I recently worked out a formula in celestial mechanics which is so beautiful, I was stunned by it for a week.
It isn’t just the unexpected simplicity of the formula. It’s the fact that it’s one in the eye for the scientists. They are always contrasting how things look (to us) with how things really are (in the scientists’ eyes), and of course the idea is that Reality trumps Appearance every time.
The beautiful one-in-the-eye-for-the-scientists formula says that the height of the tides depends on two things and two things only: what the Moon is made of (green cheese would make smaller tides than rock) and how big it looks when we look up at it. What about how big the Moon really is? And how far away it really is? Neither of these things matters in the slightest. Reality is irrelevant: it is Appearance that counts.
Beauty should be shared; and a beautiful result must have a proof that lives up to it. To me, this means a proof that is as pure and limpid and natural as the things it talks about, not line after line of algebra. Algebra does have beauty in some people’s eyes, but only because the beauty of what is underneath the algebra shines through the symbols. I’d rather try and communicate the beauty directly.