The octahedron is related to the cube by placing each point of an octahedron at the center of each face of a cube. But every point of the tetrahedron can serve as the top of the pyramid because all four points are identical. The tetrahedron is related to the octahedron by placing the six points of the octahedron in the middle of the six edges of the tetrahedron. In fact the four faces of the tetrahedron would be parallel to four of the eight faces of the octahedron. The tetrahedron can be thought of as having removed every other face from an octahedron and extending the four faces that are left to complete the closed tetrahedron.
Tetrahedrons only form in isometric minerals that have four fold rotoinversion axes.
The third dodecahedron has asymmetrical pentagonal faces. The pentagonal dodecahedron is termed a pyritohedron because it is most commonly found in the mineral pyrite . The pyritohedron is related to the cube by orienting two of the pyritohedron's faces essentially with each cube face. Two of the pentagonal faces touch bases in a line or edge that is parallel to the cube faces. This edge is responsible for the striations seen on pyrite's cubic faces when the two forms are trying to form together.