Proofs without words - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-23T10:41:56Z http://mathoverflow.net/feeds/question/8846 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8846/proofs-without-words Proofs without words Mariano Suárez-Alvarez 2009-12-14T06:04:14Z 2012-12-02T07:18:26Z <p>Can you give examples of proofs without words? In particular, can you give examples of proofs without words for <em>non-trivial</em> results?</p> <p>(One could ask <a href="http://mathoverflow.net/faq#whatquestions" rel="nofollow">if this is of interest to mathematicians</a>, 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 <a href="http://en.wikipedia.org/wiki/Autostereogram" rel="nofollow">autostereograms</a> with determination until we joyously <em>see</em> it.)</p> <p>(I'll provide an answer as an example of what I have in mind in a second)</p> http://mathoverflow.net/questions/8846/proofs-without-words/8847#8847 Answer by Mariano Suárez-Alvarez for Proofs without words Mariano Suárez-Alvarez 2009-12-14T06:05:09Z 2010-05-17T11:48:32Z <p>A proof of the identity $$1+2+\cdots + (n-1) = \binom{n}{2}$$</p> <p><img src="http://img31.imageshack.us/img31/919/triangle1y.png" alt="alt text"></p> <p>(Adapted from an entry I saw at <a href="http://demonstrations.wolfram.com/" rel="nofollow">Wolfram Demonstrations</a>)</p> http://mathoverflow.net/questions/8846/proofs-without-words/8849#8849 Answer by Mike for Proofs without words Mike 2009-12-14T06:13:11Z 2009-12-14T06:13:11Z <p>As you probably already know — there are loads of these in <a href="http://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006" rel="nofollow">Proofs without Words</a> (and II) by Roger Nelson.</p> http://mathoverflow.net/questions/8846/proofs-without-words/8851#8851 Answer by Mike for Proofs without words Mike 2009-12-14T06:29:44Z 2009-12-14T06:39:31Z <p>This is elementary as well, but one of my favorite ones :)</p> <p>$1^2 + 2^2 + \dots + n^2 = \frac13n(n+1)(n+\frac12)$</p> <p><em>(Author: Man-Keung Siu)</em></p> <p><img src="http://img163.imageshack.us/img163/6605/mankeungsiu.png" alt="alt text" /></p> http://mathoverflow.net/questions/8846/proofs-without-words/8852#8852 Answer by David Lehavi for Proofs without words David Lehavi 2009-12-14T07:05:44Z 2009-12-14T07:05:44Z <ul> <li><p>The first homotopy group of SO_3 has an order 2 element (that's a classic).</p></li> <li><p>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</p></li> </ul> http://mathoverflow.net/questions/8846/proofs-without-words/8857#8857 Answer by Alon Amit for Proofs without words Alon Amit 2009-12-14T07:53:45Z 2012-12-02T07:18:26Z <p>The cover of Peter Winkler's first book is a great proof without words of a statement which I'll leave you to guess, regarding the combinatorics of tiling a heaxagon with rhombi.</p> <p>EDIT: I think the guessing game isn't helpful. The statement is that when tiling a perfect hexagon with the appropriate kind of rhombi of various orientations, the number of tiles in each orientation is the same. The image is slightly misleading in its use of color; there ought to be just three colors, corresponding to the three orientations.</p> <p><img src="http://i.imgur.com/OHqeo.jpg" alt="Picture"></p> http://mathoverflow.net/questions/8846/proofs-without-words/8858#8858 Answer by Alon Amit for Proofs without words Alon Amit 2009-12-14T08:05:58Z 2010-05-15T13:30:46Z <p>In an attempt to push the bar towards the non-trivial, I'll mention the proof that the boundary complex of every polytope is shellable. The proof is virtually word-free but requires an actual movie rather than a still image: imagine yourself in a spaceship, taking off in a straight line from one of the facets, away from the polytope. Every once in a while a new facet is visible to you; under assumptions of general position, this provides a shelling of the complex (obviously, you need to fly off to projective infinity and come back on the other side). </p> <p>This was assumed by Euler but first proved only in 1970 by Brugesser and Mani, who said that the idea came to him in a dream. More details <a href="http://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/" rel="nofollow">here</a> (search for "shellability") or <a href="http://gilkalai.wordpress.com/2008/09/18/annotating-kimmo-erikssons-poem/" rel="nofollow">here</a>. </p> http://mathoverflow.net/questions/8846/proofs-without-words/8880#8880 Answer by Steve Flammia for Proofs without words Steve Flammia 2009-12-14T15:35:33Z 2011-07-08T03:17:07Z <p>Wikipedia has a few nice proofs of the pythagorean theorem. Elementary, but elegant. </p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/220px-Pythagoras-2a.gif" alt="Pythagorean Theorem, picture proof"></p> http://mathoverflow.net/questions/8846/proofs-without-words/8881#8881 Answer by Jason Dyer for Proofs without words Jason Dyer 2009-12-14T15:50:16Z 2009-12-14T15:58:08Z <p>The cardinality of the real number line is the same as a finite open interval of the real number line.</p> <p><img src="http://numberwarrior.files.wordpress.com/2009/11/proofwowords.png" alt="Proof without words" /></p> http://mathoverflow.net/questions/8846/proofs-without-words/8883#8883 Answer by Harrison Brown for Proofs without words Harrison Brown 2009-12-14T16:08:51Z 2009-12-14T18:57:22Z <p>There's a picture proof in the <em>Princeton Companion</em>, or alternatively on p. 340 of <a href="http://www.math.cornell.edu/~hatcher/AT/AT.pdf" rel="nofollow">Hatcher</a>, of the fact that the higher homotopy groups are abelian. Actually, here's a screenshot of the one in Hatcher (hopefully fair-use!):</p> <p><img src="http://img509.imageshack.us/img509/10/abelian.jpg" alt="alt text" /></p> <p>Here f and g are mappings (with basepoint) of $S^n$ into some space for n > 1; the picture shows a homotopy between f + g and g + f. </p> http://mathoverflow.net/questions/8846/proofs-without-words/8923#8923 Answer by Aaron Mazel-Gee for Proofs without words Aaron Mazel-Gee 2009-12-15T00:19:29Z 2009-12-15T00:19:29Z <p>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.)</p> http://mathoverflow.net/questions/8846/proofs-without-words/8932#8932 Answer by Steven Gubkin for Proofs without words Steven Gubkin 2009-12-15T01:31:55Z 2011-02-18T20:10:57Z <p>Here is the very first piece of original mathematics I ever did, in high school:</p> <p>The derivative of sine is cosine.</p> <p><img src="http://www.freeimagehosting.net/uploads/d5940f69d2.jpg" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/8937#8937 Answer by Agol for Proofs without words Agol 2009-12-15T01:52:27Z 2009-12-15T01:52:27Z <p><a href="http://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow">Sphere eversion</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/8939#8939 Answer by Darsh Ranjan for Proofs without words Darsh Ranjan 2009-12-15T02:32:12Z 2009-12-22T10:54:43Z <p>Here's a proof of the inequality of the arithmetic and geometric means in the form $$\frac{x_1^n}{n} + \cdots + \frac{x_n^n}{n} \geq x_1\cdots x_n.$$ </p> <p>Proof for $n=3$:</p> <p><img src="http://img64.imageshack.us/img64/5738/arithgeom01b.png" alt="(there should be a figure here...)" /></p> <p>The "figure" for general $n$ is similar, with $n$ right pyramids, one with an $(n-1)$-cube of side length $x_k$ as its base and height $x_k$ for each $k=1,\ldots,n$. </p> <p>(I made this in <a href="http://www.inkscape.org/" rel="nofollow">Inkscape</a>, a wonderful free-software vector drawing application. For the inequality and associated labels, I used the <a href="http://www.elisanet.fi/ptvirtan/software/textext/" rel="nofollow">textext</a> extension.)</p> http://mathoverflow.net/questions/8846/proofs-without-words/9099#9099 Answer by Igor Khavkine for Proofs without words Igor Khavkine 2009-12-16T11:32:32Z 2009-12-16T12:31:40Z <p>Duality between $\ell^1$ and $\ell^\infty$ norms.</p> <p><img src="http://img705.imageshack.us/img705/1970/pnorm1oo.gif" alt="http://img705.imageshack.us/img705/1970/pnorm1oo.gif" /></p> <p>and the reverse animation</p> <p><img src="http://img689.imageshack.us/img689/6233/pnorminfoo.gif" alt="http://img689.imageshack.us/img689/6233/pnorminfoo.gif" /></p> http://mathoverflow.net/questions/8846/proofs-without-words/9381#9381 Answer by Gil Kalai for Proofs without words Gil Kalai 2009-12-19T17:45:58Z 2009-12-19T22:15:35Z <p>Q: Can you tile <img src="http://www.cl.cam.ac.uk/~mj201/images/mutilated-chess-board2.gif" alt="alt text" /> </p> <p>with <img src="http://www.stewart.hinsley.me.uk/Fractals/Scripts/Polyplet.php?r0=3" alt="alt text" /> ?</p> <p><img src="http://numberwarrior.wordpress.com/files/2009/12/tilingproof.png" alt="Tiling proof" /></p> http://mathoverflow.net/questions/8846/proofs-without-words/9548#9548 Answer by aorq for Proofs without words aorq 2009-12-22T16:54:49Z 2009-12-22T16:54:49Z <p>I'm partial to the proof using Dandelin spheres that (certain) cross sections of cones are ellipses, where an ellipse is defined as the locus of points whose total distance to two foci is constant. It's particularly nice because it explains the foci geometrically, as well as the focus-directrix property with some more work.</p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/2/21/Dandelin1.png" alt="Dandelin spheres touch the light blue plane that intersects the cone." /></p> http://mathoverflow.net/questions/8846/proofs-without-words/11879#11879 Answer by vonjd for Proofs without words vonjd 2010-01-15T16:31:52Z 2010-06-18T08:53:41Z <p>Have a look at this document from an MIT-instructor: <a href="http://mit.edu/18.098/book/extract2009-01-21.pdf" rel="nofollow">http://mit.edu/18.098/book/extract2009-01-21.pdf</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/12024#12024 Answer by serargus for Proofs without words serargus 2010-01-16T21:42:19Z 2010-01-16T21:42:19Z <p>There are a couple of Fibonacci identities, I think. For example</p> <p>$F_0^2+F_1^2+\cdots+F_n^2=F_{n}F_{n+1}$, with $F_0=1$.</p> <p>By puting together squares of side $F_n$, one at a time, you get a rectangle of dimension $F_nF_{n+1}$: The two squares of side 1, then the square of side 2, then the square of side 3 and so on. </p> <p>Here is an image I found online</p> <p><img src="http://www.mahugh.com/images/blog/2006/06/18/goldenrectangle.jpg" alt="fibonacci_rectangle" /></p> http://mathoverflow.net/questions/8846/proofs-without-words/16064#16064 Answer by Marco Radeschi for Proofs without words Marco Radeschi 2010-02-22T15:25:59Z 2012-11-13T01:22:31Z <p>The sequence of pictures</p> <p><img src="http://img17.imageshack.us/img17/9169/img2qg.png" alt="intersection of 3 diangles"> <img src="http://img839.imageshack.us/img839/8629/img4z.png" alt="intersection of 2 diangles"> <img src="http://img255.imageshack.us/img255/5643/img5bb.png" alt="intersection of 2 diangles"> <img src="http://img844.imageshack.us/img844/4253/img6i.png" alt="intersection of 2 diangles"></p> <p>proves the area formula for spherical triangles $A=\hat{ABC}+\hat{BCA}+\hat{CAB}-\pi$.</p> http://mathoverflow.net/questions/8846/proofs-without-words/17193#17193 Answer by Sunni for Proofs without words Sunni 2010-03-05T16:59:16Z 2010-03-05T16:59:16Z <p>There a proof of Erd˝os-Mordell Inequality 'without words' is an impressive one. Please follow the link <a href="http://forumgeom.fau.edu/FG2007volume7/FG200711.pdf" rel="nofollow">http://forumgeom.fau.edu/FG2007volume7/FG200711.pdf</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/17320#17320 Answer by Tobias Hagge for Proofs without words Tobias Hagge 2010-03-06T21:55:58Z 2012-10-23T15:24:00Z <p>Also elementary, but here is a proof that</p> <p>$C_n = \binom{2n}{n} - \binom{2n}{n+1} = \frac{\binom{2n}{n}}{n+1},$</p> <p>where $C_n$ is the $n$th Catalan number.</p> <p><a href="http://utdallas.edu/~hagge/images/Catalan.pdf" rel="nofollow">http://utdallas.edu/~hagge/images/Catalan.pdf</a></p> <p>Sorry for the link; new users may not use image tags.</p> <p>Here's the image:</p> <p><img src="http://i.imgur.com/3fJi1.png" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/17328#17328 Answer by shreevatsa for Proofs without words shreevatsa 2010-03-06T23:18:28Z 2010-03-06T23:53:50Z <p><a href="#9381" rel="nofollow">This other answer</a> shows that an 8x8 board with opposite squares removed cannot be tiled with dominoes, as they are of the same "colour". But what if two squares of <em>opposite</em> colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and <a href="http://shreevatsa.files.wordpress.com/2010/03/tiling-gomory.png" rel="nofollow">this is his proof</a>.</p> <p><img src="http://shreevatsa.files.wordpress.com/2010/03/tiling-gomory.png" alt="alt text"></p> <p>(Imagine A and B are the squares removed.) The image is from Honsberger's <em>Mathematical Gems I</em>.</p> http://mathoverflow.net/questions/8846/proofs-without-words/17347#17347 Answer by Russell O'Connor for Proofs without words Russell O'Connor 2010-03-07T02:16:51Z 2010-03-07T02:16:51Z <p>Because I think proofs by picture is potentially dangerous, I'll present a link to the standard proof that 32.5 = 31.5:</p> <p><img src="http://farm1.static.flickr.com/48/152036443_ca28c8d2a1_o.png" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/17373#17373 Answer by Kumar for Proofs without words Kumar 2010-03-07T11:01:10Z 2010-03-07T11:01:10Z <p>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 -</p> <ol> <li><p>Nyquist sampling theorem?</p></li> <li><p>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 :)</p></li> <li><p>Sampling in frequency domain (comb filter) is periodicity in time domain?</p></li> </ol> <p>Here are some "audio illusions" though, for your amusement - <a href="http://www.youtube.com/watch?v=e6JSTkwXg90" rel="nofollow">http://www.youtube.com/watch?v=e6JSTkwXg90</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/19338#19338 Answer by Chris Conway for Proofs without words Chris Conway 2010-03-25T18:11:49Z 2010-05-16T09:46:01Z <p>This is apparently not was intended, but I think it qualifies. From <a href="http://en.wikipedia.org/wiki/Principia_Mathematica" rel="nofollow"><em>Principia Mathematica</em></a>: <a href="http://en.wikipedia.org/wiki/File%3aPrincipia_Mathematica_theorem_54-43.png" rel="nofollow">the proof of 1+1=2</a> (I can't include the image bc I'm a new user, but perhaps an experienced user can edit this answer for me.)</p> http://mathoverflow.net/questions/8846/proofs-without-words/19341#19341 Answer by muad for Proofs without words muad 2010-03-25T19:00:35Z 2010-03-25T19:00:35Z <p>If we have 3 circles on the plane with tangent lines, we can notice they have colinear intersection!</p> <p><img src="http://img256.imageshack.us/img256/7512/picture1dt.png" alt="Made in inkscape"></p> <p>To prove it, we can visualize the same configuration in 3D, the balls lay on a surface and rather than tangent lines we take cones: The colinearity comes from the fact that if we lay a plane ontop of this configuration it will intersect the table in a line!</p> <p>This is from 'curious and interesting geometry' and the proof is attributed to John Edson Sweet. I really like this proof because it gives a vivid example of the general idea that sometimes, to solve a problem in the most simple way you need to view it as a part of some bigger whole.</p> http://mathoverflow.net/questions/8846/proofs-without-words/24774#24774 Answer by Vaughn Climenhaga for Proofs without words Vaughn Climenhaga 2010-05-15T16:40:20Z 2012-08-27T03:19:30Z <p>This should really be a comment on Marco Radeschi's <a href="http://mathoverflow.net/questions/8846/proofs-without-words/16064#16064" rel="nofollow">answer</a> from Feb 22 involving the area formula for spherical triangles, but since I'm new here I don't have the reputation to leave comments yet.</p> <p>In reply to Igor's comment (on Marco's answer) wondering about an analogous proof for the area formula of hyperbolic triangles: there is one along similar lines, and you're rescued from non-compactness by the fact that asymptotic triangles have finite area. In particular, the proof in the spherical case relies on the fact that the area of a double wedge with angle $\alpha$ is proportional to $\alpha$; in the hyperbolic case, you need to replace the double wedge with a doubly asymptotic triangle (one vertex in the hyperbolic plane and two vertices on the ideal boundary) and show that if the angle at the finite vertex is $\alpha$, then the area is proportional to $\pi - \alpha$. That follows from similar arguments to those in the spherical case (show that the area function depends affinely on $\alpha$ and use what you know about the cases $\alpha=0,\pi$).</p> <p>Once you have that, then everything follows from the picture below, since you know the area of the triply asymptotic triangle and of the three (yellow, red, blue) doubly asymptotic triangles.</p> <p><img src="http://www.math.uh.edu/~climenha/pics/hyperbolic-triangle.jpg" alt="alt text"></p> <p>(That picture is slightly modified from p. 221 of <a href="http://books.google.com/books?id=XFkc0Yn-TE8C&amp;printsec=frontcover&amp;dq=Lectures+on+Surfaces&amp;ei=bMvuS86VK4-2zQTRufDxCg&amp;cd=1#v=onepage&amp;q&amp;f=false" rel="nofollow">this book</a>, which has the whole proof in more detail.)</p> http://mathoverflow.net/questions/8846/proofs-without-words/24828#24828 Answer by Dave Pritchard for Proofs without words Dave Pritchard 2010-05-15T22:32:52Z 2012-03-26T22:15:43Z <p>Means inequalities:</p> <p><img src="http://daveagp.files.wordpress.com/2010/11/means.png" alt="alt text"></p> <p>The image was sent to me by James M. Lawrence, grazie! See also page 53 of "Proofs without words: exercises in visual thinking, Volume 2" for a very different layout of the same 4 inequalities.</p> <p>Another one exists involving the sum $$1^3 + 2^3 + \cdots + n^3:$$</p> <p><img src="http://users.tru.eastlink.ca/~brsears/math/sumcube3.jpg" alt="alt text"></p> <p>The second image is due to <a href="http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32" rel="nofollow">Brian Sears</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/25305#25305 Answer by Vaughn Climenhaga for Proofs without words Vaughn Climenhaga 2010-05-20T01:27:35Z 2012-08-27T03:17:58Z <p>It's a long list of wonderful answers already, but I can't resist...</p> <p><em>Question</em>: Is it possible to find six points on a square lattice that form the vertices of a regular hexagon?</p> <p><em>Proof without words</em>:</p> <p><img src="http://www.math.uh.edu/~climenha/pics/hex-on-lattice.jpg" alt="alt text"></p> <p><em>Hint</em>: A square lattice is invariant under rotation by &pi;/2 around any lattice point. Use reductio ad absurdum.</p> <p><em>Credit</em>: I learned that proof from Gy&ouml;rgy Elekes during the Conjecture and Proof course in the Budapest Semesters in Mathematics, after constructing a proof of my own that used entirely too many words and made very laboured use of the fact that $\sqrt{3}$ is irrational. The picture here is my own creation (using Asymptote).</p> <p><em>Follow-up</em>: Can you find four points on a hexagonal lattice that form the vertices of a square? The proof is similar but not immediate.</p> http://mathoverflow.net/questions/8846/proofs-without-words/29027#29027 Answer by Bassam Abdul-Baki for Proofs without words Bassam Abdul-Baki 2010-06-22T02:02:17Z 2010-06-22T02:02:17Z <p><a href="http://home.comcast.net/~babdulbaki/MathProofs.html" rel="nofollow">http://home.comcast.net/~babdulbaki/MathProofs.html</a></p> <p><a href="http://home.comcast.net/~babdulbaki/PWW.html" rel="nofollow">http://home.comcast.net/~babdulbaki/PWW.html</a> (See link at bottom.)</p> http://mathoverflow.net/questions/8846/proofs-without-words/29459#29459 Answer by BlueRaja for Proofs without words BlueRaja 2010-06-25T02:45:08Z 2010-06-25T02:45:08Z <p>Conway and Soifer tried to set a record for least number of words in a mathematical paper. I've reproduced it here in its entirety.</p> <p><strong>Can n<sup>2</sup> + 1 unit equilateral triangles cover an equilateral triangle of side > n, say n + ε?</strong><br> <sup><em>John H. Conway &amp; Alexander Soifer<br> Princeton University, Mathematics<br> Fine Hall, Princeton, NJ 08544, USA<br> conway@math.princeton.edu asoifer@princeton.edu</em></sup></p> <p>n<sup>2</sup> + 2 can:</p> <p><img src="http://img231.imageshack.us/img231/9329/figure1x.png" alt="alt text"></p> <p><img src="http://img138.imageshack.us/img138/1122/figure2.png" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/31419#31419 Answer by Franklin for Proofs without words Franklin 2010-07-11T15:11:05Z 2010-07-11T15:11:05Z <p>Let $0\leq x,y,z,t\leq1$ Prove that $x(1-y)+t(1-x)+z(1-t)+y(1-z)\leq 2$.</p> <p>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.</p> http://mathoverflow.net/questions/8846/proofs-without-words/31913#31913 Answer by Daniel Miller for Proofs without words Daniel Miller 2010-07-14T22:15:56Z 2010-07-14T22:15:56Z <p>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.</p> http://mathoverflow.net/questions/8846/proofs-without-words/34813#34813 Answer by Thierry Zell for Proofs without words Thierry Zell 2010-08-07T03:37:33Z 2010-08-07T03:37:33Z <p>In the movie category, I'm surprised that no-one has yet posted a link to <a href="http://www.youtube.com/watch?v=JX3VmDgiFnY" rel="nofollow">Moebius Transformations Revealed</a>. </p> http://mathoverflow.net/questions/8846/proofs-without-words/38658#38658 Answer by muad for Proofs without words muad 2010-09-14T07:44:10Z 2010-09-14T07:44:10Z <h2>$$2 \pi > 6$$</h2> <p><img src="http://i.imgur.com/EJoJy.png" alt=""></p> http://mathoverflow.net/questions/8846/proofs-without-words/38664#38664 Answer by Alexis Monnerot-Dumaine for Proofs without words Alexis Monnerot-Dumaine 2010-09-14T08:24:03Z 2010-09-14T08:24:03Z <p>A classic one, from the late 19th century, that surprized Peano's contemporaries.</p> <p><strong>Question</strong> : "A curve that fills a plane ? You must be kidding"</p> <p><strong>Answer</strong> : </p> <p><img src="http://lh6.ggpht.com/_R1baPul19Kw/TI8wWzfn-MI/AAAAAAAABFA/0c8r1MK0MmI/Peano_curve.png" alt="alt text"></p> <p>Well, of course a formal proof was necessary, but it is still one of my favorites.</p> http://mathoverflow.net/questions/8846/proofs-without-words/40164#40164 Answer by AndrewLMarshall for Proofs without words AndrewLMarshall 2010-09-27T15:36:25Z 2010-09-27T15:36:25Z <p>from Steven Strogatz's column: <a href="http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/" rel="nofollow">http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/</a></p> <p><img src="http://graphics8.nytimes.com/images/2010/04/04/opinion/04strogatz5/04strogatz5-custom1.jpg" alt="pi"></p> http://mathoverflow.net/questions/8846/proofs-without-words/50531#50531 Answer by Kris Joanidis for Proofs without words Kris Joanidis 2010-12-28T02:42:47Z 2010-12-28T02:47:48Z <p><img src="http://y0o0y.files.wordpress.com/2010/12/composition.png?w=136&amp;h=299" alt="alt text"></p> <blockquote> <p>The composition of two continuous mappings is continuous.</p> </blockquote> <p>Bloody thing won't let me embed the image...</p> http://mathoverflow.net/questions/8846/proofs-without-words/54587#54587 Answer by Daniel Parry for Proofs without words Daniel Parry 2011-02-07T01:27:59Z 2011-02-07T02:53:34Z <p>This might be trivial but integration by parts has a nice proof without words:</p> <p><img src="http://img843.imageshack.us/img843/8315/intbyparts.png" alt="alt text"></p> <p>(Got from: Roger B. Nelsen, Proof without Words: Integration by Parts, Mathematics Magazine, Vol. 64, No. 2 (Apr., 1991), p. 130; the original link is to <a href="http://www.math.ufl.edu/~mathguy/year/S10/int_by_parts.pdf" rel="nofollow">http://www.math.ufl.edu/~mathguy/year/S10/int_by_parts.pdf</a>)</p> http://mathoverflow.net/questions/8846/proofs-without-words/54601#54601 Answer by Bob Palais for Proofs without words Bob Palais 2011-02-07T04:01:50Z 2011-02-07T04:01:50Z <p>Here are some dynamic versions:</p> <p><a href="http://www.math.utah.edu/~palais/sums.html" rel="nofollow">http://www.math.utah.edu/~palais/sums.html</a> (two of the summation formulas mentioned above)</p> <p>Several belt, plate, and tangle trick animations:</p> <p><a href="http://www.math.utah.edu/~palais/links.html" rel="nofollow">http://www.math.utah.edu/~palais/links.html</a></p> <p>A visual derivation of complex multiplication:</p> <p><a href="http://www.math.utah.edu/~palais/newrot.swf" rel="nofollow">http://www.math.utah.edu/~palais/newrot.swf</a></p> <p>Pythagoras in the Isosceles case, based on the Yale tablet:</p> <p><a href="http://www.math.utah.edu/~palais/PythagorasIsosceles.html" rel="nofollow">http://www.math.utah.edu/~palais/PythagorasIsosceles.html</a></p> <p>and the general case:</p> <p><a href="http://www.math.utah.edu/~palais/Pythagoras.html" rel="nofollow">http://www.math.utah.edu/~palais/Pythagoras.html</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/58578#58578 Answer by Chris Heunen for Proofs without words Chris Heunen 2011-03-15T22:02:59Z 2012-02-15T04:47:04Z <p>Algebraic manipulations in monoidal categories can also be performed in a graphical calculus. And the best part is that this is completely rigorous: a statement holds in the graphical language if and only if it holds (in the algebraic formulation). See for example Peter Selinger's "<a href="http://dx.doi.org/10.1007/978-3-642-12821-9_4" rel="nofollow">A survey of graphical languages for monoidal categories</a>". There are many instances, for example in knot theory studied via braided categories. The following specific example comes from Joachim Kock's book "<a href="http://books.google.com/books?id=6dZZW08Z04MC" rel="nofollow">Frobenius Algebras and 2D Topological Quantum Field Theories"</a>, and proves that the comultiplication of a Frobenius algebra is cocommutative if and only if the multiplication is commutative.</p> <p><img src="http://oi55.tinypic.com/5k58uf.jpg" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/61219#61219 Answer by thei for Proofs without words thei 2011-04-10T15:54:03Z 2011-04-10T15:54:03Z <p>I suggest the videos of Viennot explaining the bijections between different families of objects counted by Catalan numbers:</p> <p><a href="http://web.mac.com/xgviennot/Cont_Science/vid%C3%A9os.html" rel="nofollow">http://web.mac.com/xgviennot/Cont_Science/vid%C3%A9os.html</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/61223#61223 Answer by Marco Golla for Proofs without words Marco Golla 2011-04-10T16:57:19Z 2012-08-27T08:33:47Z <p>I'm quite surprised no-one pointed out this one yet:</p> <p><strong>Theorem</strong>. The trefoil knot is knotted.</p> <p><em>Proof</em>.</p> <p><img src="http://poisson.phc.unipi.it/~golla/trefoil.png" alt="alt text"> $\square$</p> <p>Some comments: a <em>3-colouring</em> of a knot diagram D is a choice of one of three colours for each arc D, such that at each crossing one sees either all three colours or one single colour. Every diagram admits at least three colourings, <em>i.e.</em> the constant ones. We'll call <em>nontrivial</em> every 3-colouring in which at least two colours (and therefore all three) actually show up. It's easy to see (one theorem, more pictures!) that Reidemeister moves preserve the property of having a nontrivial 3-colouring, and that the unknot doesn't have any nontrivial colouring.</p> <p>The picture shows a (nontrivial) 3-colouring of the trefoil.</p> <p><strong>EDIT</strong>: I've made explicit what "nontrivial" meant -- see comments below. Since I'm here, let me also point out that the <em>number</em> of 3-colourings is independent of the diagram, and is itself a knot invariant. It also happens to be a power of 3, and is related to the fundamental group of the knot complement (see Justin Robert's <a href="http://math.ucsd.edu/~justin/Papers/knotes.pdf" rel="nofollow">Knot knotes</a> if you're interested).</p> http://mathoverflow.net/questions/8846/proofs-without-words/69022#69022 Answer by leonbloy for Proofs without words leonbloy 2011-06-28T14:42:08Z 2011-06-28T14:42:08Z <p>(I'd post this as a comment to Mariano Suárez-Alvarez, but I've not enough rep). From a <a href="http://math.stackexchange.com/questions/44759/combinatorial-proof-that-binomial-coefficients-are-given-by-alternating-sums-of-s/44782#44782" rel="nofollow">ME thread</a>.</p> <p>$$\sum_{k=1}^n (-1)^{n-k} k^2 = {n+1 \choose 2} = \sum_{k=1}^n \; k = \frac{(n+1) \; n}{2}$$</p> <p><img src="http://i.stack.imgur.com/2s7sk.png" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/69032#69032 Answer by Jesko Hüttenhain for Proofs without words Jesko Hüttenhain 2011-06-28T16:46:28Z 2011-06-28T16:46:28Z <p>Interesting how everyone understands <i>&quot;proof without words&quot;</i> as <i>&quot;proof made of pictures&quot;</i>. 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 <i>&quot;words&quot;</i>, 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, <b>what is a word</b>?</p> <p>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.</p> http://mathoverflow.net/questions/8846/proofs-without-words/69594#69594 Answer by jkun for Proofs without words jkun 2011-07-06T00:39:00Z 2011-07-06T00:39:00Z <p>Can you tile an 8x8 chessboard with one corner cut off with dominoes of dimension 3x1?</p> <p><img src="http://jeremykun.files.wordpress.com/2011/06/chessboard-1-by-3.jpg" alt="alt text"></p> <p>This is a simple way to show that choosing a useful coloring can make a proof trivial.</p> <p>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 <em>that</em> hard :)</p> http://mathoverflow.net/questions/8846/proofs-without-words/69756#69756 Answer by jkun for Proofs without words jkun 2011-07-07T23:25:37Z 2011-07-07T23:25:37Z <p>Another proof of the sum of the first $n$ squares, relying on the knowledge of the formula for the sum of the first $n$ numbers:</p> <p>$1^2 + 2^2 + \dots + n^2 = n(n+1)(2n+1)/6$</p> <p><img src="http://jeremykun.files.wordpress.com/2011/06/triangle-proof.png" alt="alt text"></p> <p>This one has a similar flavor to the fabled proof by Gauss of the sum of the first $n$ numbers. It's a good follow up for students after Gauss's proof.</p> http://mathoverflow.net/questions/8846/proofs-without-words/69834#69834 Answer by Phil Isett for Proofs without words Phil Isett 2011-07-08T21:49:24Z 2011-07-08T21:49:24Z <p>Here's a proof of the area of a circle (or sector) which is different from the one posted previously.</p> <p><strong>EDIT:</strong> I was unable to embed the file, which is in pdf form. Here is a link:</p> <p><a href="http://wildpositron.files.wordpress.com/2011/04/sectorarea2.pdf" rel="nofollow">http://wildpositron.files.wordpress.com/2011/04/sectorarea2.pdf</a></p> <p>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):</p> <p><a href="http://wildpositron.wordpress.com/2011/04/05/calculating-the-area-of-a-sector/" rel="nofollow">http://wildpositron.wordpress.com/2011/04/05/calculating-the-area-of-a-sector/</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/72965#72965 Answer by Ron Maimon for Proofs without words Ron Maimon 2011-08-16T06:45:45Z 2011-08-16T06:45:45Z <p>This proves the Minkowski version of the Pythagorean theorem:</p> <p><img src="http://www.gliffy.com/pubdoc/2846023/L.png" alt="alt.text"></p> <p>$c^2 = a^2 - b^2$</p> http://mathoverflow.net/questions/8846/proofs-without-words/72977#72977 Answer by Ben A. for Proofs without words Ben A. 2011-08-16T12:23:01Z 2011-08-16T12:23:01Z <p>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 $S^1\cong\mathbb{R}/\mathbb{Z}$.</p> <p>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".</p> http://mathoverflow.net/questions/8846/proofs-without-words/75570#75570 Answer by sclv for Proofs without words sclv 2011-09-16T03:05:50Z 2011-09-16T17:26:50Z <p>A line that bisects the right angle in a right triangle also bisects a square erected on the hypotenuse:</p> <p><img src="http://www.futilitycloset.com/wp-content/uploads/2011/09/2011-09-12-half-and-half-2.png" alt="http://www.futilitycloset.com/wp-content/uploads/2011/09/2011-09-12-half-and-half-2.png"></p> <p>source: <a href="http://www.futilitycloset.com/2011/09/12/half-and-half/" rel="nofollow">http://www.futilitycloset.com/2011/09/12/half-and-half/</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/75573#75573 Answer by euklid345 for Proofs without words euklid345 2011-09-16T03:19:31Z 2011-09-16T03:19:31Z <p>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. </p> <p><a href="http://www.math.ubc.ca/~cass/Euclid/byrne.html" rel="nofollow">http://www.math.ubc.ca/~cass/Euclid/byrne.html</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/76951#76951 Answer by isomorphismes for Proofs without words isomorphismes 2011-10-01T23:31:18Z 2011-10-01T23:31:18Z <p>From Wikipedia: here is a "proof without words" of the Yoneda Lemma.</p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/b/b1/YonedaLemma-02.png" alt="alt text"></p> http://mathoverflow.net/questions/8846/proofs-without-words/80369#80369 Answer by Martin Brandenburg for Proofs without words Martin Brandenburg 2011-11-08T09:33:26Z 2011-11-08T09:33:26Z <p>Proof of the associativity law $f * (g * h) = (f * g) * h$ in the fundamental groupoid of a topological space:</p> <p><img src="http://img845.imageshack.us/img845/3563/passj.gif" alt="http://img845.imageshack.us/img845/3563/passj.gif"></p> <p>You can find more of these diagrams in J. P. May's <em>A Concise course in algebraic topology</em>.</p> http://mathoverflow.net/questions/8846/proofs-without-words/88472#88472 Answer by Roberto Mizzoni for Proofs without words Roberto Mizzoni 2012-02-15T00:00:13Z 2012-02-24T01:04:53Z <p>For $0 \lt k \lt n$,</p> <p>$$\binom{n}{k} = \frac{n}{n-k}\binom{n-1}{k}$$</p> <hr> <p>How k-subsets of [n], marked dark green in the rows, come from k-subsets of [n-1] after n-fold duplication and rearrangement:</p> <p><img src="http://i43.tinypic.com/37rcn.jpg" alt="alt text"> Exactly $n-k$ times:</p> <p><img src="http://i44.tinypic.com/2nsozk1.jpg" alt="alt text"><br> By induction, a base case, and taking $k=n$ and $k=0$ for granted: $$\binom{n}{k} = \frac{n}{(n-k)} \frac{(n-1)!}{(n-1-k)!\ k!} = \frac{n!}{(n-k)!\ k!}$$</p> http://mathoverflow.net/questions/8846/proofs-without-words/98316#98316 Answer by Franklin for Proofs without words Franklin 2012-05-29T22:27:02Z 2012-05-29T22:27:02Z <p>I just saw this proof, which is of course not mine. </p> <p><a href="http://youtu.be/whYqhpc6S6g" rel="nofollow">link text</a></p> http://mathoverflow.net/questions/8846/proofs-without-words/104871#104871 Answer by Jon Cohen for Proofs without words Jon Cohen 2012-08-16T22:01:48Z 2012-08-18T22:22:22Z <p>The pathspace of any topological space is contractible.</p> <p>Pf (as given in my homotopy theory class): slurp spaghetti. </p> http://mathoverflow.net/questions/8846/proofs-without-words/105127#105127 Answer by Marc Chamberland for Proofs without words Marc Chamberland 2012-08-20T22:39:22Z 2012-08-20T22:39:22Z <p>I like the tiling proof of the <strong>Pythagorean Theorem</strong>. The left image is credited to Al-Nayrizi and Thābit ibn Qurra (9th century) and the right by Henry Perigal (19th century).</p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pythagorean_dissections.svg/300px-Pythagorean_dissections.svg.png" alt="**"></p> http://mathoverflow.net/questions/8846/proofs-without-words/105137#105137 Answer by Marc Chamberland for Proofs without words Marc Chamberland 2012-08-21T01:28:28Z 2012-08-21T01:28:28Z <p>This is a "proof without words" by an <strong>equation</strong>, not a <strong>picture</strong>.</p> <p>Three complex numbers $a,b,c$ in the complex plane form the vertices of an equilateral triangle if and only if $~a^2 + b^2 + c^2 = ab + bc + ca$:</p> <p></p> <p>$$\hspace{-3in} 2 |a^2 + b^2 + c^2 - ab - bc - ca|^2$$ $$= ( |a-b|^2 - |b-c|^2)^2 + ( |b-c|^2 - |c-a|^2)^2 + ( |c-a|^2 - |a-b|^2)^2 .$$</p>