Dihedral groups are apparent throughout art and nature. For example, dihedral groups are often the basis of decorative designs on floor tilings, buildings, and artwork. Chemists and mineralogists study dihedral groups to classify the structure of molecules and crystals, respectively. These symmetry groups are even used in advertising for many of the world's largest companies.
Some examples of dihedral groups with which we are all familiar are seen below.
| Dihedral Group | Symbol | Our Thoughts |
|---|---|---|
| D1 |  |
Shell Petroleum uses the symbol to the left. This shell shape has no rotations (other than the identity) and has only one mirror line (vertical). Therefore, like Mickey Mouse, the figure is said to be bilaterally symmetric and it fits into the category D1. |
| D2 |  |
An example of D2 that is easily spotted is the logo for the Columbia Broadcasting System (CBS). The "eye" shape within the circle prevents the figure from being able to rotate by any rotation other than a 1/2 turn. Additionally, the figure has only two ways in which it can be reflected onto itself. |
| D3 |  |
The luxury car, Mercedes-Benz, uses a symbol with three rotations and 3 mirror lines. Therefore, the emblem is an example of D3. If we were to convert this figure into a peace sign, however, we would lose 2 of the rotations and two of the reflection lines. This would leave a D1 figure. |
| D4 |  |
The symbol for Purina is a great example of a finite figure of the category D4. It is easy to see that there are four mirror reflections of the figure (one vertical, one horizontal, and two diagonal) as well as four rotations. In other words, rotating the figure four times gives the original figure (the identity). |
| D5 |  |
The symbol for Chrysler is a great example of a finite figure of the category D5. In other words, the symbol has five rotations and five axes of reflection. |
| D8 |  |
This finite figure is a dihedral group of order 8 due to its eight reflections and eight rotations. The symmetries are created by two squares placed on top of each other and offset by 90 degrees. |
During our trip to Europe, we also found many examples of dihedral groups. Please CLICK HERE to see our photos of dihedral groups.