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

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In summary, the conversation discussed the maximum number of compartments that can be created by intersecting two cubes in 3D space. The participants proposed solutions ranging from 3 to 17 compartments, with the consensus being around 10 compartments. The concept of compartments being completely bounded by the surfaces of the two cubes was also clarified. The conversation also touched on the difficulty of visualizing the problem without a visual aid.
  • #1
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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  • #2
I got 3 but I didn't really understand the question.

The Bob (2004 ©)
 
  • #3
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
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
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?
 
  • #6
I think of 13...
 
  • #7
Still don't understand. Please help.

The Bob (2004 ©)
 
  • #8
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
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.
 

1. What is the concept of "Two Cubes in 3d"?

"Two Cubes in 3d" refers to a mathematical concept where two cubes are placed in a three-dimensional space, with one cube slightly overlapping the other. This creates an interesting visual effect and has various applications in geometry and physics.

2. How is the position of the two cubes determined in 3d space?

The position of the two cubes is determined by their respective coordinates in the three-dimensional Cartesian coordinate system. This system uses three perpendicular axes (x, y, and z) to define a point in space. The coordinates of the two cubes are used to determine their exact location in 3d space.

3. What is the relationship between the two cubes in 3d?

The two cubes in 3d have a geometric relationship where one cube is partially inside the other. This creates a shared boundary between the two cubes, and the overlapping area is known as a "face". The two cubes also have a mathematical relationship, as their coordinates and dimensions can be used to calculate various properties such as volume and surface area.

4. What are some real-world examples of "Two Cubes in 3d"?

"Two Cubes in 3d" has applications in various fields such as architecture, computer graphics, and physics. For example, architects may use this concept to create visually appealing designs, and computer graphics designers may use it to create 3d models. In physics, the overlapping of two cubes can represent the interaction between two particles in motion.

5. How does the position and orientation of the two cubes affect their appearance in 3d?

The position and orientation of the two cubes can greatly impact their appearance in 3d space. For example, if the cubes are placed far apart, they may appear to be separate objects with no visible overlap. However, if they are placed close together and at certain angles, the overlapping faces may create a visually striking effect. The orientation of the cubes can also affect how light and shadows interact with them, leading to different visual effects.

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