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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).

Bill Dubuque
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    Why this is put as broad when other similar questions are not ? –  May 20 '17 at 09:37
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    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. – Christian Blatter May 20 '17 at 12:18
  • @ChristianBlatter Exactly that. I wonder if it could be unclosed. –  May 21 '17 at 06:05
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    @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. – Martin Sleziak May 21 '17 at 06:21
  • This question is somewhat similar: Interesting “real life” applications of serious theorems. (Although perhaps not exactly the same. I am not sure whether I would vote to close as duplicate of that question.) – Martin Sleziak May 21 '17 at 06:27
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    @MartinSleziak Nope, not exactly similar, the motivation and the goals are completely different. –  May 21 '17 at 06:32
  • But like the question Martin refers to, it is too broad, is subjective, and is correctly closed. One could answer with a deep theory about how toothpicks confirm/challenge our understanding of them and their use. Similarly perhaps you are interested int the "deep mathematical theory" that explains the acitivity of an ant. So many objects, potentially too many "deep mathematical theories," – amWhy Jun 01 '17 at 21:27

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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.

Christian Blatter
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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.

Fargle
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