November 2026
A kaleidoscope is a toy made from mirrors that intersect at particular angles, and the resulting reflections create mesmorising patterns. In this month’s problem, you will investigate how reflections interact with each other.
Given a line \(\ell\) in the plane, the reflection across \(\ell\), which we will denote by \(\Gamma_\ell\), is a transformation of the plane that sends every point in the plane to another point as follows:
If \(P\) is on \(\ell\), then \(\Gamma_\ell(P) = P\) (the point isn’t affected by the reflection). If \(P\) is not on \(\ell\), then \(\Gamma_\ell(P)\) is the unique point in the plane so that \(\ell\) is the perpendicular bisector of the line segment joining \(P\) and \(\Gamma_\ell(P)\) (you may take it for granted that there is indeed such a unique point).
The next image illustrates a line \(\ell\) with points \(P\), \(Q\), and \(R\) and their reflections across \(\ell\).
If \(\ell\) is the line defined by the equation \(y = x + 1\), and \(P\) has coordinates \((0,3)\), what are the coordinates of \(\Gamma_\ell(P)\)?
Let \(P\) and \(Q\) be points in the plane and let \(\ell\) be a line. Show that the distance between \(P\) and \(Q\) is equal to the distance between \(\Gamma_\ell(P)\) and \(\Gamma_\ell(Q)\).
Let \(P\), \(Q\) and \(R\) be points in the plane. Let \(\ell\) be a line and let \(P' = \Gamma_\ell(P)\), \(Q' = \Gamma_\ell(Q)\) and \(R' = \Gamma_\ell(R)\). Show that \(\angle PQR = \angle P'Q'R'\).
Let \(\ell_1\) and \(\ell_2\) be two lines in the plane that intersect at a point \(X\). Suppose that when measured counterclockwise from \(\ell_1\), the angle formed by the two lines is \(\theta\). Describe geometrically the transformation obtained by the composition \(\Gamma_{\ell_2} \circ \Gamma_{\ell_1}\) (that is, the transformation resulting from the reflection across \(\ell_1\) followed by the reflection across \(\ell_2\)).
Let \(\ell_1\) and \(\ell_2\) be two lines in the plane that intersect at a point \(X\). Suppose that when measured counterclockwise from \(\ell_1\), the angle formed by the two lines is \(\frac{180\degree}{n}\) for some integer \(n \geq 3\). As shown below, let \(P\) be a point on \(\ell_2\) other than \(X\).
By repeatedly applying the reflections \(\Gamma_{\ell_1}\) and \(\Gamma_{\ell_2}\) to \(P\), as many times as you like and in any order, you will obtain a set of points. For example, \(P, \Gamma_{\ell_2}(P), \Gamma_{\ell_1}\circ \Gamma_{\ell_2}(P)\), and \(\Gamma_{\ell_1}\circ\Gamma_{\ell_2} \circ \Gamma_{\ell_1}(P)\) are four such points. Some of the points obtained this way will be the same. Call the set of all points obtained this way the orbit of \(P\).
Including the point \(P\), how many distinct points are in the orbit of \(P\)?
Let \(k\) be the answer to Part (i). Show that the \(k\) points in the orbit of \(P\) form the vertices of a regular \(k\)-gon. [Recall that a regular \(k\)-gon is a polygon with \(k\) equal side lengths and equal internal angles].