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Posts Tagged "geometry"

The Artful Amoeba

Mysterious Tiles from a Time When Art and Science Were Friends

Portugal_geometry_tiles_jf_200

Forces in society of late have lots of us longing for the days of the Enlightenment, smallpox, powdered wigs, ridiculously uncomfortable clothing and all. It must have been nice to live in an era when science and scientists were respected, admired, and generously funded (though often by self-funded aristocrats or by royal grants gleaned from [...]

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Observations

Glow Sticks Prove the Math Theorem behind the Famous Flatiron Building

The Pythagorean theorem projected onto the Flatiron building

How many math lovers live in New York City? It’s a tough count to make, but the Museum of Mathematics made progress at its first anniversary celebration on Thursday, December 5. With a mission to illuminate the math that permeates our day-to-day lives, the Museum of Mathematics, or MoMath, wasn’t about to waste its birthday [...]

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Roots of Unity

British Objects of Constant Width

Several British objects of constant width. Image: Evelyn Lamb.

As I wrap up a trip to the UK, I reflect on the many objects of constant width I encountered here. I’ll let Numberphile tell you a little more about objects of constant width. Almost immediately after getting off the plane at Heathrow, I got some breakfast and some change in the form of metal [...]

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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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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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Roots of Unity

Counterexamples in Origami

Ain't she a beaut? Image: Evelyn Lamb.

Surfaces are complicated. Triangles are simple. That’s an idea behind some methods of creating computer graphics and some advanced mathematics. If we have a surface, we can take a bunch of points on the surface and connect them into triangles to obtain an approximation of the surface. That’s all well and good, but how reliable [...]

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Roots of Unity

A Mathematical Thanksgiving Celebration

A more realistic picture of how Borromean rings can occur in the real world. Image: public domain, via Wikimedia Commons.

Last year, the inimitable Vi Hart made a Thanksgiving video series, describing how to imbue your holiday celebration with more mathematics. My favorite video is the one about Borromean onion rings, perhaps because I’ve been slightly obsessed with Borromean rings for a while. Borromean rings are three circles that are connected so that if you [...]

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Roots of Unity

10 Secret Trig Functions Your Math Teachers Never Taught You

A diagram with a unit circle and more trig functions than you can shake a stick at. The familiar sine, cosine, and tangent are in blue, red, and tan, respectively.

On Monday, the Onion reported that the “Nation’s math teachers introduce 27 new trig functions.” It’s a funny read. The gamsin, negtan, and cosvnx from the Onion article are fictional, but the piece has a kernel of truth: there are 10 secret trig functions you’ve never heard of, and they have delightful names like “haversine” [...]

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Roots of Unity

Strumming the Lute of Pythagoras

A drawing by Joseph Koch incorporates the Lute of Pythagoras into a portrait of Pythagoras himself. Image copyright Joseph Koch. Used with permission.

When I was at the Joint Math Meetings in January, the evocative name “Lute of Pythagoras” jumped out at me in a talk by Ann Hanson of Columbia College in Chicago. Hanson teaches a course, Math in Art and Nature, that satisfies the general math requirement for Columbia College but comes with a healthy helping [...]

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