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.
就如同各時代 幾何學學生所熟知 幾何學之父是歐幾里得。 他是一位希臘數學家。 生活在西元前約三百年 埃及的亞歷山卓。 歐幾里得是《幾何原本》 這部影響深遠著作的作者。 你覺得你的數學教科書太厚嗎? 歐幾里得的《幾何原本》有 13 冊, 並且都只談論幾何。 在《幾何原本》中, 歐幾里得建構並補足許多 先前數學家的工作, 像是畢達哥拉斯(Pythagoras)、 歐多克索斯(Eudoxus)、 希波克拉底(Hippocrates)、 還有其它人。 歐幾里得把他們的結果 用邏輯的證明系統寫下來, 這證明系統建構在 定義、共同認知、 以及他那五個有名的公設之上。 (譯註:公設即一開始就認定要是對的敍述。) 公設中其中四個 是簡單且直覺的, 比如說兩點可以決定一條線。 而那第五個公設, 則衍伸出我們要講的故事。 這第五個神秘的公設 簡單地被稱作「平行公設」。 你看,和前面四個公設不一樣, 這第五個公設敘述地 十分拐彎抹角。 這是歐幾里得的版本: 「如果一條直線與另兩條相交 並且同一側的的兩內角 加起來小於兩個直角的角度, 那麼那兩條直線 終會在那一側相交, 因此它們並不平行。」 哇,這真繞口! 這是比較簡單、大家也比較熟知的版本: 「在一平面中,給定一直線及線外的一點, 只能畫出一條新的直線 通過這點並和原本的線平行。」 歷經好幾世紀,許多數學家 試著要用其它四個公設 來證明平行公設, 但都失敗了。 在這過程中,他們開始關注於 如果第五公設實際上是錯的 邏輯上會有什麼問題。 一些數學史上 最偉大的先驅 都考慮過這個問題, 像是海什木(Ibn al-Haytham)、 歐瑪爾.海亞姆(Omar Khayyam)、 納速拉丁.圖西(Nasir al-Din al-Tusi)、 幾凡尼.歇克瑞(Giovanni Saccheri)、 鮑耶.亞諾什(Janos Bolyai)、 卡爾.高斯(Carl Gauss)、 尼古拉.羅巴切夫斯基(Nikolai Lobachevsky)。 他們都曾試驗過平行公設 錯誤時的情形, 但發現這樣新的假設 只會建構出完全不一樣的幾何學。 這些幾何學則合稱為 「非歐幾何」。 嗯,我們會把非歐幾何 的細節留到下一堂課, 但主要的不同在於我們所討論的直線 所在的曲面 它曲率的不同。 原來歐幾里得 並沒在《幾何原本》中 告訴我們完整的故事; 他只是提供了一個可能的方法 來看待宇宙。 這取決於我們怎麼看它。 平坦的表面是一種樣子, 而凹的或凸的表面 卻表現出很不一樣的特徵。 一開始這些非歐幾何 似乎有點奇怪 但很快地被發現 它也適切地 描述我們的宇宙。 在我們的星球上航行 須要橢圓幾何, 而同時埃舍爾 (譯註:M. C. Escher 為荷蘭版畫家。) 又以他的藝術 展現雙曲幾何的美。 愛因斯坦也使用非歐幾何 在廣義相對論中 來描述時間與空間 在各狀態下如何改變。 而最大的謎團在於 歐幾里得 在寫下神秘的平行公設時 是否注意到這些 不同的幾何學的存在。 我們可能永遠不會知道答案, 但很難相信 像他這麼聰明的數學家 並對幾何學了解如此透徹, 他會完全沒有注意到這件事。 也許他確實知道, 然後故意寫下這樣的平行公設, 好讓好奇的後輩們 來發現其細節內容。 如果是這樣,他也許對結果很滿意。 若沒有那些天才又求新求變的思想家, 他們可以屏棄一些先入為主的觀點 並獨立思考, 這些理論可能永遠不會被發現。 我們也應該樂於 偶爾放下既有的概念 或是物理上的經驗 來看看更廣的世界, 否則我們將錯過許多事情。