Tetrahedral vs. Cube Symmetries

(Tetrahedron inscribed in cube)

The above figure graphically demonstrates that the symmetry group of the tetrahedron (Td) is a sub-group of the symmetry group of the cube (Oh). That is, any rotation, reflection, etc., which takes the tetrahedron into itself necessarily takes the cube into itself, but not vice versa!
© 1999 R. J. Hauenstein