In part three of "What on earth has Teake been doing for the last four years?" I'll talk a little on how symmetry appears in nature. When considering this subject, it seems natural to think of symmetric things that appear in nature. Snowflakes, like the one above, are good examples, but also flowers, the wings of butterflies, and sea stars (to name a few) all show some form of symmetry. They fall into the category "things in nature that look kind of symmetric", but in fact have little or nothing to do with the physicist's concept of the symmetry of nature.
One definition of symmetry in physics is as follows: the symmetries of nature are those transformations that do not change the laws of physics. To illustrate this concept, have a look at the following four clocks:
The first clock obviously satisfies the laws of physics. We now apply two different transformations, namely a mirror operation and a time reversal operation. Both give a clock that is distinct from the original one -- you can see the dial moving in the opposite direction. So the clock is not invariant under these transformations, and thus they are not symmetries of the clock. However, there's nothing physically wrong with a clock running in the opposite direction, apart from the fact that it's broken. Therefore parity (a fancy word for a mirror operation) and time reversal are symmetries of the laws of physics of this particular system. In general parity and time reversal are broken, but that's another story.
As a side note, you can see that the combined actions of parity (C) and time reversal (T) do leave the clock invariant, so CT is a symmetry of the clock.
This concept of symmetry is quite powerful. For instance, Special Relativity can be derived by demanding that the laws of physics are invariant under translations, rotations, and boosts (which together make the Poincaré group). Modern physics is to a large extent build on similar considerations of symmetry. So the study of group theory is not just a nice mathematical exercise; it actually has some useful applications throughout the field of physics. In the upcoming blogposts on "What on earth has Teake been doing for the last four years?" I'll try to explain how I applied some fancy group theory to even fancier things like supergravity. Stay tuned!