Lorentzian Coxeter systems and Boyd–Maxwell ball packings

Abstract

In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd–Maxwell ball packings. Finally, we enumerate the Coxeter systems that generate Boyd–Maxwell ball packings.