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The illumination problem is a resolved mathematical problem first posed by Ernst Straus in the 1950s. Straus asked if a single point light source can illuminate a room has walls that are all mirrors allowing for repeated reflections, regardless of the shape of the room. Alternatively, the question can be stated as asking if a snooker table can be constructed in any required shape, such that there a point where it is impossible to pot the billiard ball from another point, assuming the ball continues infinitely rather than being subject to friction
The problem was first solved in 1958 by Roger Penrose using ellipses to form the penrose unilluminable room. He showed there exists a room 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 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. This was a borderline case, however, since a finite number of dark points (rather than regions) that are unilluminable from any given position of the point source. An improved solution was put forward by D.Castro in 1997, with a 24-sided room with the same properties.
References
- Tokarsky, George (December 1995). "Polygonal Rooms Not Illuminable from Every Point". American Mathematical Monthly. 102 (10). University of Alberta, Edmonton, Alberta, Canada: Mathematical Association of America: 867–879. Retrieved 19 December 2010.