Seventeen interactive explainers. An interactive journey from the basic building blocks of symmetry to the massive E₈ lattice and the densest possible sphere packings.
Symmetry seems infinite, but mathematicians discovered long ago that continuous symmetry has a complete classification — the finite-dimensional simple Lie algebras over ℂ fall into just four infinite families, plus five strange "exceptions." Through these interactive tools, explore the geometry of these exceptions.
Build symmetry groups, manipulate complex geometries, and discover how these bizarre mathematical objects are intimately connected to the physics of the universe, string theory, and the most efficient ways to pack spheres into high-dimensional space.
At the end of the classification, the full list consists of these nine types. The first four are infinite families parameterised by rank; the last five are singular.