A Beautiful Theorem Deserves a Beautiful Proof
January 9, 2016 1:22 PM Subscribe
Douglas Hofstadter presents a proof Napoleon's theorem (on equilateral triangles constructed on the sides of another triangle), in the form of a sonnet. (Part of a longer talk; the link should take you to 34:18 in the video.)
Not so much 'a proof of' as 'a description of' but still immense fun and thank you for posting this.
*rewinds to 0:00 in the video...*
posted by motty at 4:56 PM on January 9, 2016 [3 favorites]
*rewinds to 0:00 in the video...*
posted by motty at 4:56 PM on January 9, 2016 [3 favorites]
If I would have had this in Geometry I would have dug Geometry SOOOOOOO much more and probably stuck with it more than I did.
Or maybe I'm just older wiser and know when to stop being contrarian and STFU and not be smarter than the teacher. But no - I think this would have been cool as fuck and inspired me.
So - what happens in three dimensional space in the process of "folding" the three triangles? Is that part of the "rotation" of which he speaks? I also like the idea of rotation being somehow linked to the value of the surface area (such that rotating one way gives a positive surface area and the other way a negative surface area (at least if my drunk mind is interpreting some of what he's saying correctly)) I believe this is like a zero-point/180 point where you end up with the negative value. I assume this is related to trig where you have negative angles? I'm very very sucky at math so I only understand general concepts and what little I played with for game programming. But... visually it seems to make sense and is so damn intriguing to me...
posted by symbioid at 6:18 PM on January 9, 2016 [1 favorite]
Or maybe I'm just older wiser and know when to stop being contrarian and STFU and not be smarter than the teacher. But no - I think this would have been cool as fuck and inspired me.
So - what happens in three dimensional space in the process of "folding" the three triangles? Is that part of the "rotation" of which he speaks? I also like the idea of rotation being somehow linked to the value of the surface area (such that rotating one way gives a positive surface area and the other way a negative surface area (at least if my drunk mind is interpreting some of what he's saying correctly)) I believe this is like a zero-point/180 point where you end up with the negative value. I assume this is related to trig where you have negative angles? I'm very very sucky at math so I only understand general concepts and what little I played with for game programming. But... visually it seems to make sense and is so damn intriguing to me...
posted by symbioid at 6:18 PM on January 9, 2016 [1 favorite]
Ah, I saw this in person! It was a very cool lecture and though I couldn't follow a lot of it, it was still beautiful and I'm glad I made the trip.
posted by wym at 6:25 PM on January 9, 2016 [2 favorites]
posted by wym at 6:25 PM on January 9, 2016 [2 favorites]
Loopy though he may be, his Pulitzer Prize winning book Godel, Escher & Bach remains as salient as ever - written as an Associate Professor, no less. Hats off to a great mind.
posted by onesidys at 10:53 PM on January 9, 2016 [1 favorite]
posted by onesidys at 10:53 PM on January 9, 2016 [1 favorite]
Yes, one of my favorites. Strange and loopy.
(see what I did there?)
posted by Joseph Gurl at 11:18 PM on January 9, 2016 [3 favorites]
(see what I did there?)
posted by Joseph Gurl at 11:18 PM on January 9, 2016 [3 favorites]
Aw, he's so great. (Although I agree with motty, the sonnet isn't a proof; still, hearing it laid out in verse makes the theorem much more memorable.) Thanks for sharing this.
I'm surprised he never took geometry in high school! It might have been a really lifechanging experience for him. I wasn't developmentally ready for geometry when I took it, but the sheer niftiness of a lot of the geometric proofs I had to construct in my 8th grade geometry class hits me out of the blue sometimes even now. I think it's a subject he'd have really liked as a youngster but maybe coming to it on his own as an adult he was even better able to appreciate its pleasures.
posted by town of cats at 2:37 PM on January 10, 2016
I'm surprised he never took geometry in high school! It might have been a really lifechanging experience for him. I wasn't developmentally ready for geometry when I took it, but the sheer niftiness of a lot of the geometric proofs I had to construct in my 8th grade geometry class hits me out of the blue sometimes even now. I think it's a subject he'd have really liked as a youngster but maybe coming to it on his own as an adult he was even better able to appreciate its pleasures.
posted by town of cats at 2:37 PM on January 10, 2016
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posted by Joseph Gurl at 4:45 PM on January 9, 2016 [4 favorites]