Illumination problem

Illumination problems are a class of mathematical problems that study the illumination of rooms with mirrored walls by point light sources.

Roger Penrose's solution of the illumination problem using elliptical arcs (blue) and straight line segments (green), with 3 positions of the single light source (red spot). The purple crosses are the foci of the larger arcs. Lit and unlit regions are shown in yellow and grey, respectively.

The original formulation was attributed to Ernst Straus in the 1950s and has been resolved.[1] Straus asked if a room with mirrored walls can always be illuminated by a single point light source, allowing for repeated reflection of light off the mirrored walls. Alternatively, the question can be stated as asking that if a billiard table can be constructed in any required shape, is there a shape possible such that there is a point where it is impossible to pot the billiard ball in a pocket at another point, assuming the ball is point-like and continues infinitely rather than stopping due to friction.

The original problem was first solved in 1958 by Roger Penrose using ellipses to form the Penrose unilluminable room.[1] He showed there exists a room with curved walls that must always have dark regions if lit only by a single point source. This problem was also solved for polygonal rooms by George Tokarsky in 1995 for 2 and 3 dimensions, which showed there exists an unilluminable polygonal 26-sided room with a "dark spot" which is not illuminated from another point in the room, even allowing for repeated reflections.[2] These were rare cases, when a finite number of dark points (rather than regions) are unilluminable only from a fixed position of the point source. In 1997, two different 24-sided rooms with the same properties were put forward by G. Tokarsky and D. Castro separately.[3][4]

Solutions to the illumination problem by George W Tokarsky (26 sides) and D Castro (24 sides).

In 1995, Tokarsky found the first polygonal unilluminable room which had 4 sides and two fixed boundary points.[5] In 2016, Lelièvre, Monteil and Weiss showed that a light source in a polygonal room whose angles (in degrees) are all rational numbers will illuminate the entire polygon, with the possible exception of a finite number of points.[6]

References
  1. Weisstein, Eric W. "Illumination Problem". Wolfram Research. Retrieved 19 December 2010.
  2. Tokarsky, George (December 1995). "Polygonal Rooms Not Illuminable from Every Point". American Mathematical Monthly. University of Alberta, Edmonton, Alberta, Canada: Mathematical Association of America. 102 (10): 867–879. doi:10.2307/2975263. JSTOR 2975263.
  3. Castro, David (January–February 1997). "Corrections" (PDF). Quantum Magazine. Washington DC: Springer-Verlag. 7 (3): 42.
  4. Tokarsky, G.W. (February 1997). "Feedback, Mathematical Recreations". Scientific American. New York, N.Y.: Scientific American, Inc. 276 (2): 98. JSTOR 24993618.
  5. Tokarsky, G. (March 1995). "An Impossible Pool Shot?". SIAM Review. Philadelphia, PA: Society for Industrial and Applied Mathematics. 37 (1): 107–109. doi:10.1137/1037016.
  6. Lelièvre, Samuel; Monteil, Thierry; Weiss, Barak (4 July 2016). "Everything is illuminated". Geometry & Topology. 20 (3): 1737–1762. arXiv:1407.2975. doi:10.2140/gt.2016.20.1737.
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