金碧辉煌的圣殿 （2. Morley’s trisector theorem）
The Glorious Temple of Geometry 2. Morley’s trisector theorem
在我家的墙上，挂有两类图画，第一类是我拍摄的自然、风景照片，第二类并非什么李杜诗词之类，而是几幅相当优美（我以为）的几何图案。这几个几何定理不仅优美，而且是最近一百年左右“新近”被发现和证明的。比如下面镜框中的图案叫“Morley’s trisector theorem”，它被无数人忽视了2000多年，直到1899年才被数学家Frank Morley 揭示。我们后面细谈。
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.
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:
尺规作图（Compass-and-straightedge construction或 ruler-and-compass construction）是起源于古希腊、与欧式几何密切相关的作图法。该法使用圆规（无角度，但可无限宽）和直尺（无刻度，但可无限长），且只准许使用有限次，来解决几何作图问题。千百年来，人们使用尺规作图的原则，实现了各种简单或复杂的操作，比如下图中A、B、C分别是用尺规做线段的垂直平分线，角的平分线和正六边形，这些都非常简单。而图D是用尺规做正17边形，极其复杂，多达51步。但这个难得不可想象的做图，被德国著名数学家高斯在他大学二年级的时候攻克了。
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.
** 化圆为方问题: 求一个正方形的边长，使其面积与一已知圆的相等 【Squaring a circle (constructing a square with the same area as a given circle)】
** 三等分角问题: 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)】
** 倍立方问题: 求一立方体的棱长，使其体积是一已知立方体的二倍【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.
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.
那么，是不是可以说，从那以后，初等几何之中能够发现的规律，早已被发现和解决了呢？如果把欧氏几何比作一座金矿，经过2400年的开采，一般人似乎以为，金子早已经被开采完了。可是，欧氏几何的实际情况却不是这样的。即便是在100年前，独具慧眼的人还是能拾到金块，甚至是闪亮的“大金块”。Morley’s trisector theorem正是这样一个发现。
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.
这个定理的表述极其简单（重要的话说三遍：复杂了就不美了）：对任意一个三角形，作内角三等分线，靠近公共边三等分线的三个交点，总是连成一个等边三角形。[ In any triangle, the three points of intersection of the adjancent angle trisectors form an equilateral trangle.] 这个简单而优美的规律被人们忽视了2000多年，直到1899年被英裔美国数学家Frank Morley (1860 - 1937) 发现并证明。下面的链接显示其动态过程，颇有意思 ——
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:
Morley’s trisector theorem尽管非常明晰，证明起来却不是特别容易。最简单的方法是运用三角函数。当然也有基于欧氏几何的方法和纯代数的方法，这些网上都可以找到，难度在IMO试题之下。
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的生平也是颇有意思的。他原是英国人，家里是开瓷器店的。他本人1884年剑桥大学毕业。三年以后他来到美国，先在宾州的Haverford College任教，几年里成果颇丰，包括发现这个非常优美的平面几何定理。他后来成了约翰霍普金斯大学数学系的主任，并在1919-1920年任美国数学学会的主席。在一生中，有多达50个PhD毕业于他门下。他1937年逝世后，美国数学学会这样评价他对美国数学的贡献 –
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还是一位很优秀的棋手。他曾赢过英国著名棋手Henry Bird，现在棋谱还保留着。他甚至有一次把国际象棋世界冠军、德国人Emanuel Lasker都赢了。后者也是一位数学家。我猜想两人是随便玩玩，不是正是比赛。
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的太太是小提琴音乐家。他们育有3个儿子，个个在其行当中都很优秀。长子Christopher是一位小说家和诗人，著作颇丰；次子Felix是华盛顿邮报的编辑、撰稿人，曾获普利策奖；三子Frank Jr. 获得牛津大学数学博士学位，后与父亲合作撰写数学专著。他同时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.】