The E6 Root Polytope

I’ve been thinking about the exceptional Lie algebra E6, as a spinoff of my project on E7, and I wanted to understand its root polytope better. This is 6-dimensional polytope. Let’s climb up to it, starting with some of its 4-dimensional faces, called 4-demicubes. I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams—they’re closely related, and the difference is invisible here. I explained those here: • Symmetry and the fourth dimension: part 3, part 4, part 5, part 6. Let’s dive in! The 4-demicube lives in 4 dimensions. It has 8 vertices. You get it from a 4-dimensional cube, which has 24 = 16 vertices, by keeping every other vertex, throwing away half. That leaves 8. What are its top-dimensional faces, aka ‘facets’? Surprise: there’s only one kind! All of them are regular tetrahedra. In higher dimensions the demicube has two kinds of facet – one from the cube’s corners, one from the cube’s facets. But in 4 dimensions the two kinds happen to be the same…