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In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dih''n'' ( ''n'' ≥ 2 ). == Types== ;Chiral: *''Dn'', ()+, (22''n'') of order 2''n'' – dihedral symmetry or para-n-gonal group (abstract group ''Dn'') ;Achiral: *''Dnh'', (), ( *22''n'') of order 4''n'' – prismatic symmetry or full ortho-n-gonal group (abstract group ''Dn'' × ''C''2) *''Dnd'' (or ''Dnv''), (), (2 *''n'') of order 4''n'' – antiprismatic symmetry or full gyro-n-gonal group (abstract group ''D''2''n'') For a given ''n'', all three have ''n''-fold rotational symmetry about one axis (rotation by an angle of 360°/''n'' does not change the object), and 2-fold about a perpendicular axis, hence about ''n'' of those. For ''n'' = ∞ they correspond to three frieze groups. Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal (h) is used with respect to a vertical axis of rotation. In 2D the symmetry group ''Dn'' includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection in a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D the two operations are distinguished: the group ''Dn'' contains rotations only, not reflections. The other group is pyramidal symmetry ''Cnv'' of the same order. With reflection symmetry with respect to a plane perpendicular to the ''n''-fold rotation axis we have ''Dnh'' (), ( *22''n''). ''Dnd'' (or ''Dnv''), (), (2 *''n'') has vertical mirror planes between the horizontal rotation axes, not through them. As a result the vertical axis is a 2''n''-fold rotoreflection axis. ''Dnh'' is the symmetry group for a regular ''n''-sided prisms and also for a regular n-sided bipyramid. ''Dnd'' is the symmetry group for a regular ''n''-sided antiprism, and also for a regular n-sided trapezohedron. ''Dn'' is the symmetry group of a partially rotated prism. ''n'' = 1 is not included because the three symmetries are equal to other ones: *''D''1 and ''C''2: group of order 2 with a single 180° rotation *''D''1''h'' and ''C''2''v'': group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane *''D''1''d'' and ''C''2''h'': group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane For ''n'' = 2 there is not one main axes and two additional axes, but there are three equivalent ones. *''D''2 (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation. *''D''2''h'' ( *222) of order 8 is the symmetry group of a cuboid *''D''2''d'' (2 *2) of order 8 is the symmetry group of e.g.: * *a square cuboid with a diagonal drawn on one square face, and a perpendicular diagonal on the other one * *a regular tetrahedron scaled in the direction of a line connecting the midpoints of two opposite edges (''D''2''d'' is a subgroup of ''Td'', by scaling we reduce the symmetry). 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「Dihedral 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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