Timeline for Ingenuity in mathematics
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14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 16, 2011 at 1:56 | comment | added | roy smith | as gowers says, exactly this proof is due to Euclid, in prop. VI.31. I.e. after presenting the theory of similarity Euclid gives this generalization of Pythagoras as an application of similarity. | |
| Aug 6, 2011 at 5:32 | comment | added | roy smith | It went well and I was glad I used the more powerful similarity approach for two reasons. 1) they learned a more generally useful idea. 2) I finished in the alloted time! | |
| Aug 5, 2011 at 20:48 | comment | added | Emerton | Dear Roy, Thanks very much for your comments. I'm glad that this example has been stimulating for your class, and would enjoy hearing how it goes. (Well, I hope!) Best wishes, Matthew | |
| Aug 5, 2011 at 14:23 | comment | added | roy smith | update: Matthew, finally appreciating similarity has made me change my whole focus for the 2 week course. after aiming at a big finale presenting Euclid's tour de force proof of proposition IV.10 for constructing a pentagon, i finally realized that similarity makes the proof so much easier i can actually remember it, and can finesse two preliminary somewhat unfamiliar lemmas, as well as present an important tool, SAS similarity, to the kids. so today i will do it that way. (we do have similarity legitimately now by virtue of prop. III.35, but i had not realized its value.) thanks again! | |
| Aug 5, 2011 at 3:19 | comment | added | roy smith | Matthew, I am teaching Euclid to brilliant 8-10 year olds, and seeing how useful similarity is. If one accepts it as a postulate many theorems are easy: Pythagoras, "power of the point", construction of a regular pentagon. Perhaps the message I am getting at last from your example, is that similarity is so important as a unifying principle that it should be introduced as early as possible. In logical sequence, one might prove classical Pythagoras, develop similarity with it (via Prop.III.35 of Euclid), then give this example of how similarity generalizes Pythagoras. thank you for this, roy | |
| Dec 14, 2010 at 0:59 | comment | added | Emerton | Dear Roy, Thanks for your thoughtful remarks. I agree that the ideas in this argument are quite sophisticated (so it is perhaps as much an example of depth of ideas in mathematics at of ingenuity). Best wishes, Matthew | |
| Dec 13, 2010 at 22:52 | comment | added | roy smith | This is a beautiful argument, similar to a problem in Harold Jacobs' lovely high school geometry book, but the original theorem of Pythagoras, as it first occurs in Euclid is different. The difference is in the meaning of the word "equal", which in Euclid does not mean same numerical area. Neither the concepts of area nor of proportion were available at that point in the book, so his less sophisticated proof is essentially to display compatible finite decompositions of the three figures involved. The wonderful proof here thus uses rather more sophisticated ideas which we now take for granted. | |
| Sep 29, 2010 at 8:58 | comment | added | gowers | I first came across this argument in Induction and Analogy in Mathematics, by Polya. As he says, the proof is due to Euclid, but this beautiful way of generating it may possibly be due to Polya himself. (He says that the relevant section of the book reproduces with minor changes a note of his in the American Mathematical Monthly of 1948.) I once discussed it in a public lecture and I think it served its purpose well. | |
| Sep 29, 2010 at 7:45 | history | edited | Emerton | CC BY-SA 2.5 |
added 223 characters in body
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| Sep 29, 2010 at 5:31 | comment | added | Emerton | Dear Chandan, Regarding your comment about perspective, it is precisely the fact that, of all non-scientists, painters might be one group of people who nevertheless know geometry, which caused me to ask my parenthetical question. | |
| Sep 29, 2010 at 5:30 | comment | added | Noah Snyder | I love this argument. Even if not everyone in your talks follows it, it'll be worth it for the people who do follow it. | |
| Sep 29, 2010 at 4:56 | comment | added | Chandan Singh Dalawat | I heard this first in a lecture by Hyman Bass many years ago. By the way, aren't painters supposed to know perspective, which beget projective geometry ? | |
| Sep 29, 2010 at 3:55 | history | answered | Emerton | CC BY-SA 2.5 |