Year: 2004
Topic: Algebra Complex Numbers Geometry

Problem:
Let $n$ be a positive integer, $n \ge 2$, and put  $\theta = 2 \pi / n$. Define points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2 , \dots, n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$.  Let $R$ denote the map obtained by applying, in order, $R_1$,  then $R_2, \dots$, then $R_n$. For an arbitrary point $(x,y)$, find, and simplify, the coordinates of $R(x,y)$.

Solution:
Identify the $xy$-plane with the complex plane $\mathbb{C}$, so that $P_k$ is the real number $k$. If $z$ is sent to $z'$ by a counterclockwise rotation by $\theta$ about $P_k$, then $z'-k = e^{i\theta} (z-k)$; hence the rotation $R_k$ sends $z$ to $\zeta z + k (1-\zeta)$, where $\zeta = e^{2\pi i/n}$. It follows that $R_1$ followed by $R_2$ sends $z$ to $\zeta(\zeta z +(1-\zeta)) + 2 (1-\zeta) = \zeta^2 z + (1-\zeta)(\zeta + 2)$, and so forth; an easy induction shows that $R$ sends $z$ to \[ \zeta^n z + (1-\zeta)(\zeta^{n-1} + 2 \zeta^{n-2} + \dots + (n-1) \zeta + n). \] Expanding the product $(1-\zeta)(\zeta^{n-1} + 2 \zeta^{n-2} + \dots + (n-1) \zeta + n)$ yields $-\zeta^n - \zeta^{n-1} - \dots - \zeta + n = n$. Thus $R$ sends $z$ to $z+n$; in cartesian coordinates, $R(x,y) = \boxed{(x+n,y)}$.