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

Cluster size in bond percolation on the Platonic solids