# Two Cubes in 3d

1. Jul 23, 2004

### Jin314159

Take two cubes of the same size in 3d space. What is the maximum number of compartments that can be created by intersecting these two cubes? A compartment can be defined as a space that is completely bounded by surfaces of the two cubes, which doesn't contain another compartment within itself.

2. Jul 23, 2004

### The Bob

I got 3 but I didn't really understand the question.

3. Jul 23, 2004

### Njorl

Imagine holding them in your fingers, index finger on top, thumb on the bottom. Hold one by the faces, with a face toward you. Hold the other the same way, then rotate it by 45 degrees on any axis. Colocate the centers of mass. This gives 9 solid regions. I have been considering a second rotation, but I haven't seen a benefit yet.

Njorl

4. Jul 23, 2004

### Njorl

I think a very small second rotation combined with a slight movement from coinciding centers of mass might create a tenth solid region.

The slight rotation connects 2 pairs of regions (eliminating two), and creates two more. The slight translation then cuts apart one of the connections formed by the slight rotation, giving a 10th region.

That's as good as I can do without holding a couple of dice in my hand.

Njorl

5. Jul 23, 2004

### Gokul43201

Staff Emeritus
I can picture the 9 region solution...can't see the 10th. But I'll go out on a limb and propose 17 solid regions. I can't really "see" this...but it may happen if you line up one body diagonal (line between opposite corners) along the line joining opposite face centers with both cubes concentric.

Why does this make me wish I had tried harder to picture Brillouin Zones in 3D ?

6. Jul 23, 2004

### hemmul

I think of 13...

7. Jul 23, 2004

### The Bob

8. Jul 23, 2004

### NateTG

Bob - if you allow the cubes to intersec, then the surfaces of the cubes form a partition of space. The compartments are elements of the partition.

Fore example:
If the two cubes are perfectly superimposed, then they form one compartment.

If you then rotate one of the cubes 45 degrees through the center on an axis perpendicular to one of the faces, you will have nine compartments.

There are at most 13 'compartments', but I think the maximum is even lower than that.

9. Jul 23, 2004

### Jin314159

I got 11 compartments, but I'm not sure if that's right.

Like NateTG said, if the two cubes are perfectly superimposed, they form one compartment. If the you rotate one of the cubes 45 degrees through the center on an axis perpendicular to one of the faces, you have nine compartments. Now, if you rotate the same cube on an axis perpendicular to the first axis and a face, which also goes through the center, you get 11 regions.

Maybe this was a bad brain teaser to put up on a verbal-based forum. You really have to see the picture to get it.

10. Jul 24, 2004

### AKG

I think it's ten. Haven't proved it yet, just an initial guess. My reasoning is if you take one of the cubes, and place it "in" the other cube such that every pair of opposite corners of the first cube stick out of a pair of opposite faces of the second, then the first cube has it's 8 corners forming bounded regions. On top of that, you will have some region of the second cube that doesn't overlap the first, and finally a region where both overlap, this gives 10.

11. Jul 24, 2004

### The Bob

Cheers TG. I believe I've got it. :surprise:

12. Jul 24, 2004

### Jin314159

A cube has 8 corners but only 6 faces. Not corner will be a bounded region.

13. Jul 24, 2004

### AKG

My bad. Not sticking out of opposite faces then, but sticking out near the opposite corners of the other cube.