We first draw a 3-fold rotation axes, then construct a circle centered with the axis and cross the vertex on the buttom face.

Second, we rotate the original tetrahedron and copy it.

Set one of the vertex on the buttom face being moved around with the circle.

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

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

This constructed convex hull is a new tetrahedron which is irregular.

The height of it is 1-, where a is the side of the original tetrahedron.

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

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

We cut the original tetrahedron 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 tetrahedron rotating, which shows the projected surface.

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

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

First, the polygon with blue surfaces is duplicated by the red one with the 2-fold reflection axes.

Second, the polygon with yellow surfaces is duplicated by the red one with the other 2-fold reflection axes.

Finally, the polygon with green surfaces is duplicated by the yellow one with the 2-fold reflection axes.

Then we find the final result is the projection of a rotated tetrahedron on the original tetrahedron.