Illumination problem
Encyclopedia
The illumination problem is a resolved mathematical problem first posed by Ernst Straus in the 1950s. Straus asked if 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 if a billiard table can be constructed in any required shape, such that there is a point where it is impossible to the billiard ball in a at another point, assuming the ball continues infinitely rather than stopping due to friction
.
The problem was first solved in 1958 by Roger Penrose
using ellipses to form the Penrose unilluminable room. Using the properties of the ellipse, he showed there exists a room with curved walls that must always have dark regions if lit only by a single point source. This was a borderline case, however, since a finite number of dark (rather than regions) are not illuminable from any given position of the point source. An improved solution was put forward by D. Castro in 1997, with a 24-sided room with the same properties.
Point source
A point source is a localised, relatively small source of something.Point source may also refer to:*Point source , a localised source of pollution**Point source water pollution, water pollution with a localized source...
, allowing for repeated reflection of light off the mirrored walls. Alternatively, the question can be stated as asking if a billiard table can be constructed in any required shape, such that there is a point where it is impossible to the billiard ball in a at another point, assuming the ball continues infinitely rather than stopping due to friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
.
The problem was first solved in 1958 by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
using ellipses to form the Penrose unilluminable room. Using the properties of the ellipse, he showed there exists a room with curved walls that must always have dark regions if lit only by a single point source. This was a borderline case, however, since a finite number of dark (rather than regions) are not illuminable from any given position of the point source. An improved solution was put forward by D. Castro in 1997, with a 24-sided room with the same properties.