。兩千多年過後，數學的其他分支發展到了新的高度。於是，數學家們便使用新的數學公具，證明了正七邊形和“古希臘三大難題”，用尺規作圖是不可能解決的。就好比說飛機無論飛得多快，也不可能飛到月球上去。上述這些工作，在十九世紀上半葉，也就是距今200年左右，就已經完成了。

No matter how hard people tried, these problems could not be solved using compass-and-straightedge construction, and they could not be disproven within the scope of Euclidean geometry either. Over two thousand years later, other branches of mathematics reached new heights. Consequently, mathematicians used new mathematical tools to prove that constructing a regular heptagon and the "Three Classical Problems of Antiquity" were impossible with compass-and-straightedge construction. It's akin to saying that no matter how fast an airplane flies, it cannot reach the moon. This work was completed in the first half of the 19th century, around 200 years ago.

那麽，是不是可以說，從那以後，初等幾何之中能夠發現的規律，早已被發現和解決了呢？如果把歐氏幾何比作一座金礦，經過2400年的開采，一般人似乎以為，金子早已經被開采完了。可是，歐氏幾何的實際情況卻不是這樣的。即便是在100年前，獨具慧眼的人還是能拾到金塊，甚至是閃亮的“大金塊”。Morley’s trisector theorem正是這樣一個發現。

So, can we say that since then, all the patterns discoverable in elementary geometry have already been found and resolved? If we liken Euclidean geometry to a gold mine, it might seem that, after 2400 years of mining, people believe all the gold has been extracted. However, the actual situation of Euclidean geometry is not like that. Even a hundred years ago, astute individuals could still find nuggets of gold, even shiny "big nuggets." Morley's trisector theorem is one such discovery.

這個定理的表述極其簡單（重要的話說三遍：複雜了就不美了）：對任意一個三角形，作內角三等分線，靠近公共邊三等分線的三個交點，總是連成一個等邊三角形。[ In any triangle, the three points of intersection of the adjancent angle trisectors form an equilateral trangle.] 這個簡單而優美的規律被人們忽視了2000多年，直到1899年被英裔美國數學家Frank Morley (1860 - 1937) 發現並證明。下麵的鏈接顯示其動態過程，頗有意思 ——

The statement of this theorem is extremely simple (it's worth emphasizing three times: “The core of beauty is simplicity” as Paulo Coehlo stated): In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. This simple and elegant rule was overlooked for over 2000 years until it was discovered and proven by British-American mathematician Frank Morley (1860-1937) in 1899. The following link shows its dynamic process, quite interesting:

https://www.youtube.com/watch?v=rLuVuxrOxa8

Morley’s trisector theorem盡管非常明晰，證明起來卻不是特別容易。最簡單的方法是運用三角函數。當然也有基於歐氏幾何的方法和純代數的方法，這些網上都可以找到，難度在IMO試題之下。

Morley's trisector theorem, although very clear in its statement, is not particularly easy to prove. The simplest method involves using trigonometric functions. Of course, there are methods based on Euclidean geometry and pure algebra, all of which can be found online, and their difficulty level is below that of International Mathematical Olympiad (IMO) problems.

Frank Morley的生平也是頗有意思的。他原是英國人，家裏是開瓷器店的。他本人1884年劍橋大學畢業。三年以後他來到美國，先在賓州的Haverford College任教，幾年裏成果頗豐，包括發現這個非常優美的平麵幾何定理。他後來成了約翰霍普金斯大學數學係的主任，並在1919-1920年任美國數學學會的主席。在一生中，有多達50個PhD畢業於他門下。他1937年逝世後，美國數學學會這樣評價他對美國數學的貢獻 –

Frank Morley's life is also interesting. Originally British, he came from a family of porcelain shop owners. He graduated from Cambridge University in 1884. Three years later, he came to the United States, initially teaching at Haverford College in Pennsylvania, where he made significant contributions, including discovering this very beautiful geometry theorem. He later became the head of the mathematics department at Johns Hopkins University and served as the president of the American Mathematical Society in 1919-1920. In his lifetime, as many as 50 PhDs graduated under his supervision. After his death in 1937, the American Mathematical Society evaluated his contribution to American mathematics as follows:

"...one of the more striking figures of the relatively small group of men who initiated that development which, within his own lifetime, brought Mathematics in America from a minor position to its present place in the sun."

Frank Morley還是一位很優秀的棋手。他曾贏過英國著名棋手Henry Bird，現在棋譜還保留著。他甚至有一次把國際象棋世界冠軍、德國人Emanuel Lasker都贏了。後者也是一位數學家。我猜想兩人是隨便玩玩，不是正是比賽。

Frank Morley was also an outstanding chess player. He once defeated the renowned British chess player Henry Bird, and the game records are still preserved. He even defeated the World Chess Champion, German Emanuel Lasker, at one point. The latter was also a mathematician. I suppose they were just playing for fun, not in a formal competition.

Frank Morley的太太是小提琴音樂家。他們育有3個兒子，個個在其行當中都很優秀。長子Christopher是一位小說家和詩人，著作頗豐；次子Felix是華盛頓郵報的編輯、撰稿人，曾獲普利策獎；三子Frank Jr. 獲得牛津大學數學博士學位，後與父親合作撰寫數學專著。他同時也是一位作家和出版商。

Frank Morley's wife was a violinist. They had three sons, each excelling in their respective fields. The eldest son, Christopher, was a novelist and poet with numerous works. The second son, Felix, was an editor and contributor for The Washington Post, and he was a Pulitzer Prize winner. The third son, Frank Jr., obtained a Ph.D. in mathematics from Oxford University, later collaborating with his father on mathematical publications. He was also a writer and publisher.

【Edited from ChatGPT translation.】