The icosahedral point group

A nice feature of the package is that it knows all pint group symmetries, not only the crystallographic ones. For example, the command ShowCrystal[xx=MakeCrystal[{{0,0,1}},"I"]] shows a regular dodecahedron, the faces of which are perpendicular to the 5fold symmetry axes of the point group:

But there is more one can do. With the command Cases[Symmetry["I"], {{_, _, _}, 2 p/3, 1}][[1, 1]] one obtains one of the 3fold symmetry axis of the point group. Using this axis as the Miller index input to the MakeCrystal command produces a polyhedron with faces perpendicular to the 3fold symmetry axes of the group. Thus, ShowCrystal[{Cases[Symmetry["I"], {{_, _, _}, 2 p/3, 1}][[1, 1]]}, "I"] yields:

The same trick can be done with the 2fold symmetry axes. With ShowCrystal[{Cases[Symmetry["I"], {{_, _, _}, 2 p, 1}][[1, 1]]}, "I"] one obtains