Next: Problem 32: Bar-Magnet Polyhedra
Previous: Problem 30: Thrackles
Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? A light ray is trapped if it includes no point strictly exterior to the convex hull of the mirrors. The source point is disjoint from the mirrors. Although several versions of the problem are possible, it seems to make the most sense to treat the mirrors as open segments (i.e., not including their endpoints), but demand that they are disjoint as closed segments.
O’Rourke and Petrovici [OP01]. The question seems natural enough to have been raised earlier, but no other source is known.
Conjecture 9 from that paper: “No collection of segment mirrors can trap all the light from one source.”
In [OP01] several other conjectures are formed that imply a resolution to the posed problem. The strongest—that no collection of mirrors as above can support even a single nonperiodic ray, i.e., one that reflects forever (so is trapped) but never rejoins its earlier path—was disproved by Ben Stephens in 2002, who designed a contruction of 8 mirrors that traps a ray reflecting nonperiodically. A similar construction was discovered and described in [MSZ09], which also established that any finite number of rays can be trapped nonperiodically. Milovich [Mil04] proved that if the angles between the lines containing the mirrors are rational multiples of \pi, then all but a countable number of light rays escape. In his book on billiards, Tabachnikov says, “It is unknown whether one can construct a polygonal trap for a parallel beam of light” [Tab05], p. 116. This is in contrast to known nonpolygonal traps for such beams.
Pach’s “enchanted forest” of circular mirrors.
Presented at the Open Problem session of the 13th Canad. Conf. Comput. Geom., Waterloo, Ontario, Aug. 2001. Also, Oberwolfach, Jan. 2009.
J. O’Rourke, 28 Aug. 2001; 24 Feb. 2003; 5 Oct. 2005; 7 Sep. 2009.
David Milovich. Trapping light with mirrors. MIT Undergrad. J. Math., 6:153–180, 2004.
Zachary Mitchell, Gregory Simon, and Xueying Zhao. Trapping light rays aperiodically with mirrors. Unpublished manuscript, August 2009.
Joseph O’Rourke and Octavia Petrovici. Narrowing light rays with mirrors. In Proc. 13th Canad. Conf. Comput. Geom., pages 137–140, 2001.
Serge Tabachnikov. Geometry and Billiards, volume 30 of Mathematics Advanced Study Semesters. American Mathematical Society, 2005.