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In geometry, the complete or final stellation of the icosahedron〔Coxeter et al. (1938), pp 30–31〕〔Wenninger, ''Polyhedron Models'', p. 65.〕 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. 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. ==History== 480px * 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.〔Louis Poinsot, Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9, pp. 16–48, 1810.〕 * 1812: Augustin-Louis Cauchy made a further enumeration of star polyhedra, proving there are only 4 regular star polyhedra.〔Cromwell (1999) (p. 259)〕 * 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''.〔Wheeler (1924)〕 * 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. The complete stellation is referenced as the eighth in the book. *1974: In Wenninger's 1974 book ''Polyhedron Models'', the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W42. *1995: Andrew Hume named it in his Netlib polyhedral database as the echidnahedron〔The name ''echidnahedron'' may be credited to Andrew Hume, (developer ) of the netlib (polyhedron database ): "... and some odd solids including the echidnahedron (my name; its actually the final stellation of the icosahedron)." (geometry.research; "polyhedra database"; August 30, 1995, 12:00 am. ) 〕 (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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Final stellation of the icosahedron」の詳細全文を読む スポンサード リンク
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