In crystallography, the dyakis dodecahedron only exists in one crystal, which is pyrite. Pyrite has other forms other than the dyakis dodecahedron, including tetrahedra, octahedra, cubes and pyritohedra. Though the cube and octahedron are  in the cubic crystal system, the dyakis dodecahedron and the pyritohedon are in the isometric crystal system and the tetrahedron is in the tetrahedral crystal system.  Though the dyakis dodecahedron has 3-fold axes like the pyritohedron and cube, it doesn't have 4-fold axes but it does have order-4 vertices, as when the dyakis dodecahedron is rotated 90 or 270° along an order-4 vertex, it is not the same as before, because the order-4 vertices act as 2-fold axes, as when they are rotated a full turn or 180°, the polyhedron looks the same as before.