Thirty-two crystal classes crystallographic point groups

3 Thirty-two crystal classes (crystallographic point groups) 

The Hermann-Mauguin notation for the description of point group symmetry (in contrast to the Schonflies system used in Chapter 6) is widely adopted in crystallography. An n-fold rotation axis is simply designated as n. An object is said to possess an n-fold inversion axis h if it can be brought into an equivalent configuration by a rotation of 360°/n in combination with inversion through a 

Idealized shapes of cubic crystals with well-developed faces described by form symbols (a) p(100), fb) / 111 [, (c) r 110, and various combinations 

Some cubic crystals with faces indexed by Miller indices, (a) Cubic unit cell and labels of axes and angles (b) and (c) two habits of pyrites FeS2, class m3 (d) tetrahedrite Cu3SbS3, class 43m (e) spinel MgAl204, class m3m and (f) almandine (garnet) Fe3Al2(Si04)3, class m3m. 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important 

---

The thirty-two crystal classes (crystallographic point groups) described in Section 9.1.4 can also be classified into the same seven crystal systems, depending on the most convenient coordinate system used to indicate the location and orientation of their characteristic symmetry elements, as shown in Table 9.2.1. 

---