As any current or past geometry student knows, the father of geometry was Euclid, a Greek mathematician who lived in Alexandria, Egypt, around 300 B.C.E. Euclid is known as the author of a singularly influential work known as "Elements." You think your math book is long? Euclid's "Elements" is 13 volumes full of just geometry. In "Elements," Euclid structured and supplemented the work of many mathematicians that came before him, such as Pythagoras, Eudoxus, Hippocrates and others. Euclid laid it all out as a logical system of proof built up from a set of definitions, common notions, and his five famous postulates. Four of these postulates are very simple and straightforward, two points determine a line, for example. The fifth one, however, is the seed that grows our story. This fifth mysterious postulate is known simply as the parallel postulate. You see, unlike the first four, the fifth postulate is worded in a very convoluted way. Euclid's version states that, "If a line falls on two other lines so that the measure of the two interior angles on the same side of the transversal add up to less than two right angles, then the lines eventually intersect on that side, and therefore are not parallel." Wow, that is a mouthful! Here's the simpler, more familiar version: "In a plane, through any point not on a given line, only one new line can be drawn that's parallel to the original one." Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but weren't able to do so. In the process, they began looking at what would happen logically if the fifth postulate were actually not true. Some of the greatest minds in the history of mathematics ask this question, people like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, Giovanni Saccheri, János Bolyai, Carl Gauss, and Nikolai Lobachevsky. They all experimented with negating the parallel postulate, only to discover that this gave rise to entire alternative geometries. These geometries became collectively known as non-Euclidean geometries. We'll leave the details of these different geometries for another lesson. The main difference depends on the curvature of the surface upon which the lines are constructed. Turns out Euclid did not tell us the entire story in "Elements," and merely described one possible way to look at the universe. It all depends on the context of what you're looking at. Flat surfaces behave one way, while positively and negatively curved surfaces display very different characteristics. At first these alternative geometries seemed strange, but were soon found to be equally adept at describing the world around us. Navigating our planet requires elliptical geometry while the much of the art of M.C. Escher displays hyperbolic geometry. Albert Einstein used non-Euclidean geometry as well to describe how space-time becomes warped in the presence of matter, as part of his general theory of relativity. The big mystery is whether Euclid had any inkling of the existence of these different geometries when he wrote his postulate. We may never know, but it's hard to believe he had no idea whatsoever of their nature, being the great intellect that he was and understanding the field as thoroughly as he did. Maybe he did know and he wrote the postulate in such a way as to leave curious minds after him to flush out the details. If so, he's probably pleased. These discoveries could never have been made without gifted, progressive thinkers able to suspend their preconceived notions and think outside of what they've been taught. We, too, must be willing at times to put aside our preconceived notions and physical experiences and look at the larger picture, or we risk not seeing the rest of the story.
不论是哪个时代的几何学学生都知道, 几何学之父是欧几里得, 他是一位希腊数学家, 生活在西元前约三百年的 埃及亚历山大。 欧几里得撰写了 《几何原本》这部影响深远的著作。 你觉得你的数学教材太厚吗? 欧几里得的《几何原本》有 13 册, 并且都只谈论几何。 在《几何原本》中, 欧几里得建构并 补足了许多先前数学家的工作, 像是毕达哥拉斯 (Pythagoras)、 欧多克索斯 (Eudoxus)、 希波克拉底 (Hippocrates)、 等等。 欧几里得把他们的结果 通过被证明过的逻辑体系进行编撰, 这一体系建立在定义、 共同认知、 以及他那五个有名的公理之上 (译注:即一开始被认定为正确的叙述。) 其中四个公理非常简单易懂, 比如说两点可以决定一条线。 而那第五个公理, 则衍生出我们要讲的故事。 这第五个神秘的公理 被简单地称作“平行公理”。 可以看到,和前面四个公理不一样, 第五个公理的描述十分拐弯抹角。 欧几里得的版本是: “如果一条直线与另两条直线相交 并且同一侧的两内角 加起来小于两个直角的角度, 那么这两条直线 最终会在那一侧相交, 因此它们并不平行。” 哇,这真绕口! 而这是比较简单、众所周知的版本: “在一个平面中, 给定一直线及线外的一点, 只能画出一条通过 这点并与已知直线平行的线。” 历经好几世纪,许多数学家 试着要用其它四个公理来证明平行公理, 但都失败了。 在这个过程中,他们不禁想到 如果第五个公理实际上是错的, 逻辑上会有什么问题。 一些数学史上 最伟大的先驱都考虑过这个问题, 像是海什木 (Ibn al-Haytham)、 欧玛尔.海亚姆 (Omar Khayyam)、 纳速拉丁.图西 (Nasir al-Din al-Tusi)、 几凡尼.歇克瑞 (Giovanni Saccheri)、 鲍耶.亚诺什 (Janos Bolyai)、 卡尔.高斯 (Carl Gauss)、 以及尼古拉.罗巴切夫斯基 (Nikolai Lobachevsky)。 他们都试验过平行公理错误时的情形, 但发现这种猜测 只会建构出完全不一样的几何学。 这些几何学则合称为 “非欧几何”。 不过我们会把 非欧几何的细节留到下一堂课, 但主要的不同在于我们所讨论的 直线所在曲面的曲率不同。 原来,欧几里得并没在《几何原本》中 告诉我们完整的故事; 他只是提供了一个可能的方法 来看待宇宙。 这取决于我们看待它的视角。 平坦的表面是一种样子, 而凹的或凸的表面 却表现出很不一样的特征。 这些非欧几何一开始似乎有点奇怪, 但很快地被发现它也能恰当地 描述我们的宇宙。 在我们的星球上航行需要用到椭圆几何, 而同时埃舍尔 (译注:荷兰版画家) 又以他的艺术展现了双曲几何的美。 爱因斯坦也使用非欧几何 在广义相对论中 来描述时间与空间 在各状态下如何改变。 而最大的谜团在于欧几里得 在写下神秘的平行公理时, 是否注意到这些不同几何学的存在。 我们可能永远不会知道答案, 但很难相信 像他这么聪明的数学家 对几何学又了解得如此透彻, 怎么会完全没有注意到这件事。 也许他确实知道, 然后故意写下这样的平行公理, 好让好奇的后辈们 来发现背后的细节。 如果是这样,他也许会感到很欣慰。 若没有那些不断创新的 天才思想家能够摒弃一些 先入为主的观点并独立思考, 这些理论可能永远不会被发现。 我们也应该乐于 偶尔放下既有的概念和物理经验, 来看看更广的世界, 否则我们可能会错过许多奇妙的事情。