# Four Color Theorem

• December 26th, 2010, 12:57 PM
m84uily
Four Color Theorem
The week before Christmas break my teacher told me about this theorem. Since then I've been wondering: How many colors would be required for a map in 3 space?

Here are a few stupid solutions to the problem:

n space requires 2n colors
1 space requires 2 colors
2 space requires 4 colors
3 space requires 6 colors

n space requires 2^n colors
1 space requires 2^1 colors
2 space requires 2^2 colors
3 space requires 2^3 colors

3 space requires infinitely many colors.

Does anyone have a good idea for the answer to this question?
• December 26th, 2010, 06:08 PM
MagiMaster
http://en.wikipedia.org/wiki/Four_co...eneralizations

In 3D, you can make a set of objects that requires arbitrarily many colors. The example given is a flexible set of rods.
• December 26th, 2010, 06:27 PM
m84uily
Okay, thank you.
• July 1st, 2011, 07:00 PM
GuildCompounder
Patterned 3d spaces
I wrote an oscillating resolver in 2d to prove all combinations of map at a given size of rectangular grid map. Since the 4 color theorem is proved by other means, I guess that means it is safe to assume that the oscillating resolver damps even though it picks random colors to avoid impasse. It also must be safe to assume that it is possible to induce to infinity in both dimensions of the rectangle, the oscillating resolver will still damp to a resolution.
What are the uses of the oscillating resolver in higher dimensional patterned spaces? Isn't it then safe to prove k-coloration in n dimensions in patterned spaces with finite coloration with oscillating resolvers?