# Retroreflectors in higher dimensions?

I went to an exhibition recently on the mathematics of mirrors and saw an object they called a "retroreflector", where three square mirrors are placed orthogonal to each other. The exhibit demonstrated by using a torch that whatever angle you aimed the torch at the object, light would always bounce off the retroreflector back towards the light's origin. After thinking about this at home for some time, I figured out what appears to be a standard argument for why this is true.

Now, my question is about whether there are generalisations in higher dimensions and what they might mean. This occurred to me because in 2D, if you have two unit-length lines orthogonal to each other which act as mirrors, then they act as a 2D retroreflector.

It would seem natural to me, therefore, to have 3 "orthogonal" cubes in 4-space - the problem is that, while I understand what it means for a lightray to bounce off a line in 2-space or a surface in 3-space, I can't think of any reasonable notion of lightrays bouncing off volumes...

• Those "volumes" are hyperplanes, which are "flat" (zero thickness) in the containing space. – quasi Jul 8 '18 at 10:33

## 1 Answer

Recall that a reflection at the $xy$-plane reflects a point $(x,y,z)$ into $(x,y,-z)$. Hence reflecting at the $xy$-plane, then the $xz$-plane, then the $yz$-plane (or in any other order), maps $(x,y,z)\mapsto (x,y,-z)\mapsto (x,-y,-z)\mapsto (-x,-y,-z)$.

In the same way, reflecting at the $n$ major hyperplanes (which are $(n-1)$-dimensional) maps any point in $\Bbb R^n$ to its negative ...