Multi-Dimensional Four-color TheoremDate: 08/08/97 at 10:46:53 From: Kenny Jensen Subject: Multi-dimensional Four-color Theorem I was wondering if any work had been done on theorems like the four- color theorem for different dimensions. It seems to me that a more general theorem could be made such as: The number of colors needed to color an n-dimensional map is 2^n. This seems to work for points, lines, and planes. If there has been any work on this subject could you please point me to the sources? Date: 08/11/97 at 14:24:25 From: Doctor Ceeks Subject: Re: Multi-dimensional Four-color Theorem Hi, This is a very nice question. It's a very good idea to try to generalize theorems in ways as natural as the one you are suggesting. In this case, however, once you go to three dimensions, you can make partitions of space into regions for which you need N colors to color the regions in order that no two adjacent regions will have the same color for any N. You can make an example by starting with one ball. Now, add a ball to the picture and connect it with a thin tube to the first ball. Now, add a third ball to the picture and connect this ball with two thin tubes to the two balls already in the picture. You can keep adding balls and connecting them to all the other balls like this because there is enough space in three dimensions to work with. If the balls represent regions, since each ball is touching every other ball, you need at least as many colors as there are balls to color them. Do you understand what I'm saying? -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 08/11/97 at 18:41:45 From: Anonymous Subject: Re: Multi-dimensional Four-color Theorem Hello, Thanks for the explanation. It helped me visualize the question better than I had previously. |
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