As any current or past geometry student knows, the father of geometry was Euclid,
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，
a Greek mathematician who lived in Alexandria, Egypt, around 300 B.C.E.
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。
Euclid is known as the author of a singularly influential work known as "Elements."
欧几里得撰写了《几何原本》这部影响深远的著作。
You think your math book is long?
你觉得你的数学教材太厚吗？
Euclid's "Elements" is 13 volumes full of just geometry.
欧几里得的《几何原本》有13册，并且全部是有关几何的。
In "Elements," Euclid structured and supplemented the work of many mathematicians that came before him,
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，
such as Pythagoras, Eudoxus, Hippocrates and others.
像是毕达哥拉斯、欧多克索斯、希波克拉底等等。
Euclid laid it all out as a logical system of proof
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，
built up from a set of definitions, common notions, and his five famous postulates.
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。
Four of these postulates are very simple and straightforward, two points determine a line, for example.
其中四个公理非常简单易懂，比如说两点可以决定一条线。
The fifth one, however, is the seed that grows our story.
而那第五个公理，则衍生出我们要讲的故事。
This fifth mysterious postulate is known simply as the parallel postulate.
这第五个神秘的公理被简单地称作“平行公理”。
You see, unlike the first four, the fifth postulate is worded in a very convoluted way.
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。
Euclid's version states that, "If a line falls on two other lines
欧几里得的版本是：“如果一条直线与另两条直线相交，
so that the measure of the two interior angles on the same side of the transversal add up to less than two right angles,
并且同一侧的两内角加起来小于两个直角的角度，
then the lines eventually intersect on that side, and therefore are not parallel."
那么这两条直线最终会在那一侧相交，因此它们并不平行。”
Wow, that is a mouthful! Here's the simpler, more familiar version:
哇，这真绕口！而下面这个是更简单、更众所周知的版本：
"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."
“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。”
Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but weren't able to do so.
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。
In the process, they began looking at what would happen logically if the fifth postulate were actually not true.
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。
Some of the greatest minds in the history of mathematics ask this question,
一些数学史上最伟大的先驱都考虑过这个问题，
people like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, Giovanni Saccheri, János Bolyai, Carl Gauss, and Nikolai Lobachevsky.
像是伊本·艾尔-海什木、欧玛尔·海亚姆、纳速拉丁·图西、几凡尼·歇克瑞、鲍耶·亚诺什、卡尔·高斯以及尼古拉·罗巴切夫斯基

They all experimented with negating the parallel postulate,
他们都试验过平行公理错误时的情形，
only to discover that this gave rise to entire alternative geometries.
但发现这种猜测只会建构出完全不一样的几何学。
These geometries became collectively known as non-Euclidean geometries.
这些几何学则合称为“非欧几何”。
We'll leave the details of these different geometries for another lesson.
我们会把非欧几何的细节留到另外一堂课。
The main difference depends on the curvature of the surface upon which the lines are constructed.
主要的不同在于我们所讨论的直线所在曲面的曲率不同。
Turns out Euclid did not tell us the entire story in "Elements,"
原来，欧几里得并没在《几何原本》中告诉我们完整的故事，
and merely described one possible way to look at the universe.
他只是提供了一个可能的方法来看待宇宙。
It all depends on the context of what you're looking at.
这取决于我们看待它的视角。
Flat surfaces behave one way, while positively and negatively curved surfaces display very different characteristics.
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。
At first these alternative geometries seemed strange,
这些非欧几何一开始似乎有点奇怪，
but were soon found to be equally adept at describing the world around us.
但很快就被发现它也能恰当地描述我们的宇宙。
Navigating our planet requires elliptical geometry while the much of the art of M.C. Escher displays hyperbolic geometry.
在我们的星球上航行需要用到椭圆几何，而同时埃舍尔又以他的艺术展现了双曲几何的美。
Albert Einstein used non-Euclidean geometry as well
爱因斯坦也使用非欧几何
to describe how space-time becomes warped in the presence of matter, as part of his general theory of relativity.
在广义相对论中来描述时间与空间在各状态下如何改变。
The big mystery is whether Euclid had any inkling of the existence of these different geometries when he wrote his postulate.
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。
We may never know, but it's hard to believe he had no idea whatsoever of their nature,
我们可能永远不会知道答案，但很难相信像他这么聪明的、对几何学又了解得如此透彻的数学家
being the great intellect that he was and understanding the field as thoroughly as he did.
怎么会完全没有注意到这件事。
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.
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。
If so, he's probably pleased.
如果是这样，他也许会感到很欣慰。
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.
如果没有那些不断创新的、能够摒弃一些先入为主的观点并独立思考的天才思想家，这些理论可能永远不会被发现。
We, too, must be willing at times to put aside our preconceived notions and physical experiences
我们也应该乐于偶尔放下既有的概念和物理经验，
and look at the larger picture, or we risk not seeing the rest of the story.
来看看更广的世界，否则我们可能会错过许多奇妙的事情。