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In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure.〔Gosset, 1900〕 Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221. The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the triangle face centers of the 221, and is the same as the rectified 122. These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . == 2_21 polytope== 40px |- |bgcolor=#e7dcc3|4-faces||648: 432 40px 216 40px |- |bgcolor=#e7dcc3|Cells||1080 40px |- |bgcolor=#e7dcc3|Faces||720 40px |- |bgcolor=#e7dcc3|Edges||216 |- |bgcolor=#e7dcc3|Vertices||27 |- |bgcolor=#e7dcc3|Vertex figure||121 (5-demicube) |- |bgcolor=#e7dcc3|Petrie polygon||Dodecagon |- |bgcolor=#e7dcc3|Coxeter group||E6, (), order 51840 |- |bgcolor=#e7dcc3|Properties||convex |} The 221 has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection. 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「2 21 polytope」の詳細全文を読む スポンサード リンク
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