Abstract

This work investigates a method of counting that accounts for natural side conditions, such as treating colourings that look the same after being rotated or mirrored as equal. This method relies on group theory, an area of mathematics that studies symmetries and patterns. Group actions describe how these symmetries affect colourings. They are useful, because they allow us to treat rotated colourings as equivalent. The most important theory for my project are Burnside’s lemma and the enumeration theorem of Pólya, which give formulas for counting the number of possibilities if certain configurations are treated as being equal.

In this project, the method was applied to examples to demonstrate its use. One example is the number of colourings of the faces of platonic bodies (dice with 4, 6, 8, 12 and 20 faces), given that rotation is allowed. Other examples include counting the number of different isomers of alcohol and counting the number of different graphs – drawings with points and lines connecting them. The goal of this project was to investigate these counting methods and to find different applications for them. It was found that this method is applicable when a relevant group exists and is usually much more efficient than brute force methods.