A symmetry group is the complete set of transformations that map a shape or pattern onto itself. In geometry, the most common symmetries are rotations, reflections, translations, and glide reflections. Studying these groups helps students see that symmetry is not only visual, but also algebraic because transformations can be combined like operations.
Symmetry groups are used in art, architecture, crystallography, molecular structure, and pattern design.
For a regular polygon, the symmetry group often includes rotations around its center and reflections across mirror lines. These transformations form a group because doing one symmetry after another gives another symmetry of the same figure, there is an identity transformation, and every symmetry has an inverse. The dihedral group Dn describes the symmetries of a regular n-gon and has n rotations and n reflections, for a total of 2n elements.
Repeating patterns can have larger symmetry groups, such as frieze groups in one direction and wallpaper groups across the plane.
Key Facts
- A symmetry of a figure is a transformation that maps the figure exactly onto itself.
- The identity symmetry leaves every point fixed and is included in every symmetry group.
- For a regular n-gon, the rotation angles are 0°, 360°/n, 2(360°/n), ..., (n - 1)(360°/n).
- The dihedral group Dn of a regular n-gon has 2n symmetries: n rotations and n reflections.
- Combining symmetries is group operation composition, written as T2 ∘ T1, meaning do T1 first and then T2.
- A frieze pattern repeats in one direction, while a wallpaper pattern repeats in two independent directions.
Vocabulary
- Symmetry group
- A symmetry group is the set of all transformations that map a figure or pattern onto itself, together with the operation of composition.
- Rotation symmetry
- Rotation symmetry occurs when a figure matches itself after being turned by a certain angle around a fixed point.
- Reflection symmetry
- Reflection symmetry occurs when a figure matches itself after being flipped across a mirror line.
- Dihedral group
- A dihedral group Dn is the symmetry group of a regular n-sided polygon, containing rotations and reflections.
- Glide reflection
- A glide reflection is a transformation made by reflecting across a line and then translating parallel to that line.
Common Mistakes to Avoid
- Counting only visible mirror lines and forgetting rotations. A regular polygon usually has rotational symmetries even when no mirror line is drawn on the diagram.
- Using 360°/n as the only rotation symmetry. That is the smallest nonzero rotation for a regular n-gon, but its multiples are also symmetries.
- Thinking every symmetric shape has a dihedral group. Dihedral groups describe regular polygons and similar finite figures with rotations and reflections, not every possible pattern.
- Assuming transformation order never matters. In many symmetry groups, especially with rotations and reflections, doing A then B can give a different result from doing B then A.
Practice Questions
- 1 A regular octagon has symmetry group D8. How many rotations, how many reflections, and how many total symmetries does it have?
- 2 List all rotation angles from 0° up to but not including 360° that map a regular hexagon onto itself.
- 3 A pattern repeats horizontally and has mirror lines perpendicular to the direction of repetition. Explain why this is a frieze symmetry pattern rather than just the symmetry of one isolated shape.