Illumination problems involving mirrors have been studied for some time in the literature. The following problem, due to Erns Straus, is the most important open problem in this area:
Problem: Is it true that in every polygonal region whose walls are mirrors, there is a point q such that if we light a match at that point, the whole of P is illuminated by a straight ray from q or by a ray that bounces off the walls of P?
In a recent paper, G. Tokarsky showed that there are some polygons that contain a point for which Straus's problem is false.
G. W. Tokarsky, Polygonal rooms not illuminable from every point". American Mathematical Monthly 10, 867-879, 1995.
V. Klee, Is every polygon illuminable from some point? American Mathematical Monthly 76, 180, 1969.
Here is another problem on mirrors due to Janos Pach:
Problem: Let S be a set of disjoint circles whose boundaries represent mirrors. Is it true that if we place a light source at any point, not on any of the mirrors, a light ray always escapes to infinity?
I. Stewart, Mathematical recreations. Scientific American, 275(2), 100-103, August 1996.
A similar problem is due to J. Urrutia and J. Zaks (1991):
Problem:
Let
be a family of
If the mirrors are infinite, this problem is false. The next figure shows a counterexample due to M. Pocchiola. It is obtained by placing a light at the center of a hexagon, H together with six mirrors emanating from the vertices of H in the couterclockwise direction as shown in the next figure:
M. Pocchiola, Personal communication, August 1993.
J. Urrutia, and J. Zaks, Personal communication, August 1991.
A la página principal de problemas abiertos . A mi página.
Jorge Urrutia,
Instituto de Matemáticas, Universidad Nacional Autónoma de México urrutia@matem.unam.mx |
School of Information Technology and Engineering University of Ottawa, jorge@site.uottawa.ca |