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?

[clockwise.livejournal.com]

Your 0 and 1 dimensional examples counter your assertion that the "four color map" problem is unique to two dimensional space. In zero dimensions it is the "one color map" and in one dimension it is the "two color map", both of which are exceptionally boring.

If as you assert 3 dimensions is infinite, then 4 is infinite too (duh).

A space that loops around the edges requires additional colors. Consider this example which have classes 3 & 4 wrapping around the edges thus requiring a fifth color.

      A
      |
    |   |            ?
    |   +-------+
    |   |       |
    |   +---+   |
    | 4 |   |   |
  --+---+ 1 |   +------
        |   |   |   
<-    3 +---+ 5 | 3     ->
        |   |   |
  --+---+ 2 |   +------
    | 4 |   |   |
    |   +---+   |
    |   |       |
    |   +-------+   ?
    |   |
      |
      v

In an infinite plane the right hand case of 3 could reuse either color 1 or color 2 since the had already been enclosed, but by allowing wrapping a class that touches these can go take an alternate way around to their neighbor (5) without having to also enclose the fourth color.