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A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as ''A''5 (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product ''A''5 × ''Z''2. The latter group is also known as the Coxeter group ''H''3, and is also represented by Coxeter notation, () and Coxeter diagram . == As point group == Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups. Icosahedral symmetry is ''not'' compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are: : : These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups. The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus. Note that other presentations are possible, for instance as an alternating group (for ''I''). 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「icosahedral symmetry」の詳細全文を読む スポンサード リンク
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