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The illumination problem is a resolved mathematical problem first posed by Ernst Straus. There are several equivalent statements of the problem:
- If a room has walls that are all mirrors, is there any point at which a point light source will not illuminate another point in the room, allowing for repeated reflections.
- If a snooker table can be constructed in any required shape, is 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 solved in 1958 by Roger Penrose using ellipses. 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 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. A more rigorous solution was put forward by D.Castro in 1997, with a 24-sided room.
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.