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The question of usefulness of mathematics in everyday life is a cliche, and I am not asking that.

What are some objects*/algorithms/other curious stuff/tricks, which has surprisingly deep mathematical principle governing them ?

(*objects means concrete touchable stuff that you're likely to encounter in real life, e.g Mechanical puzzles)

Rubik's cube (Lot's of stuff from group theory and combinatorics) is a very good example, and so is the trick that you give someone a bunch of cards and tell them to pick consecutive five, and you ask them to tell the colors of them, and you tell all the card's value (Which is based on De Bruijn sequence).

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closed as too broad by Meta-мета-μετα-meta-мета-μετα, Jyrki Lahtonen May 29 '17 at 16:03

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Why this is put as broad when other similar questions are not ? $\endgroup$ – user441034 May 20 '17 at 9:37
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    $\begingroup$ One needs such a list when one has to explain to a general public why deep and difficult mathematics has an impact on our everyday life. This question is not so much about puzzles. Why it is put on hold is beyond me. This was an act in Trump style. $\endgroup$ – Christian Blatter May 20 '17 at 12:18
  • $\begingroup$ @ChristianBlatter Exactly that. I wonder if it could be unclosed. $\endgroup$ – user441034 May 21 '17 at 6:05
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    $\begingroup$ @ChristianBlatter If you think it should be reopened, one of possible things to do is is that you could cast a reopen vote. This puts the question into reopen review queue, where other users can vote whether to reopen it or leave it closed. Another alternative is to make a post in reopen request thread - ideally including some arguments why the question should be reopened. Here is related discussion in chat. $\endgroup$ – Martin Sleziak May 21 '17 at 6:21
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    $\begingroup$ @MartinSleziak Nope, not exactly similar, the motivation and the goals are completely different. $\endgroup$ – user441034 May 21 '17 at 6:32
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A prime example is computer tomography. While ordinary X-ray pictures are just photographs made with some other kind of light, CT pictures are the result of an integral transform applied to X-ray measurements having its roots in abstract harmonic analysis. This transform has been invented by Radon in 1910, but the practical application was only possible in the last quarter of the 20th century.

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There is the classical example of a pizza. The way pizza crust usually deforms is by bending without stretching--unless, I guess, the pizza is particularly doughy--so if you curve it along the axis parallel to the crust, it must stay straight along the perpendicular axis, and the pizza doesn't flop. This is because if you bend a surface without stretching, its Gaussian curvature (i.e. the product of the maximum directional curvature with the minimum one) must remain the same, in this case zero.

In addition, no perfect map can be made of the Earth, because the Gaussian curvature is positive everywhere on a globe, so there's no way to cut it into a map, which has Gaussian curvature 0, without stretching it somehow.

These both result from Gauss's "Theorema Egregium", so named because it is so surprising that an idea of curvature coming from coordinates is invariant under such transformations. I always found it neat that the pizza thing and the globe-map thing are the same thing.

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