Sunday, February 8, 2009

Greg Egan and Commutative Diagrams

A commutative diagram is a diagram illustrating how different series of functions can have the same result when composed. We draw arrows to represent functions. To follow a path of arrows means that we perform each function as we get to the corresponding arrow. If we always get the same result no matter what we started with then we say the diagram commutes. For example, in the below diagram one could label the top arrow f(x)=2x, the left arrow g(x)=x^2, the right arrow h(x)=x^2 and the bottom arrow j(x)=4x.

Now, no matter what number we start with (starting in the upper left corner) when, either path we take to get to the bottom right corner will give us the same result. The diagram reflects that h(f(x))=j(g(x)). That is, that (2x)^2= 4(x^2). Not all commutative diagrams form a square. Others might form a triangle or other shapes. Stated this way commutative diagrams might seem trivial. But they are not trivial at all. Many examples of commutative diagrams show up in higher areas of mathematics especially in algebra. Mathematicians are very fond of commutative diagrams because they are an efficient way of describing deep structure between functions or sets.

Why am I thinking about commutative diagrams? I happened to be reading a science fiction story by author Greg Egan titled Glory. The story involved an advanced, now extinct alien species, that was very skilled at mathematics. To give a taste of their mathematical prowess, Egan correctly described the idea of a commutative diagram and then stated that one of the species' theorems took the form of a commutative diagram that could be best thought of as a 7 dimension hypercube. When I read this I was amazed. This is an excellent idea for a theorem that an extremely advanced species might prove. I had read some of Egan's works before and had heard good things about others but seeing this description I was pleasantly surprised. Often when people write science fiction they merely fill in buzzwords or string together words that sound good. If all science fiction writers could write like Greg Egan I doubt that the genre would have any trouble being taken seriously.

1 comment:

PersonalFailure said...

I read that book. Now I actually understand it. Thanks!