My answer to a question on Quora: What makes a proof beautiful?
It is like the beauty of a human being, or a flower, or work of art: it is next to impossible to give a definition of any degree of precision, but we can make a list of attributes which make an object (a human being, or a flower) more beautiful than other objects of the same class.
Timothy Gowers, one of the leading mathematicians of our time, once said in one of his talks:
The following informal concepts of mathematical practice cry out to be explicated: beautiful, natural, deep, trivial, “right”, difficult, genuinely, explanatory.
Many of these informal (but instantly recognisable for every mathematician) words apply to proofs and may contribute to characterisation of a proof as “beautiful”:
natural, deep, “right”, explanatory, elegant, revealing, self-contained, concise, streamlined, well-structured, clear, unexpected, surprising, constructive, clean …
— this list (utterly random) can be easily continued.
For a human to be beautiful, it is expected that he/she has “clear eyes” (which is actually a sign of good health). The same with “clear” proofs: this is a good sign that the proof is in some sense “healthy”. On the other hand, some proofs (especially in their draft form) could be “suspicious” and “unreliable” — and even “fishy”; frequently it is lethal, and means that, under closer consideration, the proof is likely to collapse. And “clear” proof is not the same as “clean” proof —I feel the difference, but it will take some time for me to make it explicit even for myself.
In short, a book can be written on this topic — but, to the best of my knowledge, has never been.