Joho the Blog
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May 15, 2003
There was an interesting the Boston Globe recently about Four Colors Suffice, a book by Robin Wilson on the history of the famous 4-color problem: How do you prove that you only need four colors to ensure that neighboring countries are colored differently. (More important: Why is Greenland pink?) The proof (according to the article about the book that I didn't read) was the first generated by a computer that couldn't be checked by humans: in 1976, a Cray ground through every conceivable variation and found none that required more than four colors. I have a question for the mathematically inclined (i.e., people unlike me): How many colors would you need for a 3-D map? Or, if you prefer, how many colors would you need to ensure that blocks (of any shape) stacked in any arbitrary way have differently colored neighbors? I am so bad at 3D stuff that you could tell me the answer is 2 and I would believe you, just so long as you looked at me with those doe-eyes of yours. Posted
by D. Weinberger at May 15, 2003 09:35 AM
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Comments
This is something I've wondered for a while, and so far all I can conclude is that it requires at least six. Think of a map spread out on a table, and imagine the air above the map as another country. The air borders (in 3D) every country on the four color map, so you need at least five colors for the map and the air. You could then take a piece of silly putty and make it touch four countries which the air also touches, adding another color to get six. Maybe you can even keep doing this until you reach lots more colors, I'm not sure -- but it's at least six.
Posted by: Jonathan Blocksom | May 15, 2003 01:05 PM
http://mathforum.org/library/drmath/view/61105.html
shows that it is infinite.
Essentially think about a braided or woven object
made up of strands of different colors. In the braided case, each strand can touch every other strand, in the woven case, each strand touches n/2
colors. So no finite n is large enough.
Posted by: Jim Penny | May 15, 2003 08:20 PM
If you read my book, you'll see two examples of three-dimensional maps that require as many colours as you choose. One of these was produced by Frederick Guthrie back in 1880.
RJW
Posted by: Robin Wilson | August 25, 2003 08:27 PM
u guys r sshole
Posted by: Anonymous | January 13, 2005 05:35 PM