[mtbusche]

Are all the atoms in a diamond cubic crystal equivalent?

I've watched a couple of youtube videos that describe the basic cubic unit cell.  Without thinking too hard, one might guess there are three types of atoms:  the atoms at the corners (8 on the cube), the atoms at the faces (6 on the cube), and the 4 internal to the unit cube.  But it seems possible (likely?) to me that under appropriate translations and/or rotations that one could redraw the cubic lattice so that the interior atoms (for example) appear at corners or faces, etc.

Anyway, I've only ever had high school chemistry, and my interest is from a mathematical perspective.

[mjc123]

Yes, they are. Each C atom is surrounded symmetrically by 4 other C atoms in a tetrahedral arrangement. (You can draw out the  diamond structure like this, but then the cubic symmetry isn't obvious.)