We first draw a 3-fold rotation axes, then construct a circle centered with the axis and cross the three vertex on the end the three connected sides.

Second, we rotate the original cube and copy it.

Set one of the vertex on the end the three connected sides being moved around with the circle.

This is the rotated cube which will be projected on the original cube.

Select one face of the rotated cube and the center of it and then construct a convex hull composed with.

This constructed convex hull is a pyramid which is irregular.

The height of it is , where a is the side of the original cube.

This irregular pyramid is seemed as a splitter, which cut the original cube into component.

Because the interface between it and the original cube is the projection of the rotated cube, which was prooved by linear matrix equation.

We cut the original cube with the splitter which metioned in the Step 2.

The polygon with red faces is what we construct after splitting.

And the face will change when the vertex of rotated cube rotating, which shows the projected surface.

We see this resulted polygon as a component which combine a new cube because of symmetry.

We start to duplicate the component with the symmetric properties of cube.

First, the polygon with yellow surfaces is duplicated by the red one with the 3-fold reflection axes.

Second, the polygon with blue surfaces is duplicated by the yellow one with the 3-fold reflection axes.

Then we duplicate the polygon with yellow surfaces marked stripes with point symmetric.

Repeat the step above to construct the polygon with red surface.

We can find that the projection components are in pairs because of point symmetric properties of cube.

Finally, is the blue pairs of polygon.

Then we can construct the final projection of the rotated cube on the original cube with these components.