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.