The 32 crystal classes may be grouped into six crystal systems according to the following criteria and constraints (compare with table 2.2 of K&H):
| Crystal System | Rotational Symmetry Constraints | Axis Length Constraints | Axis Angle Constraints |
| Triclinic | One 1-fold or axis | No Constraints | No Constraints |
| Monoclinic | One 2-fold or (=m) axis | No Constraints | a=g=90° |
| Orthorhombic | Three perpendicular 2-fold or axes | No Constraints | a=b=g=90° |
| Tetragonal | One 4-fold or axis | a=b | a=b=g=90° |
| Hexagonal | One 6-fold, 3-fold | a=b | a=b=90°, g=120° |
| Isometric | Four 3-fold or axes | a=b=c, | a=b=g=90° |
The 'lattice constraints' of the chart above are developed in chapter 3 of K&H, and in lecture #4.
Of the six crystal systems, which has the highest number of minerals? In other words, are minerals distributed evenly among the six crystal systems, or is their a tendency toward higher or lower symmetry among crystalline materials?
The crystal classes are identified in the textbook using
Hermann-Mauguin notation (see K&H, Table 2.9). The Hermann-Maugin notation
identifies rotation or rotoinversion axes (if present) along particular
directions and mirror planes (if present) perpendicular to the same directions.
The directions for writing out the HM notation for each of the six crystal
systems as follows:
| Triclinic | no particular setting |
- |
- |
| Monoclinic | the b-axis only |
- |
- |
| Orthorhombic | the a-axis | the b-axis | the c-axis |
| Tetragonal | the c-axis | the a-axis | the [110] direction |
| Hexagonal | the c-axis | the a-axis | the [210] direction |
| Isometric | the c-axis | the [111] direction | the [110] direction |
We will define what is meant by various axes and directions in a subsequent lecture. The main point is that symmetry elements are listed in a crystal from the highest level of symmetry present (e.g., a 4-fold axis of rotation), to the lowest level present. Moreover, the symmetry of the crystal systems tends to increase from the top to the bottom of the table above.
We can also recap previous discussions by restating Neumann's Principle:
The symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal.
Thus any physical property we may measure for a mineral - e.g., hardness, thermal conductivity, refractive index, x-ray diffraction - must vary with direction in symmetrical ways such that all of the symmetry elements of the point group (and perhaps more) are present for that physical property.
Symmetry elements are also evident in the forms of crystals: In ideal crystals, each face is repeated by all of the symmetry elements of the point group of that crystal. For example, if the face (110) is present on a tetragonal crystal, the three faces symmetrically equivalent to (110) by the 4-fold axis should also be present. The set of crystal faces in a crystal class that are symmetrically equivalent are called a form for that class and are designated by braces around the Miller indices of one of the faces. For example, the four faces mentioned above may be represented by the one symbol {110}. This form is called a tetragonal prism (see Table 2.5 of K&H). All of the common forms are shown in figure 2.36 of K&H. Note that an individual crystal may have a habit, or shape, that includes several forms. Simple forms would include prisms, pinacoids, pyramids, and sphenoids; more complex forms are many-sided geometrical objects, most notably for the isometric system.