Academy
Polygon Symmetries
Level 1 - Math I (Physics) topic page in Groups.
Principle
The symmetries of a regular polygon form a group under composition. Each symmetry moves the polygon in the plane but leaves the occupied shape unchanged.
Polygon symmetry groups give concrete examples of rotations, reflections, and non-commutative composition. These ideas also appear when physical systems are invariant under rotations or mirror symmetries.
Notation
- \(n\) is the number of sides of the regular polygon.
- \(D_n\) is the symmetry group of a regular \(n\)-gon.
- \(e\) is the identity symmetry.
- \(r\) is a rotation by \(360^\circ/n\).
- \(s\) is one reflection symmetry.
- Composition means applying one symmetry after another.
- In a product such as \(sr\), the rightmost transformation \(r\) is applied first.
Method
To list the symmetries of a regular \(n\)-gon:
- List the \(n\) rotations \(e,r,r^2,\\ldots,r^{n-1}\).
- Choose one reflection \(s\).
- Compose that reflection with the rotations to get \(s,sr,sr^2,\\ldots,sr^{n-1}\).
- Count the \(n\) rotations and \(n\) reflections.
- Use composition as the group operation.
Rules
- A full turn gives the identity: \(r^n=e\).
- Reflecting twice gives the identity: \(s^2=e\).
- Reflection reverses rotation direction: \(srs=r^{-1}\).
- A regular \(n\)-gon has \(2n\) symmetries.
- Dihedral groups are generally not commutative.
Examples
For a square, the symmetry group is \(D_4\). It has four rotations:
\[e,\ r,\ r^2,\ r^3\]and four reflections:
\[s,\ sr,\ sr^2,\ sr^3\]The total number of symmetries is:
Order can matter. From \(srs=r^{-1}\), multiply on the right by \(s\). The steps are:
This shows that doing a rotation and reflection in one order can differ from doing them in the other order.
Checks
- Count \(2n\) symmetries: \(n\) rotations and \(n\) reflections.
- Do not treat reflections as rotations.
- Apply the rightmost transformation first in products such as \(sr\).
- Check whether order matters before commuting two symmetries.