How to visualize a $120^\circ$ (or $240^\circ$) rotation of a cube about its body diagonal?

Question

I'm finding rotational symmetries of a cube and have some difficulties with visualizing $120^\circ$ or $240^\circ$ rotations.

Any tips?

Answer 1

Do you see the cube in this figure?

Hint:

Projecting the cube on a plane orthogonal to the main diagonal $1-8$ you find a regular hexagon (you can see an explict construction at: Construct orthogonal projection for plane (matrix form)), and you can easily see the angles.

Answer 2

With reference to your sketch, figure out a plane orthogonal to the axis $1-8$, which is a plane $x+y+z=s$.
Start with $s=0$, it will contain only point $8$.
The plane $s=1$, will contain points $3,5,7$ clearly they are symmetric to the axis, so will be at $120°$ over that plane.
Then plane $s=2$ with points$2,4,6$, also at $120°$ among them, but phased $60°$ with respect to previous triple.
And finally $s=3$ with point $1$.

Answer 3

The $90^\circ$ rotation with the cube in its standard orientation is nice to picture:

Maybe there is something, psychologically, that's nice about vertical rotation axes?

But as Jmoravitz suggests, there is truly no substitute for holding a cube and rotating. Magnus Wenninger's book, Polyhedron Models for the Classroom is a great resource for starting with nice, paper models.