The Best Way to Pack Spheres - Numberphils

I warn you, this one is pretty nerdy.

Most of it is about the mathematical proof that spheres cannot be packed to occupy more than 74.05% of a volume. There's a lot of math and handwaving involved in that proof.

The part that fits with what we teach about material science is from about 1:30 to 6:30. After that it gets into the proof.

I am curious about one thing, though. At 4:53 he shows something that he says is the smallest part of the crystal that can repeat to form the entire crystal. I would normally think of that as the unit cell, but I've never seen that specific image as the unit cell for FCC - which is what I think is being shown. Can anybody help clarify that for me?