Deep mathematics behind common and curious objects

Question

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).

Answer 1

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.

Answer 2

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.