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Outermost stellation of the icosahedron

In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron.

Johannes Kepler in his Harmonices Mundi applied the stellation process, recognizing the small stellated dodecahedron and great stellated dodecahedron as regular polyhedra. However, Louis Poinsot in 1809 rediscovered two more, the great icosahedron and great dodecahedron. This was proved by Augustin Louis Cauchy in 1812 that there are only four regular star polyhedrons, known as the Kepler–Poinsot polyhedron. extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the complete stellation. published a list of twenty stellation forms (twenty two including reflective copies), also including the complete stellation. H. S. M. Coxeter, P. du Val, H. T. Flather and J. F. Petrie in their 1938 book The Fifty Nine Icosahedra stated a set of stellation rules for the regular icosahedron and gave a systematic enumeration of the fifty nine stellations which conform to those rules. The complete stellation is referenced as the eighth in the book. In Wenninger's book Polyhedron Models, the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W42. In 1995, Andrew Hume named it in his Netlib polyhedral database as the echidnahedron, after the echidna, or spiny anteater is a small mammal that is covered with coarse hair and spines and which curls up in a ball to protect itself.

The stellation of a polyhedron extends the faces of a polyhedron 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.

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).