Mathematics desk
< May 22	<< Apr \| May \| Jun >>	May 24 >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

May 23

Are there retroreflectors in higher dimensions?

A retroreflector in 1 dimension is a point, in 2-D a right angle works, in 3-D a corner cube works (without needing refraction), do all dimensions have one? Is there always only one per dimension that doesn't need refraction to work? Sagittarian Milky Way (talk) 18:02, 23 May 2017 (UTC)[reply]

Assuming that you mean n-dimensional Euclidean space, the answers to your first questions is 'yes': The construction of having one reflecting hyperplane per dimension such that they are all mutually perpendicular works for any number of dimensions; this is a corner of a hypercube. As to the second question, I'm pretty sure there are multiple configurations that could be regarded as retroreflectors, albeit inefficient ones. For example, in two dimensions, consider half of a regular octahedron (four adjacent edges); any ray that reflects off all four of the lines would return antiparallel to the incoming ray. Alternately, also in two dimensions, consider two reflecting lines at 45°; any ray that reflects twice off each line would return in the direction it came from. Though I have not checked the detail, I expect that a five-reflection retroreflector could be constructed in three dimensions, etc. —Quondum 06:01, 24 May 2017 (UTC)[reply]

Thanks. Since a solid can rotate on two axises at once in 4D I wasn't sure if some weird thing didn't kick in when there's enough degrees of freedom. But apparently not so hypercubes would work. Sagittarian Milky Way (talk) 00:02, 25 May 2017 (UTC)[reply]