[Someone]

To help people understand why this theorem is surprising, first rephrase it from “the candle lights the entire room (bar a finite number of points)” to “the candle can be seen from any position in the room (bar a finite number of positions)”

Next, don’t think of “room”; that puts your mind too much towards simple, almost convex structures. Instead, think of the a floor of a building where all doors are removed.

For example, take the ground floor plan of the Pentagon, with its myriad of rooms and corridors, with all doors removed, and replace all walls by perfect mirrors. Is there a spot to place a candle so that it or it’s reflection, reflection of a reflection, etc. can be seen from all locations in the pentagon, bar a finite number? The theorem says there is.

Now, feel free to make it harder: add back the doors, but don’t completely close them, keeping a rational angle with the walls the door opening is in. Feel free to make the angles as small as you like.

Next, place room dividers wherever you want, as long as they are perfect mirrors, form rational angles with the walls, and don’t completely close of some room or corridor in the Pentagon.

Do you think you’ll be able to completely shield of at least one room, wherever that candle is placed? If so, you’re mistaken.

[NKosmatos]

That would make a great 2D game... You’re given a floor plan at each level (with increased complexity/geometry) and you need to place the candle (light source) in a point where it will illuminate the whole floor.

[crawfordcomeaux]

I want to design floorplans with intentional shadow spaces.

[pbhjpbhj]

There's a good game called Lazer Maze (by ThinkFun) that is related, it concerns shining a directional Lazer though.

[anonytrary]

> bar a finite number? The theorem says there is.

I find that so interesting, as if these positions are an intrinsic mathematical property of the room. I wonder if there's a classification of rooms this way. What do rooms that have the same number of dark points have in common?

This theorem also has implications in the limits of using sound and light for surveillance. The government could ban homes with irrational angles in order to guarantee there are no dark spots for any kind of radiation surveillance.

[popcorncolonel]

Great visualization, thanks!