Here is a video that shows the nature of rectangular and circular maps.
All maps concerning the four color theorem (“regular maps” | “planar graphs without loops”) can be topologically transformed into rectangular and circular maps.
Wikipedia: “In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, every planar graph is four-colorable.”
But exactly which kind of planar graphs have to be considered?:
Here is the tranformation of a potential criminal map into a rectangular map. All regular maps can be topologically trasformed into rectangular or circular maps. See the theorem in T2.
Faces 5, 10, 15, 20, 25 are marked with a different color to simplify the reading of the map.
The computerized and colored version can be found here: Conversion of famous maps.
The string that represents this map and that can be used with the Java application to recreate this map is:
1b+, 2b+, 25b+, 27b+, 26b-, 24b-, 3b-, 23b-, 12b-, 22b-, 21b-, 20b-, 19b-, 15b-, 16b-, 18b-, 17b-, 11b-, 9b-, 5b-, 6b-, 8b-, 7b-, 2e-, 3e-, 4b-, 5e-, 6e-, 9e-, 10b-, 8e-, 12e-, 14b-, 13b-, 11e-, 15e-, 16e-, 14e-, 19e-, 25e-, 24e-, 23e-, 26e-, 22e-, 27e+, 21e+, 20e+, 18e+, 17e+, 13e+, 10e+, 7e+, 4e+, 1e+
Here is a picture of this map with transparency set to 15%:
I corrected a mistake on Tutte’s map conversion. This one is the fixed version. Face number 25 surrounding all the other faces on the original map, is the ocean of the rectangular and circular maps.
Same map converted into rectangular and circular (using different colors):
Here is a presentation I made with Prezi and a video with the application to generate maps (circular and rectangular) on Youtube.
Please, feel yourself at home and you are very welcome to take a look around, leave comments, ask questions, send ideas, corrections and so on.
Bye
And please, if my english gets too incomprehensible, tell me.
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
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.
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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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.
Four Color Theorem 