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In mathematics, the binary icosahedral group 2I or <2,3,5> is a certain nonabelian group of order 120. It is an extension of the icosahedral group ''I'' or (2,3,5) of order 60 by a cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism : of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120. It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3). The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) ==Elements== Explicitly, the binary icosahedral group is given as the union of the 24 Hurwitz units : with all 96 quaternions obtained from :½ ( 0 ± ''i'' ± φ−1''j'' ± φ''k'' ) by an even permutation of all the four coordinates 0, 1, φ−1, φ, and with all possible sign combinations. Here φ = ½ (1 + √5) is the golden ratio. In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unit quaternion group Sp(1). The convex hull of these 120 elements in 4-dimensional space form a regular 4-polytope, known as the 600-cell. 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「Binary icosahedral group」の詳細全文を読む スポンサード リンク
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