Crystal Symmetry The 32 Crystallographic Point Groups

A crystal consists of an ordered 3D repetition of a fundamental unit (the asymmetric unit), which may be one molecule or several molecules (all the same or of several kinds). This ordered array consists of a large number of unit cells (effectively, an infinite number) that fit together to fill space and 

Crystal System Lattice Symmetry Axial Cell 

However, the symmetry properties of the crystals themselves are more complex than those of the lattices, and we now turn to these. There are, of course, close connections between lattice symmetries and crystal symmetries and we shall presently bring our knowledge of lattice symmetries into use in exploring crystal symmetries. 

Associated with this change from rotoreflections (5,/s) to rotoinversions, there are other notational changes and it is most efficient to deal with them all at once. To describe point symmetry in a crystal we use the following symmetry elements and operations. 

there are five pure rotations, by 2nfn where n = 1, 2, 3, 4, and 6. Previously these axes were symbolized C, C2, C3, C4, and C6. They are 

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