Concept
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. This polyhedron is the seventeenth stellation of the icosahedron, and given as Wenninger model index 42. As a geometrical figure, it has two interpretations, described below: As an irregular star (self-intersecting) polyhedron with 20 identical self-intersecting enneagrammic faces, 90 edges, 60 vertices. As a simple polyhedron with 180 triangular faces (60 isosceles, 120 scalene), 270 edges, and 92 vertices. This interpretation is useful for polyhedron model building. Johannes Kepler researched stellations that create regular star polyhedra (the Kepler-Poinsot polyhedra) in 1619, but the complete icosahedron, with irregular faces, was first studied in 1900 by Max Brückner. 1619: In Harmonices Mundi, Johannes Kepler first applied the stellation process, recognizing the small stellated dodecahedron and great stellated dodecahedron as regular polyhedra. 1809: Louis Poinsot rediscovered Kepler's polyhedra and two more, the great icosahedron and great dodecahedron as regular star polyhedra, now called the Kepler–Poinsot polyhedra. 1812: Augustin-Louis Cauchy made a further enumeration of star polyhedra, proving there are only 4 regular star polyhedra. 1900: Max Brückner extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the complete stellation. 1924: A.H. Wheeler in 1924 published a list of 20 stellation forms (22 including reflective copies), also including the complete stellation. 1938: In their 1938 book The Fifty Nine Icosahedra, H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie 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.
