How many compartments can be created by intersecting two cubes in 3d?

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Discussion Overview

The discussion revolves around determining the maximum number of compartments that can be created by intersecting two cubes in 3D space. Participants explore various configurations and rotations of the cubes to conceptualize how many distinct bounded regions can be formed.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that rotating one cube by 45 degrees can create nine compartments.
  • Others propose that a slight second rotation and translation could potentially yield a tenth compartment.
  • One participant speculates that aligning the body diagonals of the cubes could lead to as many as 17 solid regions, although this is not visually confirmed.
  • Another participant claims to have visualized 11 compartments based on specific rotations of the cubes.
  • Some participants express uncertainty about the number of compartments, with guesses ranging from 10 to 13, and acknowledge the difficulty of visualizing the problem without a diagram.
  • There is a repeated assertion that if the cubes are perfectly superimposed, they form one compartment, and various rotations can increase the number of compartments.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the maximum number of compartments, with estimates varying from 9 to 17. Multiple competing views remain, and the discussion reflects uncertainty and differing interpretations of the configurations.

Contextual Notes

Some participants note the challenge of visualizing the problem in a verbal format, suggesting that a diagram would aid in understanding the spatial relationships involved.

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

The Bob (2004 ©)
 
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
 
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
 
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 ?

Got AutoCAD nearby?
 
I think of 13...
 
Still don't understand. Please help.

The Bob (2004 ©)
 
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.
 
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
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
NateTG said:
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.

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

The Bob (2004 ©)
 
  • #12
AKG said:
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.

A cube has 8 corners but only 6 faces. Not corner will be a bounded region.
 
  • #13
Jin314159 said:
A cube has 8 corners but only 6 faces. Not corner will be a bounded region.
My bad. Not sticking out of opposite faces then, but sticking out near the opposite corners of the other cube.
 

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