The above screencast, by David Koski, was meant to have sound.
He narrated over the phone what was going on.
I thought I could keep it in my head, but the zig-zag pattern of three tetrahedrons in the frustum is frustrating to memorize.
Start with ye regular old tetrahedron then move a face outward from its opposite vertex to make the edges phi (1.618...) times what they were.
The three zig zag tetrahedrons inside are now 1/phi, 1, phi in ratio, where 1 = the volume of the original tetrahedron.
You may generalize this result to non-regular tetrahedrons.
Constructing the Desert Tetrahedron (Dec, 2013)
Multiplying Desert Tetrahedron by PHI (Dec, 2013)