Symmetries in three dimensions
So far we have talked only about patterns in two dimensions. What we are really interested in, however, are patterns of atoms in three dimensions. First, it is clear that a three-dimensional crystal will have three primitive vectors. If we then ask about the possible symmetry operations in three dimensions, we find that there are $230$ different possible symmetries! For some purposes, these $230$ types can be grouped into seven classes, which are drawn in Fig. 30–10. The lattice with the least symmetry is called the triclinic. Its unit cell is a parallelepiped. The primitive vectors are of different lengths, and no two of the angles between them are equal. There is no possibility of any rotational or reflection symmetry. There are, however, still two possible symmetries—the unit cell is, or is not, changed by an inversion through the vertex. (By an inversion in three dimensions, we again mean that spatial displacements $\FLPR$ are replaced by $-\FLPR$—in other words, that $(x,y,z)$ goes into $(-x,-y,-z)$). So the triclinic lattice has only two possible symmetries, unless there is some special relation among the primitive vectors. For example, if all the vectors are equal and are separated by equal angles, one has the trigonal lattice shown in the figure. This figure can have an additional symmetry; it may be unchanged by a rotation about the long, body diagonal.  If one of the primitive vectors, say $\FLPc$, is at right angles to the other two, we get a monoclinic unit cell. A new symmetry is possible—a rotation by $180^\circ$ about $\FLPc$. The hexagonal cell is a special case in which the vectors $\FLPa$ and $\FLPb$ are equal and the angle between them is $60^\circ$, so that a rotation of $60^\circ$, or $120^\circ$, or $180^\circ$ about the vector $\FLPc$ repeats the same lattice (for certain internal symmetries).  If all three primitive vectors are at right angles, but of different lengths, we get the orthorhombic cell. The figure is symmetric for rotations of $180^\circ$ about the three axes. Higher-order symmetries are possible with the tetragonal cell, which has all right angles and two equal primitive vectors. Finally, there is the cubic cell, which is the most symmetric of all.  The point of all this discussion about symmetries is that the internal symmetries of the crystals show up—sometimes in subtle ways—in the macroscopic physical properties of the crystal. For instance, a crystal will, in general, have a tensor electric polarizability. If we describe the tensor in terms of the ellipsoid of polarization, we should expect that some of the crystal symmetries should show up also in the ellipsoid. For example, a cubic crystal is symmetric with respect to a rotation of $90^\circ$ about any one of three orthogonal directions. Clearly, the only ellipsoid with this property is a sphere. A cubic crystal must be an isotropic dielectric.  On the other hand, a tetragonal crystal has a fourfold rotational symmetry. Its ellipsoid must have two of its principal axes equal, and the third must be parallel to the axis of the crystal. Similarly, since the orthorhombic crystal has twofold rotational symmetry about three orthogonal axes, its axes must coincide with the axes of the polarization ellipsoid. In a like manner, one of the axes of a monoclinic crystal must be parallel to one of the principal axes of the ellipsoid, though we can’t say anything about the other axes. Since a triclinic crystal has no rotational symmetry, the ellipsoid can have any orientation at all.  As you can see, we can make a big game of figuring out the possible symmetries and relating them to the possible physical tensors. We have considered only the polarization tensor, but things get more complicated for others—for instance, for the tensor of elasticity. There is a branch of mathematics called “group theory” that deals with such subjects, but usually you can figure out what you want with common sense. 