I'm a huge fan of Tom Lehrer and have mentioned him a number of times before. I just came across an interview with him from 2000 in which he discussed his dual life as a mathematician and performer. I especially loved this quote, on the concept of "elegance" in mathematics:
I think the construction part, the math, how to say it, the logical mind, the precision, is the same that's involved in math as in lyrics.... And I guess in music too. It's gotta come out right. It's like a puzzle, to write a song. The idea of fitting all the pieces so it exactly comes right, the right word at the end of the sentence, and the rhyme goes there and not there. Mathematicians, as opposed to natural scientists, are so interested in elegance. That's the word you hear in mathematics all the time. "This proof is elegant!" It doesn't really matter what it proves. "Look at this -- isn't that amazing!" And it comes out at the end. It's neat. It's not just that it's proof, because there's plenty of proofs that are just boring proofs. But every now and then there's a really elegant proof.
I have tried many times to capture and define the euphoria of achieving some kind of mathematical beauty or elegance -- almost always in vain. Erdos said it well:
Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you.