Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes through one of the points $x_i$ it automatically passes through the other two after a number of reflections on $M$?

The question needs some formalisation for what is meant by $M \subset \mathbb{R}^d$ and “rays of light”, although it is easy to guess what they should be:

$M$ is a $(d-1)$-dimensional differentiable submanifold of $\mathbb{R}^d$.

A ray of light is a continuous function $g:\mathbb{R} \rightarrow \mathbb{R}^d$ for which $g^{-1}(g(\mathbb{R}) \cap M)$ is a discrete set. Moreover, $g$ is locally smooth, linear and unit speed at any point $t\in \mathbb{R}$ for which $g(t) \notin M$: $\left\|g'(t)\right\|=1$ and g''(t)=0. At any point $t^* \in \mathbb{R}$ for which $g(t^*)\in M$ we require $g$ to satisfy the reflection law: denoting $a=\lim_{t \to t^* -} g'(t)$ and $b = \lim_{t \to t^* +} g'(t)$ \begin{equation} b + a \perp n(g(t)) \end{equation} and \begin{equation} b - a \propto n(g(t)) \end{equation} (n is a local unit-normal vector field on $M$).

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