EENS 2110 
Mineralogy 
Tulane University 
Prof. Stephen A. Nelson 
External Symmetry of Crystals, 32 Crystal Classes 

As stated in the last lecture, there are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. These are often also referred to as the 32 point groups. We will go over some of these in detail in this lecture, but again I want to remind everyone that the best way to see this material is by looking at the crystal models in lab. HermannMauguin (International) Symbols Before going into the 32 crystal classes, I first want to show you how to derive the HermannMauguin symbols (also called the international symbols) used to describe the crystal classes from the symmetry content. We'll start with a simple crystal then look at some more complex examples. 
The rectangular block shown here has 3 2fold rotation axes (A_{2}), 3 mirror planes (m), and a center of symmetry (i). The rules for deriving the HermannMauguin symbol are as follows:

2 2 2
2 m 2 m 2 m

2/m2/m2/m If you look in the table given in the lecture notes below, you will see that this crystal model belongs to the Rhombicdipyramidal class. 
Our second example involves the block shown here to the right.
This model has one 2fold axis and 2 mirror planes. For the 2fold
axis, we write:
2 Each of the mirror planes is unique. We can tell that because each one cuts a different looking face. So, we write 2 "m"s, one for each mirror plane: 2 m m 
Note that the 2fold axis is not perpendicular to a mirror plane, so we
need no slashes. Our final symbol is then:
2mm For this crystal class, the convention is to write mm2 rather than 2mm (I'm not sure why). If you consult the table below, you will see that this crystal model belongs to the Rhombicpyramidal class. 
The third example is shown here to the right. It contains 1 4fold
axis, 4 2fold axes, 5 mirror planes, and a center of symmetry. Note
that the 4fold axis is unique. There are 2 2fold axes that are
perpendicular to identical faces, and 2 2fold axes that run through the
vertical edges of the crystal. Thus there are only 2 unique 2 fold
axes, because the others are required by the 4fold axis perpendicular to
the top face. So, we write:
4 2 2 
Although there are 5 mirror planes in the model, only 3 of them are
unique. Two mirror planes cut the front and side faces of the crystal, and
are perpendicular to the 2fold axes that are perpendicular to these
faces. Only one of these is unique, because the other is required by
the 4fold rotation axis. Another set of 2 mirror planes cuts
diagonally across the top and down the edges of the model. Only one
of these is unique, because the other is generated by the 4fold rotation
axis and the previously discussed mirror planes. The mirror plane
that cuts horizontally through the crystal and is perpendicular to the
4fold axis is unique. Since all mirror unique mirror planes are
perpendicular to rotation axes, our final symbol becomes:
4/m2/m2/m Looking in the table below, we see that this crystal belongs to the Ditetragonaldipyramidal class.

Our last example is the most complex. Note that it has 3 4fold rotation axes, each of which is perpendicular to a square shaped face, 4 3fold rotoinversion axes (some of which are not shown in the diagram to reduce complexity), each sticking out of the corners of the cube, and 6 2fold rotation axes (again, not all are shown), sticking out of the edges of the cube. In addition, the crystal has 9 mirror planes, and a center of symmetry. 
There is only 1 unique 4 fold axis, because each is perpendicular to a
similar looking face (the faces of the cube). There is only one
unique 3fold rotoinversion axes, because all of them stick out of the
corners of the cube, and all are related by the 4fold symmetry.
And, there is only 1 unique 2fold axis, because all of the others stick
out of the edges of the cube and are related by the mirror planes the
other set of 2fold axes. So, we write a 4, a
,
and a 2 for each of the unique rotation axes.
4 2 There are 3 mirror planes that are perpendicular to the 4 fold axes, and 6 mirror planes that are perpendicular to the 2fold axes. No mirror planes are perpendicular to the 3fold rotoinversion axes. So, our final symbol becomes: 4/m2/m Consulting the table in the lecture notes below, reveals that this crystal belongs to the hexoctahedral crystal class. 
The 32 Crystal Classes The 32 crystal classes represent the 32 possible combinations of symmetry operations. Each crystal class will have crystal faces that uniquely define the symmetry of the class. These faces, or groups of faces are called crystal forms. Note that you are not expected to memorize the crystal classes, their names, or the symmetry associated with each class. You will, however, be expected to determine the symmetry content of crystal models, after which you can consult the tables in your textbook, lab handouts, or lecture notes. All testing on this material in the lab will be open book. In this lecture we will go over some of the crystal classes and their symmetry. I will not be able to cover all of the 32 classes. You will, however, see many of the 32 classes during your lab work. Note that it is not easy to draw a crystal of some classes without adding more symmetry or that can be easily seen in a two dimensional drawing. The table below shows the 32 crystal classes, their symmetry,
HermannMauguin symbol, and class name. 
Crystal System  Crystal Class  Symmetry  Name of Class 
Triclinic  1  none  Pedial 
i  Pinacoidal  
Monoclinic  2  1A_{2}  Sphenoidal 
m  1m  Domatic  
2/m  i, 1A_{2}, 1m  Prismatic  
Orthorhombic  222  3A_{2}  Rhombicdisphenoidal 
mm2 (2mm)  1A_{2}, 2m  Rhombicpyramidal  
2/m2/m2/m  i, 3A_{2}, 3m  Rhombicdipyramidal  
Tetragonal  4  1A_{4}  Tetragonal Pyramidal 
_{4}  Tetragonaldisphenoidal  
4/m  i, 1A_{4}, 1m  Tetragonaldipyramidal  
422  1A_{4}, 4A_{2}  Tetragonaltrapezohedral  
4mm  1A_{4}, 4m  Ditetragonalpyramidal  
2m  1_{4}, 2A_{2}, 2m  Tetragonalscalenohedral  
4/m2/m2/m  i, 1A_{4}, 4A_{2}, 5m  Ditetragonaldipyramidal  
Hexagonal  3  1A_{3}  Trigonalpyramidal 
1_{3}  Rhombohedral  
32  1A_{3}, 3A_{2}  Trigonaltrapezohedral  
3m  1A_{3}, 3m  Ditrigonalpyramidal  
2/m  1_{3}, 3A_{2}, 3m  Hexagonalscalenohedral  
6  1A_{6}  Hexagonalpyramidal  
1_{6}  Trigonaldipyramidal  
6/m  i, 1A_{6}, 1m  Hexagonaldipyramidal  
622  1A_{6}, 6A_{2}  Hexagonaltrapezohedral  
6mm  1A_{6}, 6m  Dihexagonalpyramidal  
m2  1_{6}, 3A_{2}, 3m  Ditrigonaldipyramidal  
6/m2/m2/m  i, 1A_{6}, 6A_{2}, 7m  Dihexagonaldipyramidal  
Isometric  23  3A_{2}, 4A_{3}  Tetaroidal 
2/m  3A_{2}, 3m, 4_{3 }  Diploidal  
432  3A_{4}, 4A_{3}, 6A_{2}  Gyroidal  
3m  3_{4}, 4A_{3}, 6m  Hextetrahedral  
4/m2/m  3A_{4}, 4_{3}, 6A_{2}, 9m  Hexoctahedral 
Note that the 32 crystal classes are divided into 6 crystal systems.

Triclinic System Characterized by only 1fold or 1fold rotoinversion axis


Monoclinic System Characterized by having only mirror plane(s) or a single 2fold axis. 



The most common minerals that occur in the prismatic class are the micas (biotite and muscovite), azurite, chlorite, clinopyroxenes, epidote, gypsum, malachite, kaolinite, orthoclase, and talc. 
Orthorhombic System
Characterized by having only two fold axes or a 2fold axis and
2 mirror planes. 




Tetragonal System
Characterized by a single 4fold or 4fold rotoinversion axis. 








Note that I will not have time in lecture to cover the rest of the 32 crystal classes, that is those belonging to the hexagonal and isometric systems. These are difficult to draw, and are best left for the student to study using the textbook, pages 183207, and the crystal models in lab. 
Examples of questions on this material that could be asked on an exam
