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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.

This can also be thought of in terms of a billiard or pool table. Is there some configuration that has one spot (or more) which is impossible to hit from some other point, even if the ball bounces around forever?

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.

Roger Penrose’s solution of the illumination problem using elliptical arcs (blue) and straight line segments (green), with 3 positions of the single light source (red spot). The purple crosses are the foci of the larger arcs. Lit and unlit regions are shown in yellow and grey, respectively.

The solution was improved upon when George Tokarsky found a 26-sided room with a “dark” spot, and then two years later D. Castro offered the 24-sided improvement below. If a candle is placed at A, and you’re standing at B, you won’t see its reflection anywhere around you — even though you’re surrounded by mirrors.

If Satan plays miniature golf, this is his favorite hole. A ball struck at A, in any direction, will never find the hole at B — even if it bounces forever.

[Via Futility Closet, Wikipedia]

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