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line group
A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations.
One constructs a line group by taking a point group in the full dimensions of the space, and then adding translations or offsets along the line to each of the point group's elements, in the fashion of constructing a space group. These offsets include the repeats, and a fraction of the repeat, one fraction for each element. For convenience, the fractions are scaled to the size of the repeat; they are thus within the line's unit cell segment.
== One-dimensional ==
There are 2 one-dimensional line groups. They are the infinite limits of the discrete two-dimensional point groups C''n'' and D''n'':
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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