Symmetry (Math and Physics) Part I

Today, we begin a multi-part series on symmetry. I'm not sure how many installments there'll be, but I plan to approach symmetry from multiple perspectives. In this first posting, we'll introduce some definitions and visual illustrations of symmetry. In the next posting, we'll look at physicists' writings on the importance of symmetry for physics. Later on, we'll look at the mathematical concept of symmetry groups, the notations of which, such as SU(2) and SO(3), often show up in physics books for educated laypersons. We'll also examine Galois groups and Lie groups.

Much of this material is impenetrably way over my head, and probably capable of being understood only by mathematicians and mathematical physicists with specialized advanced training. However, I feel I can at least sketch out the major ideas.

Let's start by looking at the following diagram of a soccer field (also known as a "pitch"):


One might say the field is "symmetric" because, emanating from the vertical center line in rightward and leftward directions, the two halves of the field look identical. One could also rotate the field 180 degrees (or any whole-number multiple of that) and it would look identical to how it had looked before.

This leads us to our formal definition of symmetry: that things “appear unchanged when certain transformations are applied.” This definition comes from a site called Mathematics Illuminated, which also provides some nice visual illustrations of different types of symmetry, including mirror/bilateral and rotational.

Another web resource, this one from the University at Buffalo, presents some artistic depictions and interactive demonstrations of symmetry in physics (when the page comes up, scroll down a bit and click on the heading for the Flash presentation on symmetry).