Symmetry Operations

There are several symmetry operations that help define the crystal's outward symmetry. These operations represent the way a crystal can repeat the facets (faces) on their crystal's surface.
One way to repeat a face is with a mirror plane that can reflect a face from one side of the crystal to the other. Consequent to being reflected by a mirror plane, the reflected face must be identical but reversed in orientation. In other words, if the original face has any right handed characteristics, then the reflected face must have the same characteristics but with a left handed slant to them.

A rotational axis is a line imaginarily drawn through the crystal that acts as an axis just like the axis for a tire. A face can be repeated on a crystal when the crystal is rotated around this axis and a new face is left at various intervals during the rotation. Consequent to being rotated is that the face must be identical to the original face when the face is viewed head on. In other words, if the face has a right handed slant and is rotated, the rotated faces must keep the right handed slant.
The interval for dropping a face is determined by a division of the full turn into equal segments. For example, to drop four faces on a crystal the rotation requires a stop at every 90° and this type of rotation is called a four fold rotational axis. Rotational axes can have rotations of 1, 2, 3, 4 and 6 fold. Thus the 1 fold axis rotates the crystal in 360° intervals, the 2 fold interval is 180°, the 3 fold interval is 120°, the 4 fold interval is 90° and the 6 fold interval is 60°.

A rotoinversion axis goes one step further, by after rotating once and before dropping a face, it inverts the face through the crystal's center to the other side. The resulting face is completely flipped, i.e., up is down and right is left. The rotoinversion continues until it returns to the original starting face. Rotoinversion is constrained by the same rules for the simple rotational axes with the same folds or turns and degrees.

Finally a center is all that is left of symmetry operations to discuss. A center is simply, or perhaps not so simply, an operation that takes a face on one side of a crystal and inverts it through the center of the crystal. This has the same effect as the inversion in a rotoinversion operation in that the face is completely flipped up to down and right to left. Every point in a crystal is inverted to the other side of the crystal. Usually, a center is one operation that is all but ignored in most crystals because it is caused by the juxtaposition of other symmetry operations. However in the triclinic system it is the only possible symmetry operation except for a one fold rotational axis, which is actually nothing.

Crystallographic Axes:

Other axes mentioned are crystallographic axes that are used by crystallographers like geometric axes to plot the faces and symmetry elements and their orientations within the crystal. These axes may or may not be part of the symmetry of the crystals. But they usually are since crystallographers will often orient the crystallographic axes along the planes and axes of symmetry. There are 3 axes, a, b, and c. The axes pass through the center of the crystal and, by using them, we can describe the intersection of any given face with these 3 axes.