[sd79812]

Why is the molecule's orientation with an electric field affect polarizability?
Electrons are diffuse enough to be independent with respect to orientation and effect of electric field on polarizability?

Why is icosahedral independent of how it faces with the electric field in respect to polarizability?

"Molecular orientation with respect to an electric field can affect polarizibility (labeled
Orientation-dependent), except for molecules that are: tetrahedral, octahedral or icosahedral
(labeled Orientation-independent). This factor is more important for unsaturated molecules that
contain areas of electron-dense regions, such as 2,4-hexadiene. Greatest polarizability in these
molecules is achieved when the electric field is applied parallel to the molecule rather than
perpendicular to the molecule."

[mjc123]

You don't polarise electrons, you polarise the molecule. The distortion of the electron density depends not only on the applied field, but also on the field due to the nuclei, and its orientation relative to the applied field.

Have you studied symmetry? If two directions are symmetry-equivalent, the polarisability in those directions (or any direction in the plane defined by them) is the same. For tetrahedral, octahedral and icosahedral symmetry, all three directions are symmetry-equivalent, so the polarisability is orientation-independent.