It can be named by its Schläfli symbol as {5, 3n − 2} (dodecahedral) or {3n − 2, 5} (icosahedral).
The family starts as 1-polytopes and ends with n = 5 as infinite tessellations of 4-dimensional hyperbolic space.
The complete family of icosahedral pentagonal polytopes are: The facets of each icosahedral pentagonal polytope are the simplices of one less dimension.
Their vertex figures are icosahedral pentagonal polytopes of one less dimension.
[1] Like other polytopes, regular stars can be combined with their duals to form compounds; Star polytopes can also be combined.