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In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') does not change the object. They are the finite symmetry groups on a cone. For ''n'' = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation. == Types == ;Chiral: *''Cn'', ()+, (''nn'') of order ''n'' - ''n''-fold rotational symmetry - acro-n-gonal group (abstract group ''Cn''); for ''n''=1: no symmetry (trivial group) ;Achiral: *''Cnh'', (), (''n'' *) of order 2''n'' - prismatic symmetry or ortho-n-gonal group (abstract group ''Cn'' × ''D1''); for ''n''=1 this is denoted by ''Cs'' (1 *) and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the ''n''-fold rotation axis. *''Cnv'', (), ( *''nn'') of order 2''n'' - pyramidal symmetry or full acro-n-gonal group (abstract group ''Dn''); in biology ''C2v'' is called biradial symmetry. For ''n''=1 we have again ''Cs'' (1 *). It has vertical mirror planes. This is the symmetry group for a regular ''n''-sided pyramid. *''S2n'', (), (''n''×) of order 2''n'' - gyro-n-gonal group (not to be confused with symmetric groups, for which the same notation is used; abstract group ''C2n''); It has a 2''n''-fold rotoreflection axis, also called 2''n''-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like ''Dnd'', it contains a number of improper rotations without containing the corresponding rotations. * * for ''n''=1 we have ''S2'' (1×), also denoted by ''Ci''; this is inversion symmetry or centrosymmetry. ''C2h'', () (2 *) and ''C2v'', (), ( *22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. ''C2v'' applies e.g. for a rectangular tile with its top side different from its bottom side. 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「Cyclic symmetry in three dimensions」の詳細全文を読む スポンサード リンク
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