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I wonder if the bit about it only working for rational angles would be better phrased as it not being proven for these angles yet? It seems strange that it woul

I wonder if the bit about it only working for rational angles would be better phrased as it not being proven for these angles yet? It seems strange that it would not work for one of these if you can get arbitrarily close to it on either side and have that work. Then again there all a lot of strange things in mathematics and my only knowledge of this problem is from this article.

notfashion 7y ago
That's fair to say. Rational angles happen to make things easy and result in a problem which has connections with lots of well-developed areas of mathematics. This paper https://www.math.brown.edu/~res/Papers/intel.pdf https://www.math.brown.edu/~res/Papers/intel.pdf is mentioned in one of the Numberphile videos on the topic and it basically says this in the introduction.
anonytrary 7y ago
But you also realize the dark point is arbitrarily small? It's an exact point. You can get arbitrarily close to an exact point without ever reaching it.
learnstats2 7y ago
I also found this strange, but I think the example is much harder to understand than the article claims. By my current interpretation: If the light shines in all directions, then a regular polygon, or in fact any convex polygon, doesn't need the mirrors to illuminate the room - the candle will light the room without needing a bounce. So, if you're thinking of a convex polygon, it's not quite relevant to this example. The only difficult case would be if there was a concave polygon and the light had to bounce into the "caves" (or perhaps a series of rooms). It's possible to imagine that there are specific set-ups where, if the mirrors are set up just right, the light will be bounced back out of a cave rather than filling it up. If such a situation exists, then that bounce back would happen only with a very precise set-up: being arbitrarily close to the set-up wouldn't result in the bounce back.

notfashion 7y ago

That's fair to say. Rational angles happen to make things easy and result in a problem which has connections with lots of well-developed areas of mathematics. This paper https://www.math.brown.edu/~res/Papers/intel.pdf https://www.math.brown.edu/~res/Papers/intel.pdf is mentioned in one of the Numberphile videos on the topic and it basically says this in the introduction.

anonytrary 7y ago

But you also realize the dark point is arbitrarily small? It's an exact point. You can get arbitrarily close to an exact point without ever reaching it.

learnstats2 7y ago

I also found this strange, but I think the example is much harder to understand than the article claims. By my current interpretation: If the light shines in all directions, then a regular polygon, or in fact any convex polygon, doesn't need the mirrors to illuminate the room - the candle will light the room without needing a bounce. So, if you're thinking of a convex polygon, it's not quite relevant to this example. The only difficult case would be if there was a concave polygon and the light had to bounce into the "caves" (or perhaps a series of rooms). It's possible to imagine that there are specific set-ups where, if the mirrors are set up just right, the light will be bounced back out of a cave rather than filling it up. If such a situation exists, then that bounce back would happen only with a very precise set-up: being arbitrarily close to the set-up wouldn't result in the bounce back.