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Once I heard about a prophet that used math to foresaw with great accuracy many events of the humanity. Today I oddly realized the time between falling drops after washing cups fit the inverse square law.

What are some unexpected things accurately predicted by math?

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    Be sure to balance the unreasonable effectiveness with the unreasonable *ineffectiveness* of math in complex systems, eg in economics: http://cje.oxfordjournals.org/content/29/6/849.abstract, in biomedicine: http://www.deirdremccloskey.com/docs/fisherian.pdf2012-12-11

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The angle of the wake of a body moving steadily in deep water (e.g. a ship or a duck) is always $2\arcsin(1/3) \approx 38.9^\circ$.

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Existence of Black Holes, Pulsars, parallel universes, worm holes.

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    It's not known that parallel universes or worm holes exist, so I don't think we should classify those as "accurately predicted by math" ...2012-06-27
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Here is an interesting article about Mathematical Fortune-Telling

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Look at Good–Turing rule, the two posts in Azimuth blog:

The puzzle itself:

Suppose you go into the jungles of Ecuador and start collecting butterflies. You count the number of butterflies of each different species that you find. You get a list of numbers, something like this:

14, 10, 8, 6, 2, 1, 1, 1

Puzzle: What is the chance that the next butterfly you find will belong to a new species?

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A curious application from Fred Hoyle : Use the the prevalence of life forms on earth to deduce a new resonance in the carbon-12 nucleus.