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Posts Tagged "Euclid"

The Artful Amoeba

Origin of Mysterious Portuguese Mathematical and Geographical Tiles Revealed

tiles_mathematical_simoes_fig27_200

A few months ago I wrote about some mystifying mathematical and geographic tiles I encountered at the National Tile Museum in Lisbon, Portugal. Their accompanying label gave no clue to who had made them or why. Several readers subsequently wrote to tell me what they knew about these tiles. Thank you to everyone who did [...]

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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

What’s the Deal with Euclid’s Fourth Postulate?

An illustration from Oliver Byrne's 1847 edition of Euclid's Elements. Image: Public domain, via Wikimedia Commons.

In February, I wrote about Euclid’s parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry. I included the text of the five postulates, from Thomas Heath’s translation of Euclid’s Elements: “Let the following be postulated: 1) To draw a straight line [...]

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Roots of Unity

The Math Wars, Lewis Carroll Style

Lewis Carroll in 1863, photographed by Oscar Gustave Rejlander. Image: Public domain, via Wikimedia Commons.

In 1879, Charles Dodgson, better known as Lewis Carroll, published an odd little book called Euclid and his Modern Rivals (available for free at the Internet Archive). Though it takes the form of a play, it is a defense of Euclid’s Elements as the best textbook for geometry. Carroll’s introduction lays out his purpose and why [...]

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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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