# 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

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

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

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]