A math thought

[benlehman]

The "four color map" problem is unique to two dimensional space. Why?

I was thinking about this years ago, and I forgot it until today.

Edit: Huh. Does it apply in four dimensions or not? I thought I answered that but I didn't actually. My hunch is that for three and higher dimensions the number of mutually adjacent objects is infinite.

Let's look at this:
dimensionality -> maximum mutually adjacent objects
0 -> 1
1 -> 2
2 -> 4
3 -> infinite
4 -> ?

Second Edit: It doesn't seem to matter if the space is open (infinite in all directions) or closed (loops at the edges.) Does it?

Third Edit: So the 1d case is trivial, largely because a 1d object can only be adjacent to two other objects, which clearly limits mutual adjacency. A 2d object can be adjacent to an infinite number of other objects, a fact which I make use of in my infinite mutually adjacent 3d objects construction (see comments). Perhaps mutual adjacency in dimension n depends on adjacency of the dimension n-1? That seems tempting, especially if we're going to view dimension n as dimenion n-1 with a time axis.

Comments

[ornithoptercat.livejournal.com]

I can easily think of a situation in 3-space where you'd have four 3D shapes representing countries (or whatever) all touching one another. Or is the problem nonexistant in that there is no upper limit to the number of colors you can force it to require?

[benlehman.livejournal.com]

Yeah, you can pretty easily imagine an infinite set of mutually adjacent three dimensional objects.

The generalized four color map would be: "What is the maximum size of a set of mutually adjacent n-dimensional objects in an n-dimensional space?"

[benlehman.livejournal.com]

To elaborate on the last post, here's what such a set looks like:

Picture a stack of boxes. Stack as many as you like.

Now, nail a plank to the top box so long it stretches down to touch the bottom box (and thus, naturally, touchs all the other boxes.) Nail a plank to the next box such that it, too stretches to the bottom. Etc, etc.

Picture it with ten or so and you see how you can stack as many boxes as you like this way.

yrs--
--Ben