  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 all the
Coxeter systems that generate Boyd-Maxwell ball packings.
