Math

The Illumination Problem Is A Well-Known Mathematical Problem

This is the illumination problem: There is a room with one candle in it. The light from the candle (which radiates out in all directions) will bounce off the walls and travel around the room indefinitely. Is it possible to have a room with areas that will never be illuminated by the candle? Short answer: Yes, such rooms exist. The shapes of two examples, though, are weirder than you'd think.

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That's One Powerful Candle

First, a technicality: in this mathematical context, assume the light from the candle can travel through the room and bounce off the walls forever. If you have a regular rectangular room, a single candle that radiates in all directions will totally illuminate the room. Things get trickier when the room is not entirely convex—think of the shape of a five-point star as an example. In most cases, you'll find that the candle will bounce around enough to reach every corner to illuminate even the weirdest shaped rooms.

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This problem was initially posed in the 1950s by German-American mathematician Ernst Straus. Roger Penrose, an English mathematician and physicist, came up with a solution in 1958. He conceived of a room involving half-ellipses (semi-circles, basically) that would always have dark regions. This room is something like an oval with two sideways-lying, mushroom-shaped chunks taken out of opposite ends.

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Here's Where It Gets Weird

In 1995, mathematician George Tokarsky conceived of the first polygonal room to solve the problem—no curved lines here. His 26-sided polygonal room is weird in that it's not a little sliver in the corner of the room that's left in the dark, like Penrose's example. In Tokarsky's strange case, there are just individual points in the room that will never be illuminated, not full regions. Since '95, there have been more rooms imagined that would never be completely illuminated (keep in mind that these are theoretical constructs, not literal rooms you can stand in). However, Tokarsky was the first to design a polygonal room that has at least one dark point.

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– Numberphile

– Numberphile
Curiosity Staff