# Grothendieck's mathematical diagram.

I was going through this article which appeared in the Notices of the AMS, and in it, there's a picture which shows a mathematical diagram drawn by Grothendieck. I would be delighted if anyone could explain what that is. Thanks in advance.

EDIT: The comments below suggest that the diagram is a dessin d'enfant. And even though I am by no means an expert, I somehow feel that it may not be one. I'd love some clarification on how one may call it so. Questions I want to ask now, are

1) Is the diagram really a dessin? Or is it something else?

2) Do the shaded parts of the diagram signify something? Are there any special types of dessins which have a similar a structure?

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Diagram is on page 936. – supercooldave Jun 28 '10 at 17:22
At least the superimposed image is the mirror image of the one below. But what else? – Unknown Jun 28 '10 at 17:35
– Unknown Jun 29 '10 at 7:00

First of all, I believe that according to Grothendieck's definition of dessins d’enfants the picture (if I am looking at the right one) indeed seems to show one. At the same time you have a point that this is not one of the more interesting ones.

On the other hand it might be the very first one Grothendieck ever drew and then one could make a wild guess as to what it shows. I would venture to say that the diagram in question shows the complex conjugation of the Riemann sphere.

If I understand correctly, Grothendieck's inventing and studying dessins d’enfants was motivated by his goal of finding non-trivial elements of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb Q)$. An obvious (and the only obvious) non-trivial element is complex conjugation of $\mathbb C$, which actually extends to $\mathbb P^1_{\mathbb C}$, a.k.a. the Riemann sphere.

My guess is that this picture shows that and was perhaps the visual clue that led Grothendieck to make the definition of dessins d’enfants.

Addendum: To answer the question raised in the comments: The picture clearly shows a reflection. Complex conjugation is a reflection. I did not claim I have a proof for this, indeed, notice the words wild guess above. The only argument I can offer is that

i) it is reasonable to assume that this picture is or at least has something to do with dessins d’enfants.

ii) it is a very simple drawing for that

iii) there should still be some significance for someone to have put it in the article

iv) it is reasonable to assume that it is an early drawing of dessins d’enfants

v) it is clearly a reflection

vi) complex conjugation is a reflection and has a lot to do with the birth of dessins d’enfants.

As I said, this is a guess, but I wonder if anyone can offer anything other than a guess.

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Sandor--could you elaborate on how this picture shows complex conjugation? – Daniel Litt Mar 14 '11 at 9:12
Hmm...I assumed the two green circles were two hemispheres of the Riemann sphere (hence the two arrows in the diagram would indicate where to glue); then the red lines would be a graph embedded in the Riemann sphere. But I guess you are suggesting the two green circles are a "before and after" picture, which is why you say this is clearly a reflection? Or am I misinterpreting your interpretation? – Daniel Litt Mar 14 '11 at 17:57
Daniel, you are not misinterpreting :). On the other, I don't think those are hemispheres! I think that those two red arcs that go sort of horizontally are "big circles" of a sphere, so to me it seems that the picture shows two spheres (i.e., not hemi-). – Sándor Kovács Mar 15 '11 at 0:56