Illumination problem
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Penrose_unilluminable_room.svg/220px-Penrose_unilluminable_room.svg.png)
Illumination problems are a class of
Original formulation
The original formulation was attributed to
Penrose unilluminable room
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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Tokarsky_Castro_illumination_problem.svg/220px-Tokarsky_Castro_illumination_problem.svg.png)
This problem was also solved for
In 1997, two different 24-sided rooms with the same properties were put forward by George Tokarsky and David Castro separately.[2][3]
In 1995, Tokarsky found the first polygonal unilluminable room which had 4 sides and two fixed boundary points.[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 Odd Sided Tokarsky Unilluminable Room with 27 sides, 1996. A video showing the path of a billiard ball in this room.
See also
References
- JSTOR 2975263.
- ^ Castro, David (January–February 1997). "Corrections" (PDF). Quantum Magazine. 7 (3). Washington DC: Springer-Verlag: 42.
- JSTOR 24993618.
- doi:10.1137/1037016.
- .
- arXiv:1905.09358 [math.DS].
External links
- "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