Tuesday, January 14, 2014

Maps can be rectangularized maintaining borders

You can take any map, say this one of the southeast U.S....
And do what I call rectangularizing it maintaining borders. So the first step is to draw every border including the ones with the blank area of the map as straight lines. A peninsula such as Florida is made into a triangle.
The distinct regions, in this case states can be labeled or numbered and removed from the original map.
Now imagine you draw an infinite number of lines across this map from north to south, and group together any infinite collection of lines that cross through the same regions in the same order. For example one group will be:
Every line drawn between these two north south lines (they're supposed to be parallel but I was drawing by hand) passes through region 11, then 7, then 3, then 1. So doing this all the way across the map...
These are supposed to be parallel but I think you get the idea...
And now the next step is to do the same thing but with horizontal lines...
In this case you end with a 5x12 grid where each cell contains a part of 1 region or none at all or it is split by a diagonal line in which case you can draw extra lines and give the left half to one region and the right half to the other region. So This can be translated to a grid...

This is the rectangularized version. You can see that for instance from a part of the 8 region you can go north to get to 11 from another part you can only get to 10 from going north, etc... And I think every map can have this done to it. I'm not sure if it could lead to an easier proof of the four colour theorem or not...