The enumeration of all stellations of the icosahedron was accomplished
in 1938. The geometric constructions and combinatorial algorithms
used can easily be programmed in Mathematica. Its symbolic  and
graphic capabilities make it well suited to render the solids in
a variety of formats.  Icosahedra.m is a package for rendering all
59 stellations of the icosahedron.  Some examples for its use are
in the notebook Icosahedra.ma.

Sources:

Maeder, Roman E. 1993.
	The Stellated Icosahedra. Mathematica in Education, 3(1).

H. S. M. Coxeter, Patrick du Val, H. T. Flather, and J. F. Petrie. 1982
	The Fifty-Nine Icosahedra. Springer-Verlag.
	Originally published by Univ. of Toronto Press, 1938.