Hidden symmetries

Symmetries, you may have heard, play a crucial role in modern physics. (Leon Lederman and Chris Hill wrote a whole popular book about the subject, if you're interested.) But one of the things that makes them so interesting is that they can be hidden -- the symmetry is secretly there, even though you don't easily notice. And sometimes you may be interested in the converse situation -- it looks like there is an obvious symmetry of nature, but in fact there are tiny violations of it, which we haven't yet detected. To physicists, a "symmetry" is a situation where you can rearrange things a bit (values of quantum fields, positions in space, any of the characteristics of some physical state) and get the same answer to any physical question you may want to ask. An obvious example is, in fact, position in space: it doesn't matter where in the world ...