Most of us learn some amount of Euclidean geometry in school.
我們大多數人在學校裡學到了一些歐幾里德幾何。
We learn that the interior angles of triangles add up to 180°.
我們知道三角形的內角加起來是180°。
We learn how to prove that lines are parallel, or shapes are congruent or similar.
我們學習如何證明線是平行的，或者形狀是一致的或相似的。
Even though we live on a planet that is not flat, our everyday intuition is on a scale that makes us feel like Euclidean geometry is the natural way to think about shapes, lengths, and angles.
即使我們生活在一個不平坦的星球上，我們的日常直覺的規模讓我們覺得歐幾里得幾何是思考形狀，長度和角度的自然方式。
I think it’s a real shame that more students are not exposed to non-Euclidean geometry early in their educations, but that’s a column for another time.
我認為，更多的學生在早期教育中沒有接觸到非歐幾里德幾何，這是一個真正的恥辱，但這是另一個時間的專欄。
Suffice it to say that if one is lucky enough to encounter geometry beyond that which takes place on a perfectly flat plane, one will learn that there is much more to geometry than two-column proofs and the Pythagorean theorem.
可以這樣說，如果一個人足夠幸運地遇到在完美平面上發生的幾何學以外的幾何，人們就會了解到幾何學不僅僅是兩列證明和畢達哥拉斯定理。
The intuition we develop in Euclidean geometry does not prepare us well for non-Euclidean geometry.
我們在歐幾里德幾何學中發展的直覺並沒有為我們為非歐幾里德幾何學做好準備。
One of the delights I found when I first started studying hyperbolic geometry (one of the flavors of non-Euclidean geometry) was that many things that seem so obvious as not to require any kind of justification are flat-out wrong when we leave the flat Euclidean plane.
當我第一次開始研究雙曲幾何(非歐幾里德幾何的一種風格)時，我發現的樂趣之一是，當我們離開平坦的歐幾里德平面時，許多看起來如此明顯而不需要任何理由的事情都是徹頭徹尾的錯誤。
For example, in non-Euclidean geometry, there is no such thing as a pair of triangles that are similar but not congruent.
例如，在非歐幾里得幾何中，沒有一對相似但不全等的三角形。
The geometry of a sphere is another flavor of non-Euclidean geometry, so we can think about it on a globe.
球體的幾何是非歐幾里得幾何的另一種風格，所以我們可以在球體上思考它。
On the Earth, there is a triangle that connects the North Pole with Quito (the capital of Ecuador) and Libreville (the capital of Gabon).
在地球上，有一個三角形連接北極與基多(厄瓜多爾的首都)和利伯維爾(加蓬的首都)。
This triangle is close to being a triangle with three right angles, or 270° of internal angle.
此三角形接近於具有三個直角或270°內角的三角形。
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