This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean geometry as an example of a theory that non-mathematicians would find beautiful, but mathematicians would hesitate to classify as so.) Rota also cautions against creating a list of beautiful theorems, even though it might be a best-seller, because one cannot really experience beauty when taken out of context.
He explains, "The fact is that the beauty of a mathematical theorem is best observed when the theorem is presented as the crown jewel within a context of results of a theory. When instead mathematical theorems from disparate areas of mathematics are strung together and presented as “pearls”, then they lose their relevance, and are likely to be appreciated only by those who are already familiar with them."
This brings me to my question: Is the task of trying to introduce beauty in high school mathematics in vain? (This question resonates with one comment on my original question regarding whether high school students are capable of really appreciating proof.)
It seems to be there are two obvious answers, one of which is yes, just forget it. The other is, no, but don't expect students to get a full appreciation of beauty by the time they leave high school. The third, bolder, answer is that one can achieve beauty early if exposed to the right mathematics in the right way.
So: If answer three is correct, does anyone have any existence proofs (either from experience or the literature)?
(Also, just to be careful, in my original question, I was specifically asking for examples of beautiful proofs, while Rota is discussing theories and theorems as well. And I want to emphasize that I ask this question only because it is something that genuinely worries me, not that I fail to appreciate the thoughtful answers given to my original question.)