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In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as ''one-end-ringed'' Coxeter–Dynkin diagrams. The Coxeter symbol for these figures has the form ''k''''i,j'', where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a ''k'' length sequence of branches. The vertex figure of ''k''''i,j'' is (''k'' − 1)''i,j'', and each of its facets are represented by subtracting one from one of the nonzero subscripts, i.e. ''k''''i'' − 1,''j'' and ''k''''i'',''j'' − 1.〔Coxeter 1973, p.201〕 Rectified simplices are included in the list as limiting cases with ''k''=0. Similarly ''0''''i,j,k'' represents a bifurcated graph with a central node ringed. == History== Coxeter named these figures as ''k''''i,j'' (or ''k''''ij'') in shorthand and gave credit of their discovery to Gosset and Elte:〔Coxeter, 1973, p. 210 (11.x Historical remarks)〕 * Thorold Gosset first published a list of ''regular and semi-regular figures in space of ''n'' dimensions''〔Gosset, 1900〕 in 1900, enumerating polytopes with one or more types of regular polytope faces. This included the rectified 5-cell ''0''''21'' in 4-space, demipenteract ''1''''21'' in 5-space, ''2''''21'' in 6-space, ''3''''21'' in 7-space, ''4''''21'' in 8-space, and ''5''''21'' infinite tessellation in 8-space. * E. L. Elte independently enumerated a different semiregular list in his 1912 book, ''The Semiregular Polytopes of the Hyperspaces''.〔E.L.Elte, 1912〕 He called them ''semiregular polytopes of the first kind'', limiting his search to one or two types of regular or semiregular k-faces. Elte's enumeration included all the ''k''''ij'' polytopes except for the ''1''''42'' which has 3 types of 6-faces. The set of figures extend into honeycombs of (2,2,2), (3,3,1), and (5,4,1) families in 6,7,8 dimensional Euclidean spaces respectively. Gosset's list included the ''5''''21'' honeycomb as the only semiregular one in his definition. 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「Gosset–Elte figures」の詳細全文を読む スポンサード リンク
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