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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)

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where 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 Dec 14 '09 at 23:57
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I hope I am not alone in being (usually) unable to appreciate "proof by picture"... –  Suvrit Jul 8 '11 at 21:14
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@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 Jul 9 '11 at 12:11
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I 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-Alvarez Sep 16 '11 at 17:34
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My opinion is that almost every proof-without-words is improved by a few well-chosen words. –  Joel David Hamkins Feb 12 '12 at 0:47
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62 Answers

(I'd post this as a comment to Mariano Suárez-Alvarez, but I've not enough rep). From a ME thread.

$$\sum_{k=1}^n (-1)^{n-k} k^2 = {n+1 \choose 2} = \sum_{k=1}^n \; k = \frac{(n+1) \; n}{2}$$

alt text

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                          visual proof

This visual proof of $$\sum\limits_{n=1}^\infty \left (\frac{1}{2}\right)^{\,2n}=\frac{1}{3}$$ is from http://www.cecm.sfu.ca/~loki/Papers/Numbers/ (Visible Structures in Number Theory, by Peter Borwein and Loki Jorgenson, The American Mathematical Monthly, vol. 108, no. 5, 2002, pp. 897-910).

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A line that bisects the right angle in a right triangle also bisects a square erected on the hypotenuse:

http://www.futilitycloset.com/wp-content/uploads/2011/09/2011-09-12-half-and-half-2.png

source: http://www.futilitycloset.com/2011/09/12/half-and-half/

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  • The first homotopy group of SO_3 has an order 2 element (that's a classic).

  • The surface area of a quarter of the unit sphere is Pi via Gauss-Bonnet (My source is Ariel Shaqed - it should have been a classic, but no one I asked seems to knew it). The sphere is what you reach with a straight hand while standing still. Hold a Pencil in your hand, that's your tangent vector. Now parallel transport the pencil on a quarter sphere: it points in the opposite direction. QED

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For SO(3) has order 2 element: gregegan.customer.netspace.net.au/APPLETS/21/21.html –  Dan Piponi Dec 14 '09 at 15:29
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Place a glass on the open palm of your hand. You can, with a bit of practice, rotate the glass twice (but not once) around the vertical axis without spilling any liquid from it, and return to your original position. Each part of your body goes through a loop in SO_3. Moving from the shoulder via the arm to the glass, you get a homotopy essentially proving the theorem. I have seen dancers from somewhere in south-east Asia incorporating this move into their dance. –  Harald Hanche-Olsen Dec 14 '09 at 21:21
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Why are there so many words and so few pictures in this answer? –  David Eppstein Dec 14 '09 at 23:07
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@David: well, you can think if this answer (or of Harald's comment, which gets my emphatic upvote) as a script for the choreography which, when acted out, is a proof without words :P –  Mariano Suárez-Alvarez Dec 15 '09 at 0:00
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It doesn't feature Feynman, but here's a video of a human doing the plate trick (just after 1 minute in): youtube.com/watch?v=CYBqIRM8GiY –  Harrison Brown Dec 15 '09 at 2:42
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Proof of the associativity law $f * (g * h) = (f * g) * h$ in the fundamental groupoid of a topological space:

http://img845.imageshack.us/img845/3563/passj.gif

You can find more of these diagrams in J. P. May's A Concise course in algebraic topology.

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I just saw this proof, which is of course not mine.

link text

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I like the tiling proof of the Pythagorean Theorem. The left image is credited to Al-Nayrizi and Thābit ibn Qurra (9th century) and the right by Henry Perigal (19th century).

**

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I am surprised that no one had cited the "proof" that the rationals are countable yet. See, for example, this picture

enter image description here

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I think that the fact that the rationals are countable qualifies as non-trivial, when put in historical perspective –  Geoff Robinson Mar 19 at 19:02
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There a proof of Erd˝os-Mordell Inequality 'without words' is an impressive one. Please follow the link http://forumgeom.fau.edu/FG2007volume7/FG200711.pdf

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The idea is to prove things in ways that are obvious to different parts of your brain, right? Anyone found any "auditory proofs"? Some candidates -

  1. Nyquist sampling theorem?

  2. sin[a] + sin[b] = 2sin[(a+b)/2]cos[(a-b)/2]. If you use at and bt instead of a and b, you can translate that to show how the addition of two sine tones close in frequency can also be perceived as a modulation or "vibrato" around the centre frequency. The factor of 2 might be hard, though you can add a gain instead of 2 and show that the difference is silence when the gain is 2 :)

  3. Sampling in frequency domain (comb filter) is periodicity in time domain?

Here are some "audio illusions" though, for your amusement - http://www.youtube.com/watch?v=e6JSTkwXg90

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Are there more details on 1? 2 and 3 don't seem like proofs so much as examples, though maybe you are just putting them forth as challenges. –  j.c. Mar 7 '10 at 11:54
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There is a beautiful proof of the fact that a checkerboard with sides $2^{n}$, and one square removed can be tiled with $L$-shaped pieces formed by three squares. Given that a checkerboard of sides $2^{n-1}$ can be so tiled, then a square checkerboard of sides $2^{n}$ can be tiled by filling in the quarter in which the removed piece lies, and then placing an extra $L$-shaped tile with one square in each of the remaining three quarters.

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Rich Schwartz had on his site a great paper consisting of only a picture which proved that every right triangle admits a periodic billiard path. Unfortunately, he's since deleted it, so I can't post it here. (It shouldn't take too long for anyone interested to re-construct the proof, though.)

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I am guessing he did it by assembling four of the said right-triangles into a parallelogram. There is a path that bounces directly between the two longer sides. Mod out by the symmetry and you get a periodic path in the triangle. –  Willie Wong Mar 11 '10 at 17:04
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Can you tile an 8x8 chessboard with one corner cut off with dominoes of dimension 3x1?

alt text

This is a simple way to show that choosing a useful coloring can make a proof trivial.

This proof was also a result of the Conjecture and Proof class in the Budapest Semesters in Mathematics. It was one of the first problems encountered there, hence not that hard :)

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The pathspace of any topological space is contractible.

Pf (as given in my homotopy theory class): slurp spaghetti.

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Also elementary, but here is a proof that

$C_n = \binom{2n}{n} - \binom{2n}{n+1} = \frac{\binom{2n}{n}}{n+1},$

where $C_n$ is the $n$th Catalan number.

http://utdallas.edu/~hagge/images/Catalan.pdf

Sorry for the link; new users may not use image tags.

Here's the image:

alt text

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Do you have an explanation for the picture? I looked at it, and looked at it, and don't get it. –  Willie Wong Mar 11 '10 at 16:38
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Sorry for not noticing your question (much) earlier. The differences between adjacent terms in Pascal's triangle form another triangle which obeys the same generation rules. In my picture of that triangle, the yellow squares count some of the downward paths on a square grid which has been rotated $45^\circ$, namely those that never fall to the left of the top square. One definition of $C_n$ is that it is the number of such paths which terminate at the bottom corner of an $n \times n$ grid. –  Tobias Hagge Oct 26 '10 at 5:46
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Let $0\leq x,y,z,t\leq1$ Prove that $x(1-y)+t(1-x)+z(1-t)+y(1-z)\leq 2$.

Draw a 1x1 square and mark in consecutive sides disjoint segments starting at the vertexes of lengths $x,y,z,t$. Joining the consecutive end points of the intervals that are not vertexes of the square form four triangles, the area of the triangles is the left hand side divided by 2, the area of the square is the right hand side divided by 2.

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This proof without words has an awful lot of them! –  I. J. Kennedy Apr 30 '11 at 17:39
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This is apparently not was intended, but I think it qualifies. From Principia Mathematica: the proof of 1+1=2 (I can't include the image bc I'm a new user, but perhaps an experienced user can edit this answer for me.)

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Here are some dynamic versions:

http://www.math.utah.edu/~palais/sums.html (two of the summation formulas mentioned above)

Several belt, plate, and tangle trick animations:

http://www.math.utah.edu/~palais/links.html

A visual derivation of complex multiplication:

http://www.math.utah.edu/~palais/newrot.swf

Pythagoras in the Isosceles case, based on the Yale tablet:

http://www.math.utah.edu/~palais/PythagorasIsosceles.html

and the general case:

http://www.math.utah.edu/~palais/Pythagoras.html

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Interesting how everyone understands "proof without words" as "proof made of pictures". I read the title and immediately thought that every proof can be written without words, using just first order logic. I stopped there and thought that this is just another language, using different words - and I came to the conclusion that there can not be a mathematical proof without "words", because you have to get some information across! Sure, you can use different languages than English. But in the end, this boils down to the question, what is a word?

BTW: Unmentioned so far are category-theoretical proofs, which can sometimes be expressed very comprehensively as a sequence of diagrams. I am too lazy to look up a good example, because I already explained that I don't believe in the question.

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Formulas (first order or what not) are just words written in abbreviated form. That really does not count... –  Mariano Suárez-Alvarez Jul 6 '11 at 1:01
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So are pictures. That's my point. –  Jesko Hüttenhain Aug 9 '11 at 17:35
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I doubt there is any sense in which one can formalize the notion, but I think it is pretty clear that a proof written in the first order calculus, or any other calculus, is simply not a "proof without words". You do not believe in the question, you say, but I honestly cannot understand what that can possibly mean: there is certainly something that gets the name proof-without-words (there is even a section in the MAA Monthly dedicated exclusively to this, and it has run for decades!) and most people ---while probably not being able to explain exactly what they are--- recognize them. –  Mariano Suárez-Alvarez Sep 16 '11 at 17:33
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This proves the Minkowski version of the Pythagorean theorem:

alt.text

$c^2 = a^2 - b^2$

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From Wikipedia: here is a "proof without words" of the Yoneda Lemma.

alt text

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This answer has already been proposed, and after some discussion it was more or less agreed that this is not a proof-without-words in standard sense of the term. –  Mariano Suárez-Alvarez Oct 1 '11 at 23:45
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alt text

The composition of two continuous mappings is continuous.

Bloody thing won't let me embed the image...

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I suggest the videos of Viennot explaining the bijections between different families of objects counted by Catalan numbers:

http://web.mac.com/xgviennot/Cont_Science/vid%C3%A9os.html

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Here's a proof of the area of a circle (or sector) which is different from the one posted previously.

EDIT: I was unable to embed the file, which is in pdf form. Here is a link:

http://wildpositron.files.wordpress.com/2011/04/sectorarea2.pdf

I discussed what goes into making the proof complete to show that the map preserves area on my blog here (it requires just another picture or two, but it's essentially still only a geometric argument):

http://wildpositron.wordpress.com/2011/04/05/calculating-the-area-of-a-sector/

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This is not entirely without words, but Byrne's edition of Euclid's elements has cut down the number of words to a bare minimum.

http://www.math.ubc.ca/~cass/Euclid/byrne.html

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This is not quite in the spirit of the question... –  Mariano Suárez-Alvarez Sep 16 '11 at 17:27
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+1: Thanks for this wonderful and beautiful link (be it in the spirit of the question or not). –  Hans Stricker Sep 16 '11 at 18:03
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Of cause, this is not intuitive and it isn't elementary at all, but when I was asked to give the shortest proof of $\pi_1(S^1)=\mathbb{Z}$ I could imagine I answered $S^1\cong\mathbb{R}/\mathbb{Z}$.

I have to apologize if this becomes an example of misunderstanding the question, but I wanted to state that in my opinion "Proof without words" doesn't need to mean "Proof with picture".

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For $0 \lt k \lt n$,

$$\binom{n}{k} = \frac{n}{n-k}\binom{n-1}{k}$$


How k-subsets of [n], marked dark green in the rows, come from k-subsets of [n-1] after n-fold duplication and rearrangement:

alt text Exactly $n-k$ times:

alt text
By induction, a base case, and taking $k=n$ and $k=0$ for granted: $$\binom{n}{k} = \frac{n}{(n-k)} \frac{(n-1)!}{(n-1-k)!\ k!} = \frac{n!}{(n-k)!\ k!}$$

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