Hi, I am currently reading the Feynman Lectures on Physics, and I have just finished the chapter about the geometry and the symmetries of crystals, and there is something I do not quite understand. There are 230 different possible symmetries which are grouped into seven classes (triclinic --> cubic). My question is : why is a cubic crystal (for example) necessarily invariant under rotations, and thus necessarily an isotropic dielectric ? Couldn't we imagine a cubic lattice but with an internal pattern which is not invariant under rotation (with x-axis oriented arrows for instance) ? What I do not understand is that, for instance, we know there are cubic crystals which are not centrosymmetric, but Feynman says that every cubic crystal is invariant under rotations and thus isotropic dielectrics. If I can find cubic pattern which are not centrosymmetric, I do not see why I couldn't find cubic patterns which are not invariant under such rotations... Thanks.