Sunday, February 17, 2013

Three mathematical proofs

Hello everybody,

I recently had an exhibition in a small alternative art gallery in Geneva (Le Labo), and I took the opportunity to show three abstract comics that I recently made in which I try to convey the ideas of a mathematical proof into a totally wordless comic. I had already posted such a comic here in 2009, as you might recall:

A lot of mathematical "proofs without words" do exist (there are even two books published containing the "best" ones), where mathematical results are proved using figures (single ones or a sequence), but there is almost always some kind of notation introduced into the figures to make the proof understandable. And of course, it is assumed that the reader has some kind of mathematical education, so as to understand the implicit steps of the proof.

My aim was to try to see whether the "intrinsic power of the sequence" (so to say) was enough to get rid of any kind of use of letters or typographical symbols, and to really break down the proofs in sufficiently many steps so that anybody, without mathematical education beyond "mandatory school" (I don't know how to call it, in Switzerland, it means roughly up to 15 years old), could in principle understand the proof. Of course you'd have to have some inclination towards abstract reasoning to really go through the whole story, but I tried to make it as self-contained as I could. But actually even my mathematicians friends had problems to really follow the reasoning, so I probably failed badly in my attempts.

Anyway, I thought it could be of some interest for the people here, so I post some pictures of these three pieces. I am really a terrible photographer and one of the pieces is like 2 meteres long, so it's impossible to really read anything, but at least it gives an idea of what I'm talking about.

The first piece is almost identical to my 2009 post and shows a proof of Pythagore's Theorem due to Thabit Ibn Qurra (826-901).

The second is a proof of an identity for sums of cubes:


And the third one is a comic rendering of a very nice proof of a theorem in "elementary" geometry that is due to Roger B. Nelsen. This theorem has the astonishing particularity of being both rather new (it was discovered in 2004) and quite simple (if you compare it with today's research).

By the way, I was inspired by an incredible book of 1847 recently re-edited by Taschen: The elements of Euclid by Oliver Byrne. Anyone interested in abstract comics and/or mathematics should at least have a look at it:


  1. This is fantastic, thanks! I've been thinking a lot about abstract comics and math/science too, recently. Have you seen this post?

    Also this, where I mention another post by you:

    I love the third theorem. A totally Euclidean discovery, 2000+ years later.

    I have the three volume edition of Euclid from Dover. It has excellent commentary, and modern diagrams. Do you think there is still a reason to get the Taschen one too?

    1. Hi Andrei,

      it seems I had missed your two posts (which is strange since I come here regularly, but anyway). I'll read them with interest when I have some time.

      About Euclid, I don't know the Dover version. What makes Oliver Byrne's version so incredible is that it is the only one I know where pictures are used in every place instead of letters. Just a sample:

      The book can be read online there, but I really enjoyed the printed version as well (though I am not very fond of buying books edited by huge publishers such as Taschen).

    2. @Ibn al Rabin: Ces bandes-dessinées sont magnifiques ; la troisième m'intrigue énormément : comme Andreï, j'aimerai bien pouvoir la voir/ lire intégralement.

      @All folks : My strip Triangle, is based on the permutations of triangle vertexes (Ma bande-dessinée Triangle est basé sur les permutations des sommets du triangle).

  2. Mathieu, those images are really small! Any chance you could post larger ones, especially of the last one? I want to follow the proof, but it's unreadable.

    1. I can send you pictures in larger format, but it is quite difficult to follow either since some details in the original painting are rather small. Anyway, I'll send you an email with these.

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    3. And I should add that the original Roger B. Nelsen proof can be found on the page linked below. You'll see that my piece is just an elaboration of this proof.

    4. Thanks! What I find fascinating about this is the re-visualization of geometry, so to speak--an argument that geometry can function visually only (or primarily). A large part of the history of math has been so anti-visual, reducing geometry to equations, functions, as if the visual had to be purified away. I think the ideological basis of that move really needs to be addressed by a philosopher of mathematics.

  3. in the absence of words (explanations) , I can explore/ contemplate things like this visually and perhaps even reach understandings/ thoughts that are outside the strict fences of mathematics/ geometry and verbal language.... these pictures give the visual mind something to do rather than something to look at. and if teaching methods were geared more to the visual mind ....

  4. Wonderful as usual Mathieu! I hope I see you here:!/events/496219717088191/?fref=ts

  5. Idn Al Rabin: I really like these. I've been using geometric forms in my work lately but not for any real purpose like this.

    For what it's worth, there's a pdf of the Byrne edition of Euclid for free at the Internet Archive: I downloaded it over the summer as a future reference and then forgot about it until I saw this post.

  6. Mark: I don't think I'll be able to come to your exhibition, unfortunately, but it sure seems quite an interesting one.

    Derik: Thanks for the pdf of Byrne edition of Euclid. I really think it's a wonderful book.


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