A general connected simple graph supports a rich geometry. This is determined by a quadratic form arising from the Mutation Game, and generalizing the Cartan matrix for ADE graphs. An associated root system is finite and classical for the ADE graphs, unbounded but still mostly classical for the ADE~ affine cases, and largely unexplored for the rest. The associated groups of symmetries are called Weyl groups and were studied also by Coxeter in the ADE situations (and some others which we don't discuss).
The story is motivated by fascinating connections with sphere packing and kissing problems, Hurwitz quaternions and some remarkable lattices in higher dimensions. We will be looking at some novel ideas for understanding the geometry coming from Rational Trigonometry.
This lecture will be one of several in a series roughly on ADE graphs, but will be largely self-contained.