Illumination problem
Illumination problems are a class of mathematical problems that study the illumination of rooms with mirrored walls by point light sources.
![](../I/Penrose_unilluminable_room.svg.png.webp)
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]
![](../I/Tokarsky_Castro_illumination_problem.svg.png.webp)
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
- Weisstein, Eric W. "Illumination Problem". Wolfram Research. Retrieved 19 December 2010.
- 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.
- Castro, David (January–February 1997). "Corrections" (PDF). Quantum Magazine. Washington DC: Springer-Verlag. 7 (3): 42.
- Tokarsky, G.W. (February 1997). "Feedback, Mathematical Recreations". Scientific American. New York, N.Y.: Scientific American, Inc. 276 (2): 98. JSTOR 24993618.
- 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.
- 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.
External links
- "The Illumination Problem - Numberphile", on YouTube by Numberphile, Feb 28, 2017