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In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.〔Gosset, 1900〕 Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences. The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is constructed by points at the tetrahedral centers of the 321, and is the same as the rectified 132. These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . == 321 polytope== |- |bgcolor=#e7dcc3|5-faces||6048: 4032 2016 |- |bgcolor=#e7dcc3|4-faces||12096 |- |bgcolor=#e7dcc3|Cells||10080 |- |bgcolor=#e7dcc3|Faces||4032 |- |bgcolor=#e7dcc3|Edges||756 |- |bgcolor=#e7dcc3|Vertices||56 |- |bgcolor=#e7dcc3|Vertex figure||221 polytope |- |bgcolor=#e7dcc3|Petrie polygon||octadecagon |- |bgcolor=#e7dcc3|Coxeter group||E7, (), order 2903040 |- |bgcolor=#e7dcc3|Properties||convex |} In 7-dimensional geometry, the 321 is a uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplex. For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection. The 1-skeleton of the 321 polytope is called a Gosset graph. This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and Coxeter-Dynkin diagram: . 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「3 21 polytope」の詳細全文を読む スポンサード リンク
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