mathematics

# Geometry Problem

Say that you have an empty transparent cube. Then you put a reflective sphere inside – the biggest possible one. And then the next biggest possible reflective sphere. Or 8 of them, or any number of them. And so on.

You may consider a supertasking to do it, or just say that this is your transparent cube, packed with distinct reflective spheres of different sizes. There is no room for another sphere inside, no matter how small it would be.

Now, you beam an ideal ray toward one of the cube’s outer square’s midpoint. It will just reflect from the biggest sphere. The angle of incidence may even vary, doesn’t matter.

Now you beam a ray in the direction of one of the cubes main diagonals – from outside toward the center. Will it:

1. bounce back immediately after reaching the corner
2. bounce back after some finite time
3. exit at the opposite corner after some finite time
4. exit somewhere else after some finite time
5. never come out again
6. This is an illegal or ill defined question

What say you?

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## 6 thoughts on “Geometry Problem”

1. Olga says:

It will bounce directly. Every point on the surface of the cube is a point where the surface of a sphere is tangent to the cube. Otherwise you could fit in another smaller sphere. And at the very corners there are infinitely small spheres that are tangents to the corners, which the light will bounce off of.

2. There is no sphere containing corner points. All spheres are of a finite (nonzero) radius. I should have been more clear about that.

For the purpose of this problem only non-degenerated spheres count. Those which are not 1 point, but infinitely many points.

3. Olga says:

Then you can’t fill the cube. Or to be precise, then there is no last iteration of adding spheres, so there is no well defined behavior after the last step.

• Yes, this is the problem. You CAN’T fill the cube with spheres, so that the every last point is covered. But you CAN fill the cube, so that there is no finite volume of space inside the cube left.

Which is a subtle difference. But very well known, and assumed that it’s very well understood, too.

But it isn’t, really. Those pesky rays in the form of a moving point, spoil the whole game. But not only the game of this geometrical problem. No, no.

The whole infinity game is malade. It’s deeply related to the Yablo’s question, I wrote about here already.