Cut up a cube into pieces that form 3 regular tetrahedra?

Question

Everyone knows that a regular tetrahedron fits inside a cube, and that the volume of the tetrahedron is 1/3 that of the cube.

(For a picture, see this question or this Google image search.)

The question is, once you've cut the corners off the cube to make that tetrahedron, what's a simple way to cut up those four corner pieces so that their parts can be rearranged to form two additional regular tetrahedra?

Answer 1

You can't do it.

This is because of Hilbert's third problem: https://en.wikipedia.org/wiki/Hilbert%27s_third_problem

(At least, I think this is true.)

Answer 2

How are you defining "simple"? Anyone who has taken calculus knows that the pieces can be atomized into miniscule microcubes, and these can be reassembled into any shape having the same volume. I suggest your problem statement be tweaked to exclude Newtonian atomization. Conceputally, atomization is simple...it is the basis for plastics molding and metal castings.