There are six symmetries of an equilateral triangle: three reflections, and three rotations (thinking of the identity as one the rotations).
Label the three reflections \(s_1\), \(s_2\) and \(s_3\). Label the identity by \(e\), rotation by \(120^{\circ}\) as \(r_1\), and rotation by \(240^{\circ}\) clockwise as \(r_2\).
What are the inverses of \(r_1\) and \(s_1\)?
Think about the symmetries of an equilateral triangle. We label rotation by \(120^{\circ}\) as \(r_1\), and reflection in the vertical median by \(s_1\).
Is applying \(r_1\), then \(s_1\) the same as applying \(s_1\), then \(r_1\)?
Let \(n\ge3\) be a positive integer. A regular \(n\)-gon is a polygon with \(n\) sides where every side has the same length, and every angle is the same. For example, a regular \(3\)-gon is an equilateral triangle, and a regular \(4\)-gon is a square.
What symmetries does a regular \(n\)-gon have, and how many?