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  • 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 filled 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,

  • Janos 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.

  • Well, 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 that Euclid did not tell us

  • the entire story in Elements;

  • he 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 a bit 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 the way that space time

  • becomes work in the presence of matter

  • as part of his General Theory of Relativity.

  • The big mystery here is whether or not Euclid

  • had any inkling of the existence of these different geometries

  • when he wrote his mysterious postulate.

  • We may never know the answer to this question,

  • but it seem hard to believe

  • that 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 intentionally wrote the Parallel Postulate in such a way

  • as to leave curious minds after him

  • to flush out the details.

  • If so, he's probably quite pleased.

  • These discoveries could never have been made

  • without gifted, progressive thinkers

  • who are able to suspend their preconceived notions

  • and think outside of what they have 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.

As any current or past geometry student knows,

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B2 中上級

TED-ED】ユークリッドの不可解な平行命題 - Jeff Dekofsky (【TED-Ed】Euclid's puzzling parallel postulate - Jeff Dekofsky)

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    wikiHuang に公開 2021 年 01 月 14 日
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