Lists of stellations of polyhedrons are given particular rules like in the book [*The Fifty Nine Icosahedra*](http://en.wikipedia.org/wiki/The_fifty_nine_icosahedra) which follows "Miller's Rules". 


There seems to be no "correct" ruleset to use, so [more stellations are still being discovered](http://www.steelpillow.com/polyhedra/icosa/lost/lost.html) using alternate rules.


However, I have read no mention of the stellations being infinite; unlike cumulations there does seem to be some finite restriction to the total. Is this true? If not, which ruleset(s) will produce an infinite number of stellations?



Are there infinite sets of stellations of polyhedra?
Let me quote from "In search of the lost icosahedra" the paper I mentioned above:


"Stellations of a polyhedron are obtained by extending some of its edges or faces until they intersect at a distance from the original polyhedron. One way of studying stellations is to consider the planes in which the faces of the polyhedron lie, that is, its face planes. The face planes of the regular icosahedron intersect eachother (see Appendix) to dissect space into numerous regions, of which 473 are finite cells. These cells come in just 12 shapes which form layers around the original icosahedron, itself the innermost cell. The set of cells of a given shape comprises part or all of a layer, with icosahedral symmetry. The various stellations can be obtained by selecting different combinations of these cell sets. Because there are 12 types of cell and we are not interested in the 'empty' combination, there are 212 - 1 = 4,095 possible combinations."


So in this case or any other case we will be limited to a finite number of cells. Even if we ignored the 12 types and considered all types of cells there would still be 2^473-1
types of combinations. In general there will be a finite number of regions formed by the face planes of a polyhedron and this will result in a limitation on the number combinations to a finite although possibly very large number.


