How many symmetries does a regular \(n\)-gon have?

In \(D_n\), what does \(r\) usually denote?

How many symmetries does a square have?

For a regular pentagon, what is the angle of the basic rotation \(r\)?

List the rotational symmetries of a regular hexagon using powers of \(r\).

For a regular triangle, list the elements of \(D_3\) using \(r\) and one reflection \(s\).

In \(D_4\), simplify \(r^6\).

In products such as \(sr\), which transformation is applied first?

Use \(srs=r^{-1}\) and \(s^2=e\) to show that \(sr=r^{-1}s\).

In \(D_5\), simplify \(sr^7\).

Explain why \(D_4\) is generally not commutative using \(sr=r^{-1}s\).

How many rotations and reflections are in \(D_7\), and what is the total order?

In \(D_6\), simplify \((sr^2)^2\).

For a square, distinguish the four rotations from the four reflections in \(D_4\).

In \(D_n\), show that every element can be written as either \(r^k\) or \(sr^k\).

For which \(n\) does the relation \(sr=rs\) hold in \(D_n\)?

A regular polygon has \(18\) symmetries. How many sides does it have, and how many reflections?

A student says all symmetries of a polygon are rotations because the shape ends up occupying the same region. Diagnose the error.

Prove from \(srs=r^{-1}\) that \(sr^k=r^{-k}s\) for positive integers \(k\).

Explain why polygon symmetries form a group under composition.