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As a math student, it is hard for me to explain the beauty of math to non-mathematicians. So I wonder, how can one explain the elegance of abstract mathematics to somebody without a special background in math, but who is genuinely interested in understanding the joy that comes with it?

To make this clear: I am not asking about how to present knowledge in an understandable way so that your friend actually knows what a manifold is for the rest of her/his life. It's more about giving a VERY general idea of why some crazy people (called mathematicians) would consider an induction proof elegant or even beautiful. This should be adressing adults who got through high school, so I suppose more advanced concepts than those for a children's book or these first cool math experiences are desirable. Something that might take an hour to explain but is worth the effort.

What beautiful theorems or ideas do you know of that make sense intuitively to non-mathematicians when explained in the right way? Which of them are obviously beautiful, even if one doesn't know all the details about it?

Justin P.
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  • I like Hall's matching theorem. It's a surprising result, has nothing to do with numbers (which dispels a common misconception) and the argument is very aesthetic. – HallaSurvivor Aug 06 '20 at 18:50
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    Perhaps my logician is showing, but a high level picture of the incompleteness theorems can also be fun. At the level of "you find a way to encode 'this sentence is not provable'. If it's true, it's not provable, and if it's false you can prove false... Both of these are scary" – HallaSurvivor Aug 06 '20 at 18:52
  • For me integral Apollonian packing, together with the minimum background, is simply amazing. – user376343 Aug 06 '20 at 18:55
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    Perhaps this could prove useful to you: https://mathoverflow.net/questions/47214/how-to-present-mathematics-to-non-mathematicians – Omar Shaaban Aug 06 '20 at 18:59
  • Since you mentioned induction , I really like the false proof of showing "every people on earth have the same age" ! by induction . – user-492177 Aug 06 '20 at 19:02
  • It is very difficult to explain this to a non mathematician. There are some theorems and proofs which anyone can understand, but not many people will actually appreciate it. – Mark Aug 06 '20 at 19:14

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