I am trying to divide octahedron into congruent parts. I found octahedron inside tetrahedron sided by four smaller tetrahedrons. I found some division here to 12 congruent parts. I can divide octahedron to four congruent parts by cutting it by fartest-away vertices.
Does there exist some sort of general theory about dividing the octahedron or other regular polyhedrons such as tetrahedrons?