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The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule , the notation is often sufficient and commonly used for spectroscopy .
In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: C n (for cyclic) indicates that the group has an n-fold rotation axis. C nh is C n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation.
Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. In Schoenflies notation, the reflective point groups in 3D are C nv, D nh, and the full polyhedral groups T, O, and I.
In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space 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 D n includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the ...
The 7 point groups (crystal classes) in this crystal system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation, and mineral examples, if they exist. [2] [9]
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. Example symmetry subgroup tree for dihedral symmetry: D 4h, [4,2], (*224)
John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion .