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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Compound of three octahedra ： ウィキペディア英語版 Compound of three octahedra In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut ''Stars''. ==Construction== A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra.〔.〕 The eight cube vertices are the same as the eight points in the compound where three edges cross each other.〔 Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1:√2.〔 The remaining octahedron edges cross each other in pairs, within the interior of the compound; their crossings are at their midpoints and form right angles. The compound of three octahedra can also be formed from three copies of a single octahedron by rotating each copy by an angle of π/4 around one of the three symmetry axes that pass through two opposite vertices of the starting octahedron.〔.〕 A third construction for the same compound of three octahedra is as the dual polyhedron of the compound of three cubes, one of the uniform polyhedron compounds. The six vertices of one of the three octahedra may be given by the coordinates and . The other two octahedra have coordinates that may be obtained from these coordinates by exchanging the ''z'' coordinate for the ''x'' or ''y'' coordinate.〔〔 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「Compound of three octahedra」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.