  The Platonic solids is the name traditionally given to the five regular
convex polyhedra, namely the tetradron, the octahedron, the cube, the
icosahedron and the dodecahedron. Perhaps strongly boosted by the towering
historical influence of their namesake, these beautiful solids have, in well
over two millenia, transcended traditional boundaries and entered the stage in
a range of disciplines. Examples include natural philosophy and mathematics
from classical antiquity, scientific modeling during the days of the european
scientific revolution and visual arts ranging from the renaissance to
modernity. Motivated by mathematical beauty and a rich history, we consider the
Platonic solids in the context of modern quantum mechanics. Specifically, we
construct Bell inequalities whose maximal violations are achieved with
measurements pointing to the vertices of the Platonic solids. These Platonic
Bell inequalities are constructed only by inspecting the visible symmetries of
the Platonic solids. We also construct Bell inequalities for more general
polyhedra and find a Bell inequality that is more robust to noise than the
celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on
the tension between mathematical beauty, which was our initial motivation, and
experimental friendliness, which is necessary in all empirical sciences.
