Map Coloring on a Torus

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Heawood proved that seven colors are sufficient to color a map on the surface of a torus so that no two bordering areas share the same color.


A torus can be constructed by stretching a square until the two pairs of opposite edges can be glued together. Each map used in this Demonstration is a square pattern whose top and bottom match and whose left and right edges match.


Contributed by: Erik Mahieu (July 2012)
Open content licensed under CC BY-NC-SA



The hex pattern used in this Demonstration was adapted from [1].

The tile pattern used is based on [2].

For an extensive study about map coloring on a torus, see [3].

For a different pattern used for seven-coloring a torus, see [4].


[1] Wikipedia. "Heawood Conjecture." (Jul 17, 2012)

[2] Wikipedia. "Torus with Seven Colors." (Jul 17, 2010) _seven _colours.svg.

[3] John Leech, "Seven Region Maps on a Torus," The Mathematical Gazette, 39(328), 1955 pp. 102–105.

[4] Izidor Hafner. "Seven-Coloring of a Torus" from the Wolfram Demonstrations Project—A Wolfram Web Resource.

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