Can an irregular tetrahedron be inscribed in a cuboid or prism?

Question

A regular tetrahedron can be inscribed in a cube in a way that the tetrahedron edges are diagonals of the cube faces.

Is it possible to similarly inscribe any irregular tetrahedron in a cuboid or prism?

Image source: http://mathworld.wolfram.com/RegularTetrahedron.html

Answer 1

Affine transformations of the cube allow inscribing absolutely any tetrahedron in a parallelepiped, the unit cell of the triclinic crystal system. Now, for the three opposite pairs of edges of a tetrahedron, consider the vectors formed as cross products of each pair. Extending the previous line of reasoning, we see that

• a tetrahedron may be inscribed in a parallelogram prism, unit cell of the monoclinic crystal system, iff one vector is orthogonal to the other two
• it may be inscribed in a cuboid, unit cell of the orthorhombic crystal system, iff the vectors are mutually orthogonal.