WOLFRAM|DEMONSTRATIONS PROJECT

A 2-pire Map

​
country 1
1
country 2
2
opacity
0.2
The four-color map theorem states that any map can be colored with four colors so that no two neighboring regions share a color. For a proof that four colors are necessary, consider a tetrahedron. If the faces are numbered 1 to 4, any two of these faces touch each other.
Some countries, such as the United States, consist of more than one region. On a world map, Alaska, Hawaii, and the 48 states should all have the same color. Define an
m
-pire as a country with
m
disconnected regions. How many colors are needed to color a world of
m
-pires? In 1890, Percy Heawood determined that
6m
colors suffice. For 2-pires, he also showed that 12 colors were necessary.
This Demonstration shows a 12-color 2-pire map. Any two selected 2-pires share a border.