Illumination problem
Illumination problems are a class of mathematical problems that study the illumination of rooms with mirrored walls by point light sources.
Original formulation
[edit]The original formulation was attributed to Ernst Straus in the 1950s and has been resolved. Straus asked whether 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 hit the billiard ball at another point, assuming the ball is point-like and continues infinitely rather than stopping due to friction.
Penrose unilluminable room
[edit]The original problem was first solved in 1958 by Roger Penrose using ellipses to form the Penrose unilluminable room. He showed that there exists a room with curved walls that must always have dark regions if lit only by a single point source.
Polygonal rooms
[edit]This problem was also solved for polygonal rooms by George Tokarsky in 1995 for 2 and 3 dimensions, which showed that 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.[1] 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 1995, Tokarsky found the first polygonal unilluminable room which had 4 sides and two fixed boundary points.[2] He also in 1996 found a 20-sided unilluminable room with two distinct interior points. In 1997, two different 24-sided rooms with the same properties were put forward by George Tokarsky and David Castro separately.[3][4]
In 2016, Samuel Lelièvre, Thierry Monteil, and Barak 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.[5] In 2019 this was strengthened by Amit Wolecki who showed that for each such polygon, the number of pairs of points which do not illuminate each other is finite.[6]
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The first polygonal Tokarsky Unilluminable room with 4 sides, 1995. A video showing the path of a billiard ball in this room.
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The Original Tokarsky Unilluminable Room with 24 sides, 1995. A video showing the path of a billiard ball in this room.
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An Unilluminable room with 20 sides, 1996. A video showing the path of a billiard ball in this room.
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An Odd Sided Tokarsky Unilluminable Room with 27 sides, 1996. A video showing the path of a billiard ball in this room.
See also
[edit]References
[edit]- ^ 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. doi:10.2307/2975263. JSTOR 2975263.
- ^ Tokarsky, G. (March 1995). "An Impossible Pool Shot?". SIAM Review. 37 (1). Philadelphia, PA: Society for Industrial and Applied Mathematics: 107–109. doi:10.1137/1037016.
- ^ Castro, David (January–February 1997). "Corrections" (PDF). Quantum Magazine. 7 (3). Washington DC: Springer-Verlag: 42.
- ^ Tokarsky, G. W. (February 1997). "Feedback, Mathematical Recreations". Scientific American. 276 (2). New York, N.Y.: Scientific American, Inc.: 98. JSTOR 24993618.
- ^ 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.
- ^ Wolecki, Amit (2019). "Illumination in rational billiards". arXiv:1905.09358 [math.DS].
External links
[edit]- "The Illumination Problem – Numberphile", on YouTube by Numberphile, Feb 28, 2017
- "Penrose Unilluminable Room Is Impossible To Light", on YouTube by Steve Mould, May 19, 2022
- "The mushroom's shape does not matter in Penrose's unilluminable room", on YouTube by Nils Berglund, Aug 13, 2022
- "The Tokarsky original unilluminable room with 24 sides", on YouTube by George Tokarsky, Jun 16, 2022
- "Egyptian hieroglyphs: An Odd Tokarsky unilluminable room", on YouTube by George Tokarsky, Jul 15, 2022
- "Eureka! The first polygonal unilluminable room", on YouTube by George Tokarsky, Jul 29, 2022
- An interactive demonstration, on Wolfram demonstrations project