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A math thought
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.
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.
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The basic two dimensional 4 color scenario looks something like this:
| 3 | +---+ 1 | 2 +----- +---+ | | 4Within the unbound plane there is no region that you can add that touches all 4 of these regions without breaking one of the existing borders. Color 2 is thus free to be used elsewhere within the cube and you can not connect regions 1, 3, and 4 with a new color and not enclose one of them in doing so (thus creating a new "open" color).
However, if we are allowed to wrap then it becomes possible to connect the classes without forming enclosures.
Observe:
5 color with 1 dimension of wrapping:
| 3 | ---+ +---+------ 5 | 1 | 2 | 5 -> ---+ +---+------ | | 46 color with 2 dimensions of wrapping:
A A | | | | 6 | 1 | +------+ 3 | +--+---+ +---+------ <- 5 | | | | 5 -> ---+ +--+ 2 | +--- 6 | | | | 6 -> | +---+--+--- | 1 | | | 4 | | v vNote that 3 and 4 share a boarder someplace around the wrap.
Is that clear? It's getting complicated enough that the notation is a little odd. I'm pretty sure I didn't cheat...
I don't /think/ you can get more than 5 colors with the tube, or 6 colors on the surface of a sphere.
no subject