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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.)

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Beauty lies in the eye of the beholder---so what is this question really trying to ask? –  Suvrit Sep 20 '11 at 8:32
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Beauty is more abundant in high school, also significant at a university level, and goes away as the time goes by... –  Fernando Muro Sep 20 '11 at 8:35
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The pythagorean theorem is pretty beautiful if you ask me –  Michael Kissner Sep 20 '11 at 8:46
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@Suvrit: Rota says that there is a common mistake regarding beauty to consider it as a sort of light-bulb flash, and this mistake has consequences including one I mentioned, of thinking one can convey beauty by compiling a list of beautiful theorems. I guess what I'm fishing for is (a) whether Rota is right, and (b) if so, could another consequence could be that it is useless to try to convey beauty at the high school level? I personally don't think it is useless, but I want to think through Rota's concerns. –  Manya Sep 20 '11 at 11:17
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@Manya: I withhold judgement on Rota since, as mentioned in my answer, I have not read the article yet, and I certainly believe that the man has earned the benefit of the doubt. Still, on the surface, these statements strike me as very naive, if only because they attempt to make categorical statements about aesthetic values which, unlike science or mathematics, are very poorly suited to such absolutes. –  Thierry Zell Sep 20 '11 at 11:41
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closed as not constructive by Suvrit, Felipe Voloch, Andrew Stacey, quid, Douglas Zare Sep 20 '11 at 11:34

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2 Answers

Julia Setslink text

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Is beauty possible even at bee-level? We like this Julia set because the feeling of being in a meadow gives us pleasure. But flowers look the way they do because insects think they are beautiful... –  Johan Wästlund Sep 20 '11 at 12:26
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I have not read's Rota's paper, so I am not sure what point he is trying to make. Though I have glanced at the abstract, which begins with:

It has been observed that whereas painters and musicians are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work.

Maybe this statement is explained and supported in the main text, but, at first glance, this difference in attitude does not strike me as obvious (I'm pretty sure I've heard more discussions about technical aspects from mathematicians than from artists).


To come back to your question, the fact that the untrained and the trained experience beauty differently is by no means specific to mathematics: the trained benefit from having a rich context that dramatically alters their appreciation of the aesthetic aspects of a work; I cannot think of a discipline where this would fail to be true.

What does it mean for high school math? Well, just because I've lost some of my enthusiasm for Euclidean geometry since high school, because I've discovered more beautiful things still, it does not mean that the nature of the aesthetic emotions I feel for Euclidean geometry is any different from the emotions I feel for other beautiful mathematics. And if the intensity has somewhat waned in the case of Euclidean geometry (and I'm not even sure it has), it was certainly very powerful back in high school.

So, basically: who cares if the beauty that one shows in high school math class does not meet the standards of beauty of professional mathematicians? What if it makes these folks go "meh"? The teacher is not talking to them, but to the students! So, as long as the students experience the same kind of feeling of mathematical beauty, the work is done: they know there is more to this, that mathematical beauty is out there.


To finish, a couple of tangential remarks:

  • I'm not sure I buy the whole premise anyway. To me, some of the most beautiful ideas in math are commonly seen at the high-school level: they are beautiful because they are so simple yet so powerful (coordinates, change of variables,...). Though, on the other hand, I'm not sure it is possible to appreciate the beauty of these ideas without the benefit of hindsight.
  • I have purposefully stayed away from the question: What proportion of students can one reach with that?, in part because I have no idea how to answer the question, and also because this discussion would be so system- and country-dependent that MO is not the place for it. But I think this should be part of your introspection.
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Thanks for your response. –  Manya Sep 20 '11 at 11:45
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