# Why are the Dynkin diagrams E6, E7 and E8 always drawn the way they are drawn?

The Dynkin diagrams of type ADE are ubiquitous in mathematics as solutions of various classification problems. The diagram E6 is usually drawn by five dots in a row with a sixth dot above the third, see for example here. There would be many other ways to draw the diagram E6, for example the sixth dot below the five dots, or just a capital

E.

Is there a reason for drawing that diagram in that particular shape beside or is that just a confirmed habit?

• I don't necessarily know what I'm talking about, but the interpretation of the vertices of Dynkin diagrams given by John Baez here might be a clue: math.ucr.edu/home/baez/week180.html – Qiaochu Yuan Apr 9 '10 at 16:01
• The only information in the Dynkin diagram is the incidence relations: which node connects to which node. How you choose to embed that graph in the plane is of no consequence. – José Figueroa-O'Farrill Apr 9 '10 at 16:09
• I would disagree that only the connections between the nodes are important. The symmetries of the diagram are also fundamental, and the way we typically draw the diagrams exhibits that symmetry, and in the case of the $E_n$ diagrams, how the cases $n = 7, 8$ extend the more symmetric $E_6$ diagram. – Matthew Stover Apr 9 '10 at 16:18
• @Matthew: I do not understand what you are saying. There is no more information in the Dynkin diagram than in the Cartan matrix. Symmetries of a graph are permutations of the vertices which preserve the incidence relations. How you draw the diagram in the plane is inconsequential. – José Figueroa-O'Farrill Apr 9 '10 at 16:55
• Of course, you're right; one can figure out the automorphisms without the classical picture. I was merely saying that they're always drawn as they are because that's the picture for which the automorphisms stare you in the face. I only meant to argue for aesthetic optimization, not deeper mathematical content. – Matthew Stover Apr 9 '10 at 17:11

The question as stated is not really helpful, but it's worth pointing out that the ADE and other graphs/diagrams evolved over a couple of decades in different countries. The graphs, which encode at first the Coxeter data for finite (or more general) reflection groups, go back at least to Coxeter's 1934 paper. In Witt's 1941 paper these now familiar graphs reappear when the reflection groups are unified with the classification of root systems for semisimple Lie algebras over $\mathbb{C}$. Here the vertices correspond to simple roots, not just simple reflections, so the notation has to incorporate some length information. Dynkin's fundamental 1952 papers used the resulting "Dynkin diagrams" with extra labels 0, 1, 2 at vertices to classify efficiently the nilpotent orbits. Along the way a number of different choices were made about adding edges and arrows or making some vertices darker to distinguish lengths. Bourbaki's 1968 treatise is by now the easiest standard source to follow.
I disagree with the universality of your question, but I agree that the diagrams are often drawn in similar ways. They are drawn that way because they are easy to draw that way, and there isn't a good reason to deviate from what we are taught. I have seen both of your proposed alternatives for the $E_6$ diagram in the literature, and I might even say that your first alternative is the most common drawing I've seen.