This article will demonstrate that any random configuration of lines on a two dimensional surface, whether curved or not, can be coloured using 4 colours, without the occurrence of two sections of the surface with the same colour touching.
An actual configuration of countries can easily be displayed on a sphere shell, which is the real form of countries. Only, they are easier to work with if they are displayed on a circle. The projection works as follows: One takes a sphere shell as if it was made out of rubber, and one pokes a hole in it somewhere. One stretches the hole until its circumference is greater than that of the sphere. Then one flattens the sphere in order to make it lie in between the hole, and creates a circle this way. For clarity (easier visualisation) I shall give an illustration.