E8 is the largest of the five exceptional Lie groups — a 248-dimensional object of staggering symmetry, often called the most beautiful structure in mathematics. Its 240 "root vectors" live in 8 dimensions; projected onto the special Coxeter plane they fall into eight perfect concentric rings of 30. This picture is computed live: the 240 roots are generated, the Coxeter element is built from reflections, and its rotation plane is found by diagonalizing an 8×8 matrix in your browser.
why mathematicians revere this object
A root system is the symmetry skeleton of a Lie group — the set of vectors describing how its generators rotate into each other. E8's root system has 240 vectors in 8 dimensions, all the same length, arranged so that reflections in them permute the whole set. They come in two families: the 112 vectors (±1, ±1, 0,0,0,0,0,0) in all positions, and the 128 vectors ½(±1,…,±1) with an even number of minus signs.
The mesmerizing image above is the Coxeter plane projection (Petrie polygon). E8's Coxeter number is h = 30, and projecting the 240 roots onto the 2-plane in which the Coxeter element acts as a rotation by 2π/30 sorts them into 8 concentric 30-gons — the 8 reflecting the rank, the 30 reflecting h. Nothing about this layout is hand-placed: the simple roots generate eight reflections, their product is the Coxeter element, and its slowest rotation plane (the eigenvectors for eigenvalue 2cos(2π/30)) is extracted by a Jacobi eigensolver running here, then every root is projected onto it.
The web of edges connects roots at 60° to each other (inner product 1) — the structure of the Gosset polytope 4₂₁, whose 240 vertices are the roots and whose 6,720 edges are these connections.
Where E8 shows up: it's the symmetry of the densest known sphere packing in 8 dimensions — proven optimal by Maryna Viazovska in 2016, work that won her the Fields Medal in 2022. It governs heterotic string theory (the gauge group E8 × E8). In 2010 its 248-dimensional structure was glimpsed in a real material — a cobalt-niobate quantum magnet at criticality showed excitation masses in the golden ratio predicted by E8 symmetry. Garrett Lisi's (controversial) "An Exceptionally Simple Theory of Everything" tried to fit all known particles and forces into E8. And computing its full character table in 2007 — a 453,060 × 453,060 matrix, 60 GB of answers — was a landmark of computational mathematics. Few objects sit at the crossroads of geometry, number theory, and physics like E8.