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Lists of stellations of polyhedrons are given particular rules like in the book 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 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?

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3 Answers 3

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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.

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re: 2^473-1 types of combinations. At least it isn't as bad as Ramsey theory! –  Jason Dyer Nov 5 '09 at 11:03

If a stellation is formed by vertices, edges, and polygons that lie in the planes of the faces of the original polyhedron, then there are only finitely many vertices, edges, and faces to be used (subsets of a finite arrangement of planes) and therefore finitely many stellations.

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This begs the question: why does it seem the stellations are being enumerated piecemeal? Why not drop all the rules and get a full count? –  Jason Dyer Nov 4 '09 at 17:41

The new rules are a modification of Miller's rule the concept of false vertices and edges allows the extension of existing definitions including more stellations the concerns involved in Miller's rules are still a consideration. In the article here:

http://www.steelpillow.com/polyhedra/icosa/searchlost/searchlost.htm

the author is not interested in abandoning the rules but improving them. Some of the rules are important for instance those insuring symmetry. I think under all these rules the number of stellations is finite.

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