Conversion of famous maps

These maps have been topologically tranformed into circular and rectangular maps. I will add other maps on this page from time to time.

The theorem in T2 proves that all regular maps (basically all maps of interest to the four color theorem) can be topologically tranformed into circular of rectangular maps.

Even if they seem to be so different in shape, they are actually the same map. To manually verify that they are the same map, just select any face and follow the border of that face clockwise (or counterclockwise). All neighbours appear in the same order and also all other topological attributes (related to the coloring problem) are exactly the same between the two representations of the same map.

Potential criminals (B)

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+

Same map converted into circular and rectangular:

Tutte’s map

1b+, 8b+, 9b+, 15b+, 16b+, 24b+, 22b-, 20b-, 21b-, 18b-, 19b-, 17b-, 11b-, 13b-, 15e-, 14b-, 12b-, 10b-, 9e-, 4b-, 7b-, 6b-, 3b-, 2b-, 4e-, 5b-, 3e-, 11e-, 10e-, 13e-, 12e-, 14e-, 16e-, 8e-, 7e-, 6e-, 5e-, 18e-, 20e-, 22e-, 23b-, 21e-, 19e-, 24e+, 23e+, 17e+, 2e+, 1e+

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 circular and rectangular (using different colors):

Map from Wikipedia

1b+, 2b+, 9b+, 8b-, 3b-, 5b-, 7b-, 4b-, 2e-, 3e-, 5e-, 6b-, 4e-, 8e-, 7e-, 9e+, 6e+, 1e+

Same map converted into circular and rectangular (using different colors):


3 Responses to Conversion of famous maps

  1. Sidey Timmins says:

    At the US census we have unusual requirements:

    1) we may want to color non-contiguous areas that are the same entity with the same color
    2) when a single new area is added to a previously colored map we want the old coloring to be preserved with just a new color for the new area. This could be done with one more color but then the map would look silly. So we need to keep at least 1 degree of freedom at each node.
    3) when we have a pie chart with more than 4 pie slices, we have to use 5/6/.. colors depending upon the number of pie slices at that node. ( We consider they touch.)

    Would conversion to circular or rectangular help for these problems?

    Do you have code to perform 4/5/6/ .. coloring and the circular/rectangular conversions which you can share?

    Thank you and I really enjoy your website and way of showing things!


  2. stefanutti says:

    Your is the first comment ever to the website. Thanks!

    I’m not sure to have understood the problem you are facing. Can you send a picture showing an example of the maps you need to color? A picture is worth a thousand words and my english is not good enough to understand all shadows of the english language.

    The algorithm I used is a brute force algorithm. Starting from the face number one, I try to color each face in sequence, verifying all neighbors. I pre-assigned all faces with a palette of four colors. Every time I assign a color to a face, I remove that color from the palette of that face. When all colors have been used for a face with no luck, I reset the palette and move back one face. You can see the application I have on youtube ( The source code and the application can be downloaded from Sourceforge (see the website “software” tab for info). It is a single jar that can be executed using the Java Virtual machine.

    I was already thinking to modify the algorithm I implemented to work after having forced some selected faces to have a pre-assigned and fixed color. I think this can be used to represent non-continuous areas, as in your case … if I have understood it right.

    Let my know, I’ll be happy to help … if I can … and if time permits!


  3. stefanutti says:

    Hi Sidey, I am just curious. Did you solve the problem you were facing?

    Bye, Mario


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