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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