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In geometry, a Petrie polygon for a regular polytope of ''n'' dimensions is a skew polygon such that every (''n''-1) consecutive sides (but no ''n'') belong to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belong to one of the faces.〔Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 () (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161)〕 For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, ''h,'' is Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes. == History == John Flinders Petrie (1907–1972) was the only son of Egyptologist Sir W. M. Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability. In periods of intense concentration he could answer questions about complicated four-dimensional objects by ''visualizing'' them. He first noted the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: :One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; infinite but free of false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex.〔H.S.M. Coxeter (1937) "Regular skew polyhedral in three and four dimensions and their topological analogues", Proceedings of the London Mathematical Society (2) 43: 33 to 62〕 In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H.T. Flather to produce The Fifty-Nine Icosahedra for publication.〔H. S. M. Coxeter, Patrick du Val, H.T. Flather, J.F. Petrie (1938) ''The Fifty-nine Icosahedra'', University of Toronto studies, mathematical series 6: 1–26〕 Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car while attempting to cross a motorway near his home in Surrey.〔H.S.M. Coxeter (1973) Regular Polytopes, 3rd edition, page 32〕 The idea of Petrie polygons was later extended to semiregular polytopes. 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「Petrie polygon」の詳細全文を読む スポンサード リンク
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