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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 Icosian calculus ： ウィキペディア英語版 Icosian calculus The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations. Hamilton’s discovery derived from his attempts to find an algebra of "triplets" or 3-tuples that he believed would reflect the three Cartesian axes. The symbols of the icosian calculus can be equated to moves between vertices on a dodecahedron. Hamilton’s work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. He also invented the icosian game as a means of illustrating and popularising his discovery. == Informal definition == The algebra is based on three symbols that are each roots of unity, in that repeated application of any of them yields the value 1 after a particular number of steps. They are: :$$ \begin \iota^2 & = 1, \\ \kappa^3 & = 1, \\ \lambda^5 & = 1. \end Hamilton also gives one other relation between the symbols: :$\backslash lambda\; =\; \backslash iota\backslash kappa.\backslash ,\backslash !$ (In modern terms this is the (2,3,5) triangle group.) The operation is associative but not commutative. They generate a group of order 60, isomorphic to the group of rotations of a regular icosahedron or dodecahedron, and therefore to the alternating group of degree five. Although the algebra exists as a purely abstract construction, it can be most easily visualised in terms of operations on the edges and vertices of a dodecahedron. Hamilton himself used a flattened dodecahedron as the basis for his instructional game. Imagine an insect crawling along a particular edge of Hamilton's labelled dodecahedron in a certain direction, say from $B$ to $C$. We can represent this directed edge by $BC$. *The icosian symbol $\backslash iota$ equates to changing direction on any edge, so the insect crawls from $C$ to $B$ (following the directed edge $CB$). *The icosian symbol $\backslash kappa$ equates to rotating the insect's current travel anti-clockwise around the end point. In our example this would mean changing the initial direction $BC$ to become $DC$. *The icosian symbol $\backslash lambda$ equates to making a right-turn at the end point, moving from $BC$ to $CD$. 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「Icosian calculus」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.