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In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an ''8-ic semi-regular figure''.〔Gosset, 1900〕 Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, . The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421, and is the same as the rectified 142. These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: . == 421 polytope== |- |bgcolor=#e7dcc3|6-faces||207360: 138240 69120 |- |bgcolor=#e7dcc3|5-faces||483840 |- |bgcolor=#e7dcc3|4-faces||483840 |- |bgcolor=#e7dcc3|Cells||241920 |- |bgcolor=#e7dcc3|Faces||60480 |- |bgcolor=#e7dcc3|Edges||6720 |- |bgcolor=#e7dcc3|Vertices||240 |- |bgcolor=#e7dcc3|Vertex figure||321 polytope |- |bgcolor=#e7dcc3|Petrie polygon||30-gon |- |bgcolor=#e7dcc3|Coxeter group||E8, () |- |bgcolor=#e7dcc3|Properties||convex |} The 421 is composed of 17,280 7-simplex and 2,160 7-orthoplex facets. Its vertex figure is the 321 polytope. For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection. As its 240 vertices represent the root vectors of the simple Lie group E8, the polytope is sometimes referred to as the E8 polytope. The vertices of this polytope can be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop. 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「4 21 polytope」の詳細全文を読む スポンサード リンク
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