  In this paper, we explore the geometry and the arithmetic of a family of
polytopal sphere packings induced by regular polytopes in any dimension. We
prove that every integral polytope is crystallographic, and we show that there
are 11 crystallographic regular polytopes in any dimension. After introducing
the notion of Apollonian section, we determine which Platonic crystallographic
packings emerge as cross-sections of the Apollonian arrangements of the regular
4-polytopes. Additionally, we compute the M\"obius spectrum of every regular
polytope.
