into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. The Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation. The Du Val symbol of the complete stellation is H, because it includes all cells in the stellation diagram up to and including the outermost "h" layer. As a simple polyhedron As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2. The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio When regarded as a three-dimensional solid object with edge lengths a, φa, φ2a and φ2a (where φ is the golden ratio) the complete icosahedron has surface area and volume As a star polyhedron The complete stellation can also be seen as a self-intersecting star polyhedron having 20 faces corresponding to the 20 faces of the underlying icosahedron. Each face is an irregular 9/4 star polygon, or enneagram. Since three faces meet at each vertex it has 20 × 9 / 3 = 60 vertices (these are the outermost layer of visible vertices and form the tips of the "spines") and 20 × 9 / 2 = 90 edges (each edge of the star polyhedron includes and connects two of the 180 visible edges). When regarded as a star icosahedron, the complete stellation is a noble polyhedron, because it is both isohedral (face-transitive) and isogonal (vertex-transitive). See also Kepler–Poinsot polyhedron List of Wenninger polyhedron models Notes References Brückner, Max (1900). Vielecke und Vielflache: Theorie und Geschichte. Leipzig: B.G. Treubner. . WorldCat English: Polygons and Polyhedra: Theory and History. Photographs of models: Tafel VIII (Plate VIII), etc. High res. scans. A. H. Wheeler, Certain forms of the icosahedron and a method for deriving and designating higher polyhedra, Proc. Internat. Math. Congress, Toronto, 1924,