What's the E8 Picture?

I've gotten a fair number of requests for a plain explanation of what the "E8 picture" represents.

Here's the story.

The Lie group E8 has a "root system" associated to it that consists of 240 points in 8-dimensional space. Similarly, the Lie group E7 has a root system of 126 points in 7-dimensional space.

These 240 points are tightly packed together in a highly symmetric way. In fact, this configuration has a total of 696,729,600 symmetries. Contrast this with what happens if you take the 8 points at the corners of a 3-dimensional cube. These 8 points have "only" 48 rotational and reflective symmetries.

Of course we can't really visualize any objects in 8 dimensions, but we can draw 2-dimensional projections of them. For example, if you imagine shining a flashlight on a cube, the shadow it casts would (depending on how you orient the cube) look like a hexagon. And if you orient the cube just right, the shadow it casts would look like a regular hexagon--a six sided figure with all sides of equal length and all angles of equal measure. Taking it one step further, if you imagine the cube as a wire frame -- 8 points together with links that connect along edges of the cube, then the projection would look like 6 dots at the corners of a hexagon, plus another dot in the center, plus lines connecting nearby dots.

What I've done with the root system of E8 is completely analogous. I picked the "just right" direction to shine a light on these 240 points so that the 2-dimensional shadow they cast is as symmetric as possible. If you study the figure, you'll see that it has 60 symmetries: 30 rotations and 30 reflections. The 240 points wind up in 8 concentric rings of 30 points each--those are the black dots.

The lines in the figure are projections of a wire frame. Imagine, back in the 8-dimensional space where the 240 points live, having a line that connects every point to its nearest neighbors among the other 239. One of the amazing things about these 240 points is that each point has 56 nearest neighbors--they really are tightly packed!

So in the picture, there should be 56 lines emanating from each of the black dots. But some of the lines cast shadows on top of each other, so in fact you see fewer. For example, there are only 28 visible edges emanating from each of the points on the outermost ring.

The colors of the lines are of no technical importance--I simply chose them to provide better contrast.

I should add that a version of this picture was first drawn (without the aid of a computer!) in the 1960's by Peter McMullen. Scanned versions of his black-and-white drawing were also published as part of the recent media coverage, including the New York Times.

John Stembridge

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This page last modified Tue Apr 10 11:28:29 EDT 2007