Tuesday, September 29, 2009

Symmetry

This is the first post of what eventually should become the "What on earth has Teake been doing for the last four years?"-series. Brace yourself: it's about maths and physics. Run while you still can!

I'll try to keep things simple by starting of with a concept that is as mundane as it is fascinating: symmetry. Not only makes it our world round, but it’s also what makes it go round. From the perfect circular wheels on our bikes and cars that deliver an enjoyable ride, to the error-correction protocols that keep e-mails from turning into junk; it’s literally all around us.

So what is symmetry exactly? A symmetry is an action on an object that, once you’re done performing it, does not change that object. It's a somewhat abstract definition, but take for example the triangle, which has 6 symmetries.  There are two different rotations (over 120˚ and 240˚), three reflections, and finally the action of doing nothing at all (yes, that's also a symmetry). You can try them out in the following applet. Clicking on the arrows causes the triangle to rotate and reflect.


This is all pretty straightforward, right? But things start to get interesting we you keep track of the effect of the different rotations and reflections. Let's paint the corners so we can see where they end up:


One thing you'll notice is that doing twice a clockwise rotation is equal to doing one counter-clockwise rotation. The same is true for any other combination of actions -- it will always yield the net effect of one single reflection or rotation. It might also happen that the triangle ends up in the original configuration, but remember that doing nothing is also a symmetry.

The combined actions describe what mathematicians call a group. A group is a set of elements plus a rule of multiplying those elements. Let's call the set G and the multiplication rule "•". Then the precise definition of a group (in which a, b are elements of the set G) is the validity of the following four statements:
  • Closure.
    The result of the operation ab is also in G.
  • Identity element.
    There exists an element 1 in G, such that for all elements a in G, the equation 1a = a1 = a holds.
  • Inverse element.
    There exists an element a-1 in G such that aa-1 = a-1a = 1.
  • Associativity.
    The equation (ab) • c = a • (bc) holds.
These four set of rules are called the group axioms. They might sound a bit abstract, but in fact, they're not. Let's have a look at our triangle again.


The symmetry actions are labeled as follows:
  • 1: identity element ("doing nothing at all").
  • y: counter-clockwise rotation by 120˚.
  • p: clockwise rotation by 120˚.
  • R: reflection in the top vertex.
  • G: reflection in the lower-right vertex.
  • B: reflection in the lower-left vertex.
We already noticed that the closure axiom holds. The existence of the identity element is also pretty obvious. What about the identity axiom? This indeed also holds: every action has an inverse. The reflections are their own inverse, whereas the rotations are each other's. The last thing to check is associativity. It's a bit harder to verify, but believe me, it holds too.

Just for completeness sake, here's one last version of the triangle applet. This one includes the group multiplication table, which keeps track of what happens when you first do the action in the first row, followed by the action in the first column. (If you didn't believe me on the validity of associativity you can use this table to check it.)


What I've shown you so far is that the symmetries of the triangle can be described in terms of the mathematical concept of a group. The importance of group theory lies in the fact that any symmetry you can think of can be described as a group, and that conversely all groups describe a symmetry.

By now the answer to the question "What on earth has Teake been doing for the last four years?" will hardly come as a surprise: it's group theory. More on that in part two of this series!