I’ve been thinking about the exceptional Lie algebra E6, as a spinoff of my project on E7, so I want to get a good mental picture of the E6 root polytope. This is 6-dimensional polytope. Let’s climb up to it, starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking a 4-dimensional cube, or tesseract, and removing every other corner.

I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams: they’re closely related, and...