Mirrors

Table of contents

• Illuminating a polygonal room of mirrors
• A problem on a forest of circular mirrors
• Mirrors and shadows

Illuminating a polygonal room of mirrors:

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.

1. G. W. Tokarsky, Polygonal rooms not illuminable from every point". American Mathematical Monthly 10, 867-879, 1995.

2. V. Klee, Is every polygon illuminable from some point? American Mathematical Monthly 76, 180, 1969.

A problem on a forest of circular mirrors:

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?

1. I. Stewart, Mathematical recreations. Scientific American, 275(2), 100-103, August 1996.

Mirrors and shadows:

A similar problem is due to J. Urrutia and J. Zaks (1991):

Problem: Let $$M_n be a family of disjoint line segments that represent mirrors which reflect light on both sides. Let p be any point on the plane that is not aligned with at least one mirror in $$M_n. Is it true that if we place a light source at p there is always a section of the plane with area greater than 0 that is not illuminated?

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:

1. M. Pocchiola, Personal communication, August 1993.

2. J. Urrutia, and J. Zaks, Personal communication, August 1991.

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