All regular maps can be topologically transformed into rectangular of circular maps (see T2).
Here is the computerized and colored version: Conversion of famous maps UPDATED
Tutte’s map conversion (2/2)
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Tag Archives: Wolfgang Haken
Tutte’s map conversion (1/2)
CORRECTED!
All regular maps can be topologically transformed into rectangular of circular maps (see T2).
This is the conversion of Tutte’s map, made by hand, into a rectangular map. The colored and computerized version will follow.
These conversion are done just for fun and have not theoretical interest. The theorem in T2 proves that all regular maps (basically all maps of interest to the four color theorem) can be converted into circular of rectangular maps.
Tutte’s map on the left and the map made of rectangles right below it, are exacly the same map. Face number 25 correspond to the surrounding area (the ocean) of the rectangular map.
Converted a map taken from Wikipedia
Conversion of famous maps UPDATED
Original image taken from:
http://en.wikipedia.org/wiki/Four_color_theorem
Same image converted into circular and rectangular (using different colors):
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How to convert maps into rectangular maps
I am converting some famous maps into rectangular maps. You can check some converted maps here: Conversion of famous maps.
First steps are explained here: First steps.
Just another site about the four color theorem?
The four color problem has already been proved by Kenneth Appel and Wolfgang Haken back in 1976 (http://en.wikipedia.org/wiki/Four_color_theorem). Since it was the first major problem that required a computer to be completed, many people are still trying to find a simple and elegant, human checkable, “pencil and paper” proof of the problem … if any.
The approach used here to solve the problem is based on two results which, I think (please, please, please verify), haven’t been considered so far.
- Only maps with all faces with five or more edges can be considered when searching for a proof of the four color problem, as proved in T1 (T1 was already known by Kempe in the year 1879 – CORRECTED! 02/Apr/2011)
- All regular maps, no matter the complexity, can be topologically transformed into “circular maps” or “rectangular maps”, as proved in T2
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I would really like someone to verify that the approach I used so far is correct. In computer programming it is believed that is almost impossible to find errors if you are called to test your own software and this is not different for ideas. So please help!
If you like, try the application I created to generate circular and rectangular maps … and to color them automatically. The software can be found here:
From command line, just use the java command:
- jar -jar ct-ui-swixml-VERSION-jar-with-dependencies.jar (no installation required)
Contact me for any help!
Four Color Theorem 