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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

• 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 TED-Ed euclid parallel geometry line mysterious

【TED-Ed】Euclid's puzzling parallel postulate - Jeff Dekofsky

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wikiHuang posted on 2013/12/12
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