## Enumeration of all Possibilities

Our task is now to find all combinations of facets that lead to a valid stellation of the icosahedron. First, we have to define what we shall consider a valid stellation. Coxeter gives five conditions due to J. C. P. Miller:

- (i)
- The faces must lie in the twenty bounding planes of the icosahedron.
- (ii)
- The parts of the faces in the twenty planes must be congruent. Those parts lying in one place may be disconnected, however.
- (iii)
- The parts lying in one plane must have threefold rotational symmetry, with or without reflections.
- (iv)
- All parts must be accessible (they must lie on the outside of the solid).
- (v)
- Compounds that can be divided into two sets each of which has the full symmetry of the whole are excluded.

Condition (i) is automatically satisfied by our construction. Condition (iii) guarantees that the stellation has icosahedral symmetry. Those stellations having the full icosahedral symmetry (including reflections) are called *reflexible*. Those that have only rotational symmetry are called *chiral*.

A stellation is described by a set of facets. Since facets have threefold rotational symmetry, this choice satisfies condition (iii). This set of facets is the part of the face of the stellation that lies in one plane. By condition (ii) this ``layout'' is the same in each of the twenty planes. Conditions (iv) and (v) restrict the selection of subsets of the facets. All valid subsets can be found by geometrical reasoning. We will not repeat these considerations here but refer the reader to Coxeter's paper. The stellation itself consists of the faces contained in the facets in all twenty planes.

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