Stellations of the Icosahedron

The only regular polyhedron whose stellations were not completely known in the last century is the icosahedron. The enumeration of all stellations of the icosahedron was accomplished in 1938 by Coxeter and Du Val. The results were published in the University of Toronto Studies (Mathematical Series). The paper contained isometric drawings by Petrie and planar drawings by Du Val. The fourth author of that paper, Flather, prepared models of all 59 solids. Their whereabouts are explained in the preface of a 1982 Springer-Verlag reprint of the 1938 paper [Coxeter82]. Much of the material in this article is taken from that source. All code is part of the package Icosahedra.m.

The subdivision of the plane of an icosahedral face by the intersections of all other planes (except the opposite plane, which is parallel to it) is shown below. This graphic is the value of the variable lineGraphics from Icosahedra.m.

  Show[ lineGraphics ];

The following graphic leaves out the outermost points and allows a better view into the crowded central triangle.

  Show[ lineGraphicsInner ];

The lines so constructed divide the plane into fifteen different kinds of triangular or quadrilateral polygons, not counting the infinite exterior ones. The vertices of these polygons are the points a through k. Since the stellation should have icosahedral symmetry, the polygons in one plane that occur in this stellation must have threefold rotational symmetry. We combine sets of polygons with this symmetry into facets. The polygons in one facet are obtained from a single polygon by successive rotation by 120 degrees. Following Coxeter we number the facets 0 through 14. Number 0 is the innermost triangle. The other facets have either threefold rotational or sixfold dihedral symmetry and consist of three polygons each. The chiral facets, that is, those with only rotational symmetry, come in sets of two, one set being the mirror image of the other one. These two sets are labeled 2_l and 2_r, for example. In some stellations, the two related chiral facets occur together. Their union can be treated like a single facet with dihedral symmetry. It is, therefore, denoted by 2 and consists of all six polygons from 2_l and 2_r.

Here are facets 0 through 12. Not shown are the outer ones, No. 13 (consisting of the triangles bl_i dl_i c_i and br_i c_i dr_i) and No.14 (consisting of the triangles a_i dr_i+1 dl_i). By leaving out the large outer facets, we can show more details of the innermost facets. The graphic object is the value of the variable faceGraphicsInner. It uses the package Graphics`Legend` to produce the legend of color indices.

  Show[ faceGraphicsInner ];

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