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Roots of Unity

In Which Omar Khayyam Is Grumpy with Euclid

A portrait of Omar Khayyam. Image: Atilin, via Wikimedia Commons.

My math history class is currently studying non-Euclidean geometry, which means we’ve studied quite a few “proofs” of Euclid’s fifth postulate, also known as the parallel postulate. I’ve written about this postulate before. There are many statements that are equivalent to the parallel postulate, including the fact that parallel lines in a plane are equidistant. This [...]

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Roots of Unity

Chasing the Parallel Postulate

If you'd like to see these again, accept the parallel postulate. Image: Webber, via Wikimedia Commons.

Euclidean geometry, codified around 300 BCE by Euclid of Alexandria in one of the most influential textbooks in history, is based on 23 definitions, 5 postulates, and 5 axioms, or “common notions.” But as I mentioned in my recent post on hyperbolic geometry, one of the postulates, the parallel postulate, is not like the others. [...]

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Roots of Unity

Hyperbolic Quotes about Hyperbolic Geometry

This Hungarian postage stamp does not portray János Bolyai. For more information about finding the "real face of János Bolyai," click the picture to read Tamás Dénes's article on the subject.

“The treatise itself, therefore, contains only twenty-four pages—the most extraordinary two dozen pages in the whole history of thought!” “How different with Bolyai János and Lobachévski, who claimed at once, unflinchingly, that their discovery marked an epoch in human thought so momentous as to be unsurpassed by anything recorded in the history of philosophy or of [...]

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