Fundamental motifs and parity within the crystallographic point groups

This paper analyzes the Hasse diagram, or family tree, of the 3D crystal classes, also called geometric crystal classes. The 32 point-group classes are partitioned into seven crystal systems. In this paper, the structures of these systems are analyzed, leading to a new understanding of the relationships among and within them. The point groups, including their subgroups up to conjugacy, appear in six structural motifs in the Hasse diagram or family tree. Each motif has a parity - even or odd - that determines its structure. In three dimensions, the odd motifs are called monads, trigonals and cubics, and the even motifs are called dyads, tetragonals and hexagonals. Of the 32 classes of 3D point groups, 29 have a well defined parity, in that they appear in either an even or an odd motif. In contrast, the three monoclinic point groups are 'ambidextrous', in that they appear in two motifs, one of each parity. An analysis of the ten 2D point groups reveals an analogous structure, except for the presence of an ambidextrous crystal system. The striking structural uniformity of the motifs across the Hasse diagram confirms that they are essential building blocks of the crystallographic point groups.