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In 1995, Tokarsky found the first polygonal unilluminable room which had 4 sides and two fixed boundary points. 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
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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. In 2019 this was strengthened by Amit Wolecki who
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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. These were rare cases, when a finite number of dark
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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
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arcs (blue) and straight line segments (green), with 3 positions of the single light source (red spot). The purple crosses are the
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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
98:. He showed that there exists a room with curved walls that must always have dark regions if lit only by a single point source.
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showed that for each such polygon, the number of pairs of points which do not illuminate each other is finite.
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at another point, assuming the ball is point-like and continues infinitely rather than stopping due to
234:(10). University of Alberta, Edmonton, Alberta, Canada: Mathematical Association of America: 867–879.
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Solutions to the illumination problem by George W. Tokarsky (26 sides) and David Castro (24 sides)
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Tokarsky, George (December 1995). "Polygonal Rooms Not
Illuminable from Every Point".
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of the larger arcs. Lit and unlit regions are shown in yellow and grey respectively.
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271:(1). Philadelphia, PA: Society for Industrial and Applied Mathematics: 107–109.
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Tokarsky, G. W. (February 1997). "Feedback, Mathematical
Recreations".
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The first polygonal
Tokarsky Unilluminable room with 4 sides, 1995. A
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The mushroom's shape does not matter in
Penrose's unilluminable room
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Lelièvre, Samuel; Monteil, Thierry; Weiss, Barak (4 July 2016).
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An Odd Sided
Tokarsky Unilluminable Room with 27 sides, 1996. A
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The
Original Tokarsky Unilluminable Room with 24 sides, 1995. A
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Fashion, Faith, and
Fantasy in the New Physics of the Universe
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Hadwiger conjecture (alternate formulation with illumination)
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Wolecki, Amit (2019). "Illumination in rational billiards".
466:"Egyptian hieroglyphs: An Odd Tokarsky unilluminable room"
456:"The Tokarsky original unilluminable room with 24 sides"
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Tokarsky, G. (March 1995). "An Impossible Pool Shot?".
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328:(2). New York, N.Y.: Scientific American, Inc.: 98.
431:"Penrose Unilluminable Room Is Impossible To Light"
195:showing the path of a billiard ball in this room.
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163:showing the path of a billiard ball in this room.
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90:The original problem was first solved in 1958 by
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476:"Eureka! The first polygonal unilluminable room"
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175:An Unilluminable room with 20 sides, 1996. A
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687:Penrose interpretation of quantum mechanics
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417:"The Illumination Problem – Numberphile"
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307:(3). Washington DC: Springer-Verlag: 42.
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213:References
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806:(brother)
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785:Related
480:YouTube
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