Checkpoint 1.3.3.
For the image in the figure below, identify two unique mirrors and the duplicate mirrors generated by the fourfold rotational symmetry.
Section 1.3 Hermann-Mauguin Notation
A variety of notations are used to describe the symmetries of polyhedra. One used in chemistry is Hermann-Mauguin which identifies the various shapes crystals can take. This section presents how to read and write the symmetries in this notation.
Subsection 1.3.1 Relationships between Symmetries
The notation is based on identifying three types of symmetry (rotation, reflection, and reflection through a point) and identifying any relationship between the rotations and reflections. In this notation the three symmetries are called: rotation, mirror, and inversion or rotoinversion (thought of as a rotation and inversion combined).
There are two relationships between symmetries that are needed for Hermann-Mauguin notation. The first is not double counting symmetries. Consider the following example. There is a fourfold rotational symmetry around the vertical axis (in blue). There is also a twofold rotationa symmetry about an axis that goes from the middle of one square side to the middle of the opposing square side (in blue). But there are four pairs of sides (including drawing the arrow the opposite direction). We count this as a single twofold rotational symmetry, because the other three are the result of the fourfold rotation rotating the first side symmetry into the others. That is if you rotate 90° around the vertical axis, the blue rotation axis moves into the green one. Only unique axes will be recorded. There is another distinct twofold rotational symmetry, namely the one from the middle of an edge to the middle of the opposing edge. There are four versions of this which are the result of the fourfold rotation, so we count only the one version.
Mirrors may also be duplicated on account of rotational symmetries. This same shape has two pairs of mirrors that are mapped to each other. Each blue mirror is mapped to a green mirror by the rotaton, so we count only two unique mirrors from these. There is another mirror not shown: it is the xy-plane (top and bottom are mirror images).
Another relationship between symmetries is when a rotational axis is perpendicular to a mirror. This has a special notation in Hermann-Mauguin notation. In the shape we have been using the fourfold rotation is perpendicular to the mirror through the middle (bottom to top) as shown below.
Checkpoint 1.3.6.
In the figure below identify a rotation axis and a mirror that are perpendicular to each other.
Subsection 1.3.2 Identify Symmetries for Hermann-Mauguin
Now that we understand how the symmetries are related, we can identify all of them and find the Hermann-Mauguin notation for it.
