10 ms·
Einstein's boyhood proof of the Pythagorean theorem (2015)
- Bitcoin_McPonzi 9y agoAnd yet, parents of children with learning disabilities will often falsely claim that Einstein was a dull child who didn't talk and failed his courses. This, of course, has no basis in fact. (For example: http://www.myspecialneeds.ie/blog/index.php/albert-einstein-had-a-learning-disability/ http://www.myspecialneeds.ie/blog/index.php/albert-einstein-... ) I'm not sure why advocates for the disabled need to make up lies about successful people.
- JasonFruit 9y agoWhile that's true, I'm not sure it contributes to the discussion of this article.
- thethirdone 9y agoIf you use noscript, the images of the proof will not appear making it nearly impossible to follow.
- tyingq 9y agoSympathetic, but noscript compatibility has been out the window for a decade at least. You can't reliably book travel, interact socially, read email, buy basic retail items, etc, without being extremely selective about providers. I'm with you philosophically, but that ship sailed long ago.
- S4M 9y agoThe thing is that there is no particular reason for non animated images to require javascript.
- deleted 9y ago[deleted]
- philtar 9y agoLazy loading
- subroutine 9y agoIs there a browser check that can be performed so a fallback can be implemented in the case of noscript?
- acqq 9y agoThis tag works on all browsers, no nned for a browser check: https://www.techonthenet.com/html/elements/noscript_tag.php https://www.techonthenet.com/html/elements/noscript_tag.php
- deathanatos 9y agoThey're <10KiB a piece, most of them. If the New Yorker would use the proper format of SVG here (or even PNG — the diagrams are all JPEGs), they would likely compress even better; most of them also include a ton of whitespace either side of the actual image. Testing one of them, the size falls by ~28% if you correct for all this. On mobile, where my connection is less reliable, I also have a strong distaste for lazy loading: it squanders all the time available from when I started reading until now. Then, when the image is finally in view, then we start the long haul of fetching it, and risk that I've again lost good connectivity.
- NamTaf 9y agoConversely, I like that it doesn't waste my data if I only get 1/2 way through an article and have to close it because I'm out and about and the world around me may interrupt me reading something on my phone.
- madez 9y agoWell, I'm rather extremely selective instead of doing something I consider wrong.
- tyingq 9y agoSure. That path, though, is degrading over time. Surfing the web without JS is becoming less, not more, achievable.
- acqq 9y agoIt's very easy to have JS only turned on only on a few selected sites. Interested parties should check uMatrix. It tells me that the page in question otherwise executes the scripts from the following sites: https://cdn.accelerator.arsdev.net/ https://player.cnevids.com/ https://c.amazon-adsystem.com/ https://segment-data.zqtk.net/ https://js-sec.indexww.com/ https://cdn.yldbt.com/ https://www.googletagservices.com/ https://cdn.optimizely.com/ https://assets.adobedtm.com/ https://d1z2jf7jlzjs58.cloudfront.net/ https://pixel.condenastdigital.com/ https://tag.bounceexchange.com/ https://cdn.optimizely.com/ https://assets.adobedtm.com/ There are still the trackers that work with the other methods that it can't prevent. But at least one still can be selective with the JS.
- jwilk 9y agoArchived copy that doesn't require JS: https://archive.fo/PkO3L https://archive.fo/PkO3L
- madez 9y agoWelcome to nginx! If you see this page, the nginx web server is successfully installed and working. Further configuration is required. For online documentation and support please refer to nginx.org. Commercial support is available at nginx.com. Thank you for using nginx.
- pharrington 9y agoAssuming your comment isn't in error, that message is being generated by a proxy somewhere between you and archive.fo.
- jwilk 9y ago(2015)
- S4M 9y agoThank you. I edited the title.
- SebNag_ 9y agoTrying to figure out the proof of a² + b² = c² by myself, without looking up the solution, was somehow exiting. Being exposed to a riddle and trying to find the solution is kinda cool. However, not solving it after 10 minutes left me feeling a bit dumb... :)
- jacobolus 9y agoThis is a general problem with mathematics education. See http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf Real (non-trivial, non-obvious) problems that someone hasn’t seen before can take hours, days, weeks, years, whole careers, or sometimes centuries to solve. Some of them later turn out to be impossible (and for many we still just don’t know). Real math education would have students grappling with relatively open-ended problems that take significant amounts of rumination and some cleverness to solve. It would explicitly encourage/reward close critical reading, creative brainstorming, planning, strategic thinking, generalization and specialization, executive control (e.g. time management), error checking, and clarity of exposition (including when asking for help after being stuck). There would be no shame in throwing out incorrect hypotheses, asking for clarification, getting stuck on a problem, making subtle mistakes which could serve as good examples for future improvement, etc. But skill and stamina at such work must be trained slowly, starting from an early age. The problem is that current (US) math education instead pre-chews everything, assigns students lists of exercises almost identical to what they saw someone solve before, and mostly tests memorization/recall and willingness to do the same trivial task over and over for hours despite being terribly bored, under purely extrinsic motivation. For people used to such math homework, the standard response to a single problem which takes >5 minutes to work through is to give up.
- Retric 9y agoDepends on what the goals are. I don't think most counties want a large fraction of their population becoming mathematicians. They basically want most of their population able to use math not discover new areas.
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- jordigh 9y agoI find this style a bit frustrating to read. I just want to know what his proof is, but there's so much fluff in the article that the actual meat is kind of hidden. So anyway, "his" proof is to drop a height to the hypotenuse and use similar triangles. Ah, right, I know this proof and I remember it as the canonical proof my grade 9 geometry textbook presented.
- deleted 9y ago[deleted]
- mlevental 9y agothis is the New Yorker not an maa article. it's supposed to be entertaining for a typical New Yorker reader. I swear man some people are so tone deaf it's frustrating: the whole world doesn't cater to people like us. at least try to make at attempt to empathize. edit: besides it's literally labeled steps 1-5 (i had no trouble skipping the "fluff" and finding the proof).
- jordigh 9y agoI'm sure other people like this style of article. I'm just telling you I don't. Even the steps 1-5 are a bit fluffy for me. If you had just told me "drop a height to the the hypotenuse and use similar triangles" I would have known what was meant, more succinctly. Do you have a suggestion on where to get unfluffed math news? I sometimes look at the AMS notices or Terry Tao's blog, but they don't cover quite the same breadth as things like Quanta magazine. I don't know where to get the big news for major events from an unfluffed source. Usually by the time the news has hit Quanta magazine or the NYT, it has already been circulating somewhere else long enough to be old news. I just don't know where else to look.
- mlevental 9y agohttp://www.scimagojr.com/journalrank.php?category=2603 http://www.scimagojr.com/journalrank.php?category=2603
- jungletime 9y agoI remember watching a documentary once, and it kind of made the case that Einstein wasn't an isolated genius, but more like Mark Zuckerberg. Basically organizing and putting together other scientists work,then unfairly getting all the credit. Anyone seen this documentary before? Its been years since I watched it, so memory is vague.
- malkarouri 9y agoFor what it's worth, organizing other scientists work from a new perspective and having a unified framework is the highest form of genius a scientist can achieve. That is why for example Einstein is credited with relativity rather than Poincare.
- moomin 9y agoA friend of mine used to describe this as a “one-line proof”. Never heard it ascribed to Einstein before, though.
- krick 9y agoI'm surprised the article doesn't begin by describing the weather and how Einstein was dressed while he was publishing this essay or whatever… Or how poor triangles felt while "Einstein continued his love affair with the Pythagorean theorem all his life". Unusually scarce description for the newyorker.
- blueline 9y agowhy are you making a comment that's completely irrelevant to the source text to attack a perception of the new yorker's writing style that isn't even present?
- pash 9y agoEinstein’s 1949 article in the Saturday Review is “Notes for an Autobiography” [0]. It’s worth reading. 0. https://archive.org/details/EinsteinAutobiography https://archive.org/details/EinsteinAutobiography
- herodotus 9y agoThe Ontario Science Centre once had a "proof" that was made using wooden strips that were about an inch wide and perhaps a quarter of an inch deep. There was a right angled triangle with squares on each side. It looked just like typical diagrams except that (a) it was mounted on the wall, (b) the triangle and the squares were covered with glass, (c) there was exactly enough coloured liquid inside to fill the square on the hypotenuse. Every minute or so, the triangle rotated in such a way as to allow all the water from the square on the hypotenuse to flow into the squares on the other two sides. Of course, the fit was exact. I remember watching the exhibit, thinking about it, and suddenly feeling a greater depth of understanding, not of geometry, but of trigonometry! Of course, not a proof, but a brilliant exhibit. (Here is a similar demo on youtube: https://www.youtube.com/watch?v=CAkMUdeB06o https://www.youtube.com/watch?v=CAkMUdeB06o)
- abtinf 9y agoThat is a great demo. Why isn't it a proof? In a sense, isn't a concrete fact of reality better than an abstract proof?
- tempestn 9y agoHad to stop reading long enough to see if I could prove it myself using similar triangles. This is definitely cleaner than what I came up with though, which involved creating a similar triangle with side b1 set equal to side c of the original.
- signa11 9y ago> Einstein’s predictions, during a solar eclipse in 1919—was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.” for another side of eddington, see 'empire of the stars' [https://www.amazon.com/Empire-Stars-Obsession-Friendship-Betrayal/dp/061834151X https://www.amazon.com/Empire-Stars-Obsession-Friendship-Bet...] quite a fun read, i picked it up on a whim, and could not just put it down over the course of a 8hr train ride :)
- mjburgess 9y agoYou may wish to include the entire sentence: > When Arthur Eddington—the British astrophysicist who led the team that confirmed Einstein’s predictions, during a solar eclipse in 1919—was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.” It's hard to understand without it.
- mayankkaizen 9y agoIf you are alluding to some nuance, I probably missed it.
- skj 9y agoIn the first quote it is implied that Einstein is being asked the question. Then it isn't clear who Eddington was. The questioner, I thought at first. Then no, a phone typo of Einstein? I found it confusing and the larger quote cleared it up.
- deleted 9y ago[deleted]
- Hasz 9y agoEvery time Pythagoras is brought up, I can't help but think of the cult of Pythagoras as well. One of the tenants is the refutation of beans. For what reason, I have no idea. I've always found it strange that a mind so capable was also equally capable of such folly.
- EGreg 9y agoWhat?
- icebraining 9y agoApparently Pythagoras had a following (a cult) and one of the rules was not eating beans: https://classicalwisdom.com/cult-of-pythagoras/ https://classicalwisdom.com/cult-of-pythagoras/ Or maybe that meant not voting, since the Greeks used colored beans for voting.
- Rapzid 9y agoPerhaps he didn't want them eating the beans and losing their vote.
- rocqua 9y agoThe same cult found a proof that sqrt(2) is irrational, and kept it hidden because they believed natural numbers and their ratio's to be everything. Then some dude Hippasus found the same proof and talked about it openly. The pythagorean cult murdered him for it. (I checked wikipedia [1] and apparently, historians disagree on whether this is true) [1] https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus
- chx 9y agoSince we are on HN and talking of the Pythagorean theorem, I must mention E. W. Dijkstra's proof. https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E... he proves the generalization sgn(α + β - γ) = sgn(a² + b² - c²) also by using similar triangles. Incredibly neat.
- elzr 9y agoWhoa, that IS neat. Thanks for sharing! A key part of his reformulation is π = α + β + γ (the sum of the internal angles of a triangle equal 2 right angles). That statement is, funny enough, equivalent to the parallel postulate. Going on a tangent now: lately I've been thinking that the "dead horse" of the Pythagorean theorem is actually trying to tell us that flat (as opposed to fractal!) dimensions come from composable self-similarity (squares).
- chx 9y agoYeah if the parallel postulate is out then this obviously doesn't stand. Starting from the North Pole, walking down the prime meridian to the equator then turning right and walking along the equator finally turning right to the north again at the right point so you walk through New Orleans creates a triangle on a sphere with three right angles...
- logicallee 9y agoThis is the most succcinct proof, it's an ancient Chinese visual proof: https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras7.gif https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras... It's not labelled this way but notice that the BIG squares are identical, having sides of a+b. Only the four triangles are rearranged. Just subtract the four identical triangles. In the second arrangement, a c^2 area is present in addition to the same four triangles. In the first arrangement this had been rearranged to an a^2 and a b^2 area. It's important to assure yourself the triangles are the same and the big square is the same - there is no tiny hidden sliver or something, and right angles are preserved, there are no shenanigans. second explanation of same: http://www.math.toronto.edu/colliand/notes/pythagoras.html http://www.math.toronto.edu/colliand/notes/pythagoras.html
- ndr 9y agoThis is beautiful. To be honest the second image with the tilted c² is enough. From that picture alone you can figure the outer square has the same area as the inner square + 4 times the triangle area: (a+b)² = c² + 4(ab/2) a² + b² + 2ab = c² + 2ab a² + b² = c²
- logicallee 9y agoThanks. Your proof works, I could follow it. Yours does use some elementary algebra[1], which wasn't used in the Chinese geometric proof. I wonder if ancient Chinese mathematicians could simplify (a+b)^2 symbolically, or even did symbolic algebra this way? (When I referred to "ancient Chinese visual proof" I wasn't bullshitting, but I didn't find a source with the exact 2-part picture, though this makes same claim using same pictures: http://www.researchhistory.org/2012/10/24/earliest-evidence-of-pythagoras-theorem/ http://www.researchhistory.org/2012/10/24/earliest-evidence-... ) I'm sure they knew that the area of triangles, which you also use, is (ab/2) - but the purely visual proof needs nothing more than the knowledge that the area of a square is the square of the length of its sides. (And I guess some obvious facts like that no matter how you divide an area the sum of the areas of its parts will be same - the reason I mention tiny slivers is in some fake geometric proofs this intuitive knowledge is abused. For example, see this excellent description: https://en.m.wikipedia.org/wiki/Missing_square_puzzle https://en.m.wikipedia.org/wiki/Missing_square_puzzle Before you open the solution, you can look at the trick for as long as you want, you won't figure it out.) [1] https://en.m.wikipedia.org/wiki/FOIL_method https://en.m.wikipedia.org/wiki/FOIL_method
- garyrob 9y agoCan anyone point to a proof that for similar non-isosceles right triangles, the area is proportional to the square of a side? Can that be generalized to other shapes?
- jonsen 9y agoWhen you scale a figure, say by doubling all distances, the area quadruples. Scale by factor k and the area goes up by a factor k^2. So if the area proportion between the triangle and the square is T:S, and we scale both figures by the same factor k, we get areas k^2* T and k^2* S respectively. But k^2* T:k^2* S = T:S.