  The main objective of this paper is to study the size of a typical cluster of
bond percolation on each of the five Platonic solids: the tetrahedron, the
cube, the octahedron, the dodecahedron and the icosahedron. Looking at the
clusters from a dynamical point of view, i.e., comparing the clusters with
birth processes, we first prove that the first and second moments of the
cluster size are bounded by their counterparts in a certain branching process,
which results in explicit upper bounds that are accurate when the density of
open edges is small. Using that vertices surrounded by closed edges cannot be
reached by an open path, we also derive upper bounds that, on the contrary, are
accurate when the density of open edges is large. These upper bounds hold in
fact for all regular graphs. Specializing in the five~Platonic solids, the
exact value of (or lower bounds for) the first and second moments are obtained
from the inclusion-exclusion principle and a computer program. The goal of our
program is not to simulate the stochastic process but to compute exactly sums
of integers that are too large to be computed by hand so these results are
analytical, not numerical.
