Rotating cube

Here's an investigation opportunity that came up as a student was playing with my new favorite toy.

A cube is rotating about its main diagonal:

I came up with a few questions to challenge students at different levels (What is the length of the main diagonal? What type of curve appears in the silhouette? What is the volume of the rotated solid?) but wondered what else we could ask about this. I haven't solved the latter two - maybe you can.