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Can you give examples of proofs without words? In particular, can you give examples of proofs without words for *non-trivial* results?

(One could ask if this is of interest to mathematicians, and I would say yes, in so far as the kind of little gems that usually fall under the title of 'proofs without words' is quite capable of providing the aesthetic rush we all so professionally appreciate. That is why we will sometimes stubbornly stare at one of these mathematical autostereograms with determination until we joyously *see* it.)

(I'll provide an answer as an example of what I have in mind in a second)

2where possible could people also either note the image source or explain/provide a link to a "how to" for constructing the associated diagram? I think that such would also be helpful for folks` – Carter Tazio Schonwald – 2009-12-14T23:57:12.797

9I hope I am not alone in being (usually) unable to appreciate "proof by picture"... – Suvrit – 2011-07-08T21:14:50.717

14@Suvrit: I hope I am not alone in being most often unable to appreciate "proof by word" until I've read it at least twenty times and wrestled with it for many days per page! – WetSavannaAnimal aka Rod Vance – 2011-07-09T12:11:51.877

@Mariano - sorry, I'm blind in one eye, so I never did get those autostereograms. Great question, though! – WetSavannaAnimal aka Rod Vance – 2011-07-09T12:13:06.860

@Rod Vance -- I'm sure you are not alone; I think my difficulty with "pictures" lies in my lack of geometric abilities. – Suvrit – 2011-07-09T18:40:12.763

This question gives Mariano the sixth and seventh golden badges!! Congratulations!! – Gil Kalai – 2011-08-26T11:34:57.723

1Can we close this as being no longer relevant? The answers trickling in now are not really proofs without words. – David Roberts – 2011-09-16T04:03:06.457

2I am actually quite fond of this question, David! I tend to make comments on answers that are not relevant, and they have a tendency to get deleted after that. – Mariano Suárez-Álvarez – 2011-09-16T17:34:58.787

@Mariano - ok. Just a thought. Other 'borderline' (read: not particularly research-level) big-list questions that have been extremely popular are slowly getting closed, but if this one is still getting good answers, then good. – David Roberts – 2011-09-16T23:30:18.587

41My opinion is that almost every proof-without-words is improved by a few well-chosen words. – Joel David Hamkins – 2012-02-12T00:47:24.703

@one eyed and blind in one eye people curious about autostereograms: http://i39.tinypic.com/14nzlt0.gif [animated gif based on http://en.wikipedia.org/wiki/File:Stereogram_Tut_Random_Dot_Shark.png, licensed CC BY-SA 3.0]

– Roberto Mizzoni – 2012-02-16T03:28:45.3133Why has this question been closed? How can it be "no longer relevant"? (I mean: the longer people contribute examples, the better.) (BTW: I

dohave an astonishing example.) – Hans Stricker – 2013-03-15T17:53:02.947I can't post since I do not have enough rep (and I probably won't ever have as I am not such a math guy) but I was surprised not to find the proof of pythagorean theorem through similar triangles (by far my favourite) : http://is.gd/dLjpjQ.

– matovitch – 2014-06-21T15:12:57.353It is funny that most people understood "proof without words" as "proof with pictures only"; I guess it is not hard to find proof with computations only! (Though, these ones are in average less seducing...) By the way, do you have in mind other ways to "prove" things than words, pictures and computations? – user56097 – 2014-10-07T23:47:06.000

6There is no such thing as a "proof without logic," and since words are usually the best tool for conveying logical relations, I'm going to have to reject the idea of "proof without words." Sorry, -1. – goblin – 2015-01-23T03:14:01.777

3@goblin, I am afraid that you have completely misunderstood the concept. The idea is pictures which have the rather amazing capability of immediately suggesting on the mind of the viewer the idea of a proof. How on earth you managed to get from the rather well-known idea involved in this question to «proofs without logic» is a mystery to me. – Mariano Suárez-Álvarez – 2015-01-23T03:55:53.287

@MarianoSuárez-Alvarez, oh the concept is well-known alright; these useless so-called "proofs without words" are all over YouTube, usually paired with a lot of downvotes, and rightly so. Its sad that so much effort went into discovering these beautiful arguments, and then producing pictures and even animations to illustrate the idea, only to have all that hard work spoilt by this proof without words nonsense. How much better those so-called "proofs" would have been with a few premises, some inferences, and a conclusion. – goblin – 2015-01-23T22:45:27.980

Also, I am afraid you have misunderstood me. If I were to post a truth-tree for some logical tautology, well that would be a literal example of a "proof without words"; but, you would surely reject it as a non-example. Hence what you really mean is not "proof without words" but "proof without logic." – goblin – 2015-01-23T22:46:24.603

3If you cannot tell the difference between a proof-tree and a proof without words in the tradition of, say, the AMM Monthly, then that is clearly a limitation of yours. I would rather you start a meta thread, or a blog, instead of further polluting this thread with what is clearly rather orthogonal chatter. – Mariano Suárez-Álvarez – 2015-01-23T22:52:36.347

@WetSavannaAnimalakaRodVance I want to upvote your comment again and again... – math137 – 2016-01-04T22:29:52.987

1

Some proofs without words can be found on Math.SE among posts tagged proof-without-words.

– Martin Sleziak – 2016-12-04T08:06:07.303i recall that Mathematics Magazine, a more elementary cousin of mathematical Monthly (and also published by A.M.A.) used to have a fairly regular feature titled Proofs Without Words.... with some surprising pictures. – DanielWainfleet – 2016-12-15T12:59:52.557

2

Can someone with enough rep submit that 1/4 + (1/4)^2 + (1/4)^3 + ... = 1/3?

– Dan Dascalescu – 2017-02-17T07:44:35.647Unfortunately i don't have enough reputation to submit https://upload.wikimedia.org/wikipedia/commons/8/8b/Visualized_estimate_for_the_Cauchy_condensation_test.png

– Daniel Charry – 2017-12-24T03:46:44.040