Application of the four-colour theorem to the surfaces of polyhedra

The four-colour map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the map so that no two adjacent regions have the same colour. Two regions are considered to be adjacent if they share a common boundary that is not a corner (a point shared by three or more regions). The theorem was proposed in the 1850s and became the first theorem to be proved by computational methods in the 1970s. Despite the theorem being true, some geopolitical maps require more than four colours (if, for example, some regions are not contiguous) and the theorem has never been of great interest to mapmakers. This paper describes the theorem and explores how it could be extended to three dimensions. We restrict our study to the colouring of the surfaces of three-dimensional polytopes or polyhedra, specifically those that are convex. An analysis of the relationship between two-dimensional maps and three-dimensional surfaces is presented with regard to the minimum number of colours required. Visual examples are provided for regular polyhedral of increasing number of polygonal faces.