Illumination problems are a class of

mathematical problem A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...

s that study the illumination of rooms with

mirror A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...

ed walls by point light sources.

Original formulation

The original formulation was attributed to

Ernst Straus Ernst Gabor Straus (February 25, 1922 – July 12, 1983) was a German-American mathematician of Jewish origin who helped found the theories of Euclidean Ramsey theory and of the arithmetic properties of analytic functions. His extensive list of c ...

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 A billiard table or billiards table is a bounded table on which cue sports are played. In the modern era, all billiards tables (whether for carom billiards, pool, pyramid or snooker) provide a flat surface usually made of quarried slate, ...

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 A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being played. Various particular ball pro ...

at another point, assuming the ball is point-like and continues infinitely rather than stopping due to

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...

Penrose unilluminable room

The original problem was first solved in 1958 by

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...

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

This problem was also solved for

polygonal In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

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 ''points'' (rather than regions) are unilluminable only from a fixed position of the point source. In 1997, two different 24-sided rooms with the same properties were put forward by George Tokarsky and David Castro separately. In 1995, Tokarsky found the first polygonal unilluminable room which had 4 sides and two fixed boundary points. 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 showed that for each such polygon, the number of pairs of points which do not illuminate each other is finite.

References

External links

"The Illumination Problem – Numberphile"
on

YouTube YouTube is a global online video sharing and social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the second most ...

Numberphile ''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...

, Feb 28, 2017
"Penrose Unilluminable Room Is Impossible To Light"
on

by Steve Mould, May 19, 2022 *
The mushroom's shape does not matter in Penrose's unilluminable room
, on

by Nils Berglund, Aug 13, 2022
"The Tokarsky original unilluminable room with 24 sides"
on

by George Tokarsky, Jun 16, 2022
"Egyptian hieroglyphs: An Odd Tokarsky unilluminable room"
on

by George Tokarsky, Jul 15, 2022
"Eureka! The first polygonal unilluminable room"
on

by George Tokarsky, Jul 29, 2022 * A
interactive demonstration
on

Wolfram demonstrations project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

{{Roger Penrose, state=collapsed Mathematical problems Dynamical systems