  In this paper we aim to determine the baryon numbers at which the minimal
energy Skyrmion has icosahedral symmetry. By comparing polyhedra which arise as
minimal energy Skyrmions with the dual of polyhedra that minimize the energy of
Coulomb charges on a sphere, we are led to conjecture a sequence of magic
baryon numbers, B=7,17,37,67,97,... at which the minimal energy Skyrmion has
icosahedral symmetry and unusually low energy. We present evidence for this
conjecture by applying a simulated annealing algorithm to compute energy
minimizing rational maps for all degrees upto 40. Further evidence is provided
by the explicit construction of icosahedrally symmetric rational maps of
degrees 37, 47, 67 and 97. To calculate these maps we introduce two new methods
for computing rational maps with Platonic symmetries.
