Proofs without words

  • 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)

    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`

    I hope I am not alone in being (usually) unable to appreciate "proof by picture"...

    @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!

    @Mariano - sorry, I'm blind in one eye, so I never did get those autostereograms. Great question, though!

    @Rod Vance -- I'm sure you are not alone; I think my difficulty with "pictures" lies in my lack of geometric abilities.

    This question gives Mariano the sixth and seventh golden badges!! Congratulations!!

    Can we close this as being no longer relevant? The answers trickling in now are not really proofs without words.

    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 - 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.

    My opinion is that almost every proof-without-words is improved by a few well-chosen words.

    @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]

    Why has this question been closed? How can it be "no longer relevant"? (I mean: the longer people contribute examples, the better.) (BTW: I *do* have an astonishing example.)

    I 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.

    It 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?

    There 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, 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.

    @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.

    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."

    If 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.

    @WetSavannaAnimalakaRodVance I want to upvote your comment again and again...

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

    i 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.

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

    i think having both would be a plus...visuals would help concretize the idea that might be hard to grasp from the "well-chosen words"

    Technically, ten pages full of formulas count as a "proof without words", right?

    @MarianoSuárez-Álvarez Dear Dr. Suárez-Álvarez, you are a much-valued contributor on this site, no doubt. Much more important than I am, for example. But when you said "that is clearly a limitation of yours", I think you sent off the discussion in the wrong direction. If you ask a technical question and use some unfamiliar notions in it, what do people tell you? Define these notions. Here you have not defined what you mean by proofs without words.

    That by itself does not mean this is a bad question, it is actually a pretty good question, but if you can not give a complete definition of a "proof without word" that is the limitation of the question, not of the people who recognize it as a somewhat ambiguous question. In this case, there can be quite clear community consensus on what is a proof without word, but still it is you who failed to prove a definition of a proof without word, and only because of the existence of community consensus this question is answerable at all.

    I think you should apologize to goblin here, because he raised a valid point about a particular limitation of your question, and you made an ad hominem attack against him (quite successfully, because of your well-established reputation here most probably). An appropriate reply to him might have been "Yes, this is an ambiguous question, but while I can not provide a complete definition, I believe there is fairly clear community consensus on what a proof without words is, so I think the question is fine", in my opinion.

  • A proof of the identity $$1+2+\cdots + (n-1) = \binom{n}{2}$$

    proof without words

    (Adapted from an entry I saw at Wolfram Demonstrations, see also the original faster animation)

    This proof was discovered by Loren Larson, professor emeritus at St. Olaf College. He included it along with a number of other, more standard, proofs, in "A Discrete Look at 1+2+...+n," published in 1985 in The College Mathematics Journal (vol. 16, no. 5, pp. 369-382, DOI: 10.1080/07468342.1985.11972910, JSTOR).

    Fantastic! I'm reminded of a question I wrote for a high-school math competition way back when, which hinged on a similar diagram that counts solutions to a + b + c = n in nonnegative integers. (Imagine a third, horizontal, line.) The problem was something like to count the solutions to a + b + c = 100 with a, b, c bounded above by some numbers close to 100; I still don't know an elegant way to solve the problem other than to see it from the triangle.

    This one is just awesome!

    This is a beautiful proof. I like it very much. But I wonder what status proofs such as this one have in the eyes of people who think that mathematics is about deducing theorems from axioms. If you try to reduce such things to axioms the beauty vanishes.

    I gave this result +1, despite the fact that I am a formalist and do not regard this single image as a proof, because I think it is a very good proof **virus** to which experienced mathematicians are susceptible.

    @Johann, people who thing that mathematics is about deducing theorems from axioms have such a mistaken idea of what the mathematical activity is thar their judgment is more or less irrelevant :D

    @Johann: some days of the week, I am such a person; and from that point of view, the picture is a beautifully clear encoding of a certain bijection, and the formal construction of the bijection itself is a very beautiful proof. No beauty is destroyed!// I strongly believe that a proof with a clear intuition should _also_ be clear as a formal proof. If not, either (usually) our formalism isn't as good as it could be, or (occasionally) our intuition really is overlooking some non-trivial subtleties.

    That's beautiful!

    Am I the only one who doesn't understand this "proof" at all?

    @mathreader - the yellow dots are the sum of the first n numbers. Choosing two of the n+1 blue dots uniquely specifies a yellow dot in a bijective fashion.

    Steve: there are n blue dots. :)

    @Mariano: Then help me... what would be a nonmistaken idea of the mathematical activity? (so far I have no other way of seen what math is...)

    Damn, the picture's link won't load. Can anyone find another?

    @Adam: Only if $n=7$

    This beautiful proof warrants proper attribution. It was discovered by Loren Larson, professor emeritus at St. Olaf College. He included it along with a number of other, more standard, proofs, in "A Discrete Look at 1+2+...+n," published in 1985 in The College Mathematics Journal (vol. 16, no. 5, pp. 369-382).

    +1 Everybody has already noted this but I just have to add my vote that this proof is stunning.

    I wish you had shared this question to Math. S.E. :-)

    http://i.imgur.com/HCfGOYp.gif An animated version, which makes it a bit more clear.

    The article that @BarryCipra mentions is available on JSTOR at http://www.jstor.org/stable/10.2307/2686996.

    The image link seems to be broken now.

    This is still broken. Or perhaps this is without words *and* without image and thus the topvoted one ;-)

    I edited it to substitute the broken image link to one in the comments. It's not optimal, but until the original link is restored it is better than nothing.

    I really still didn't understand this one. But I found another visual explanation for it http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/triNbProof.html

    I preferred the original static image (+1), which made the idea clear. The chaotic animated image (now -1) is mainly irritating---I cannot have it on my screen for any length of time. Perhaps it would become acceptable if the animation was slowed by a factor of ten or so.

    I find the underlying argument extremely nice but the picture (especially the animated one) does not really work for me without any words. The animation makes me think that is has some meaning that the yellow dots are visited in a certain order.

    @NieldeBeaudrap sorry if this is a simple question, but what do u mean by "proof virus"? Not sure if I understood ur comment.

    How this can be a prove? Since its only displayed for $n=7$.

    Gauss discovered a solution to this sum at the age of 9. His solution is an example where a proof with words is more intuitive than a proof without words. http://math.stackexchange.com/a/1017743/202346

    Could someone please edit the image to slow down the animation by a factor of 10 or so? The current chaotic form is beyond irritating.

    With help from Jeremy Kun, I have posted a slower version of the animation (20% speed), which I think makes a much more pleasant experience.

    I don't known why You edit many time @MarianoSuárez-Álvarez

    I cannot see why this is considered a "proof without words": It takes a lot of words to state what it's a proof of - and to give hints how it does prove it. (Only the experts see it at first sight - but aren't proofs without words targeted more to the **non**-experts?)

  • Because I think proof by picture is potentially dangerous, I'll present a link to the standard proof that 32.5 = 31.5:

    "proof" that 32.5 = 31.5

    An animation of the above is:

    enter image description here

    (This work has been released into the public domain by its author, Trekky0623 at English Wikipedia. This applies worldwide.)

    There does not seem to be any necessity for the particular 'path in the relevant configuration space' that was used by the author of the above animated gif. This may be seen as an argument against including an animation.

    I think it is just as easy to introduce some kind of logical gap in a written proof as in a graphical one.

    @Steven: I think there is some truth to your claim, but I don't agree fully. First, we may notice that most proofs rely much more on writing than on pictures, and so mathematicians have developed a better radar for "written gaps". Second, there is a very strong sense in which written proofs may be formalized and checked by computer. Picture proofs, unless they share quite a bit of the "discrete" character of written proofs, usually are not amenable to such treatment. (And the notions of discreteness I can think of pretty much ensure that the picture proof could be turned into words.)

    +1 for "the standard proof that $32.5 = 31.5$." Made me laugh. :)

    @Pietro: “there is a very strong sense in which written proofs may be formalised”? Formalisation is a highly non-trivial task, and typically depends on quite a lot of mathematical background. What affects the difficulty is not whether the proof is written or graphical, but whether it’s detailed or highly abstracted. Formalising a good proof-by-picture is no harder than formalising a high-level written proof. Insofar as there’s a difference, I’d say it’s just that written proofs *can* be made detailed enough that formalising them is straightforward, whereas picture proofs perhaps can’t.

    It might be noted that the success of the illusion partly depends on the fact this uses Fibonacci numbers (it is a coincidence I guess that the next newest answer is also about Fibonacci numbers!).

    This argument against proofs by picture is itself a proof by picture.

  • The cardinality of the real number line is the same as a finite open interval of the real number line.

    Proof without words

    I suppose this picture can also be adapted to obtain the stereographic projection proof that a sphere is a manifold?

    I could swear I've seen the exact picture you speak of somewhere.

    Also, I drew the above image myself. Feel free to use in whatever you like.

    @Jason Dyer: What software did you use? Inkscape?

    I usually use Inkscape for my vector-based needs, but this was just done with my Smartboard presentation software.

    This picture shows not only that they have same cardinality but that they are homeomorphic.

  • It is known (see this other answer) that an 8x8 board in which squares at opposite corners have been removed cannot be tiled with dominoes, as the removed squares are of the same "colour". But what if two squares of different colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof:

    Image of board

    (Imagine A and B are the squares removed.) The image is from Mathematical Gems I by Ross Honsberger.

    What I like about this example is that there seems to be no straightforward proof _without_ the picture; the crux of the proof's idea is specifically this picture.

    Very nice. I guess the crux of the proof is that, when $mn$ is even, $P_m\times P_n$ is a Hamiltonian bipartite graph?

    Well, I'd call that a generalization, not the crux of the proof. :-) Staying concrete, for the question about the specific case of $m=n=8$, the crux of this proof (that this graph *is* Hamiltonian) is this picture. Similar pictures can be draw whenever $mn$ is even, sure.

    A complete result (guessed not shown) is for m or n odd : Any mxn board with 1 square removed has a neighborhood graph that has an hamiltonian cycle.

  • There are a couple of Fibonacci identities, I think. For example

    $F_0^2+F_1^2+\cdots+F_n^2=F_{n}F_{n+1}$, with $F_0=1$.

    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.

    Here is an image I found online

    fibonacci_rectangle

    Really exceptional!

    I think that there *is* a nice pictorial proof for this fact, but I don't think this is it. It's a proof for a specific $n$. To make it a general proof, the inductive step needs to be illustrated.

    @Max: The inductive step is easy to figure out, since the rectangle above contains the rectangles from previous steps.

  • This is elementary as well, but one of my favorite ones :)

    $1^2 + 2^2 + \dots + n^2 = \frac13n(n+1)(n+\frac12)$

    (Author: Man-Keung Siu)

    alt text

    There's an analogous proof that the integral of n^2 from 0 to x is x^3/3. It can be obtained from this proof by smoothing out the stepped pyramids into actual pyramids.

    I think very few people have enough spatial imagination to figure out what happens exactly in the area where the three pieces come together, or could easily depict the structure seen from the opposite end. For me the *picture* is not convincing at all (I'd rather say the formula convinces me the picture is correct than the other way round). However maybe playing with an actual model would be quite convincing.

    @Mark - I think if you just think about the width of each step at each level, you will be able to see that they do all fit together. Just counting back along a given row or column shows you that it all fits.

    A variant of Mike's construction for $\sum_{k=1}^n k^2$, easier to visualize (I'm going to try a proof-without-words, without pictures). Take $6$ copies of each parallelepiped of size $k \times k \times 1$. Glue them together so as to make the four lateral walls of a parallelepiped of (external) size $k \times (k+1) \times (2k+1)$. Do this for k from 1 to n, forming a collection of bracelets. Insert each one in the next, like matrioskas, getting a whole parallelepiped of size $n\times(n+1)\times(2n+1)$.

    @Michael Lugo: The continuous version of this proof is "elementary geometry": the volume of a pyramid is one third of its height times the area of its ground surface!

  • It's a long list of wonderful answers already, but I can't resist...

    Question: Is it possible to find six points on a square lattice that form the vertices of a regular hexagon?

    Proof without words:

    alt text

    Hint: A square lattice is invariant under rotation by π/2 around any lattice point. Use reductio ad absurdum.

    Credit: I learned that proof from György 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).

    Follow-up: Can you find four points on a hexagonal lattice that form the vertices of a square? The proof is similar but not immediate.

    +1 for the "Conjecture & Proof" shout-out. Best, course, ever!

    Igen, nagyon jó.

    Beautiful! Yes! (I had to add "Yes!" to make my comment long enough).

    köszönöm, Google Translate.

    I really like this image and proof. The same idea works for other regular polygons, and I made images for pentagons, heptagons and so on, which you can find on my blog post at http://jdh.hamkins.org/no-regular-polygons-in-the-integer-lattice/. (The code accept $n$ as input, and makes the image for an $n$-gon.)

    And here is my post on the hexagonal lattice: the only regular polygons to be found are triangles and hexagons. http://jdh.hamkins.org/no-regular-polygons-in-the-hexagonal-lattice/

  • There's a picture proof in the Princeton Companion, or alternatively on p. 340 of Hatcher, of the fact that the higher homotopy groups are abelian. Actually, here's a screenshot of the one in Hatcher (hopefully fair-use!):

    Hatcher p. 340

    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 $g + f$.

    The above diagrams show an application of the interchange law, a more general expression of the Eckmann-Hilton argument, for double categories or groupoids. Here is a more general picture

    dbgxmod

    which shows that the interchange law for a double groupoid implies the second rule $v^{-1}uv= u^{\delta v} $, where in the picture $a=\delta v$, for the crossed module associated to a double groupoid, taken from the book advertised here. There are many $2$-dimensional rewriting arguments which are essential to the results of this book.

    I've heard that term, but I've never quite understood how the diagram is supposed to prove the more general abstract nonsense theorem. But if you can explain it, that's what community wiki's for! :D

    The "more general picture" seems to be broken.

  • This might be trivial but integration by parts has a nice proof without words:

    enter image description here

    (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 http://www.maa.org/sites/default/files/Roger_B04151._Nelsen.pdf).

    @Daniel, I've turned the PDF into a PNG, and inserted the relevant part. I did keep the URL to the PDF for reference. Thanks, by the way!

    The same picture also gives an interesting formula for the integral of an inverse function!

    I guess this proof works only when $f$ and $g$ are both increasing?

  • 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.

    Dandelin spheres touch the light blue plane that intersects the cone.

    Yes, this one is beautiful.

    Indeed. And there are also similar visual proofs for the hyperbola and the parabola

    How does the picture explain the invariance of the total distance to two foci? I don't see it ; I haven't done geometry in a while though, I'm guessing it's some triviality... refresh my memory please? :)

    @PatrickDaSilva: $PF1 = PP1$ because tangents to a circle/sphere have equal length. The total distance is thus equal to $PF1 + PF2 = PP1 + PP2 = P1P2$, which is constant.

    @aorq : Riiiight. Thanks for the clarification! It is indeed very visual.

    I've learned this and related proofs from Hilbert and Cohn-Vossen (but these proofs still originated mostly with Dadelin).

    I was confused by the perspective. In case anyone is having the same problem: the perspective is from a point that is (below the apex $S$ of the cone but) _above_ the base of the cone (circle $k2$) and the bottom half-sphere ($G2$). I mistakenly thought we were looking up through $k2$ into the inside of the cone -- I think this is because in my browser, at least, the the circle $k2$ gets _thicker_ when it passes behind the ellipse and should if anything get _thinner_. It's generally a nice drawing, though, and a nice proof.

    There are more gems in Dandelin's figure but I need words : Let circles $k_1,\; k_2$ lie in planes $J_1.\; J_2 $ respectively. Let the ellipse be in plane $J_E$. Then $J_1\cap J_E ,\; J_2\cap J_E$ are the directrices.. Let line $l_P$ be the tangent to the ellipse at $P,$ with $l_P\subset J_E.$ Let $l_P$ meet $J_1,\;J_2$ at $Q_1\;,Q_2$ respectively. For $ i=1,2 , $ triangles $Q_iPP_i ,\; Q_iPF_i$ are congruent (as $Q_iP_i.\; Q_iF_i$ are tangent to sphere $G_i$), So angles $Q_iPP_i = Q_iPF_i$ . But $Q_1PQ_2$ and $P_1PP_2$ are lines. So angles $Q_2PF_2=Q_2PP_2=Q_1PP_1=Q_1PF_1.$

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