Wikipedia:Reference desk/Archives/Mathematics/2017 May 23

= 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)


 * 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)


 * 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)