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# Elegance

Ask any scientist or mathematician, and this is the quality that they would love their solution to have. They want the result to be elegant, simple, and intuitive.

To give you an example, I remember doing a problem in my calculus class which involved using a bunch of trigonometric functions. Naturally, the integral kind of exploded as I worked on it, and the result was super-complicated. However, after applying a bunch of different identities and swapping sines and cosines, the answer came back as simply tangent of theta.

When I got this result, I immediately knew I was right. The result was just too perfect after all that work for it not to be true (of course, this is a bias). Additionally, the answer made me feel good. It was a nice answer to look at, particularly after all the work required to get there.

This underscores our tendency in science and mathematics to revere simple answer. Consequently, we tend to “dress up” our equations and concepts in order to make them much more compact than they are in reality. I have two examples to illustrate the point.

First, in physics (particularly, wave motion), there’s the notion of forced oscillations for a spring or other kind of object feeling some sort of oscillation. The illusion of “dressing up” the equation was so strong in this sense that I felt moved enough to create a small comic of it:

Even as my teacher talked about this equation, she looked sheepish. As soon as we saw the whole equation, we could see why (and this was only the steady state solution).

The second example comes from the recent World Science Festival, where I watched the panel on gravitational waves. During this panel at around the thirty minute mark, the moderator (Brian Greene) walked through some of the equations of general relativity, and showed just how complicated these equations can be. Despite looking relatively (sorry!) simple, the equations are just being dressed up to cover their complexities. There’s nothing necessarily wrong about this, but it does illustrate how equations in science and mathematics can be a bit more challenging than they appear. This is all done in the name of elegance. If we can make an equation more compact, we will do it.

Often, we seek the elegant answer, wanting to have something simple after working through a bunch of mathematics. This leads us to covering up the complexities of many equations, which make them difficult to understand while looking in from the outside.

Perhaps we should embrace a little more complexity?