In general the main characteristic feature of a flexagon is that it has the appearance of a polygon which may be flexed in order to display pairs of faces, around a cycle, in cyclic order. Another characteristic feature is that faces of individual polygons, known as leaves, which make up a face of a flexagon, rotate in the sense that different vertices move to the centre of a main position as a flexagon is flexed from one main position to another. The visible leaves are actually folded piles of leaves, called pats. Sometimes pats are single leaves. Alternate pats have the same structure. A pair of adjacent pats is a ‘sector’.

A convenient mathematical framework for the analysis of flexagons is presented in this chapter, together with explanations of special technical terms. A straightforward geometric approach, without equations, is used. Geometric descriptions are used for three main purposes: firstly, to map the dynamic behaviour of flexagons, secondly to analyse their structure, and thirdly as the basis of recipes for the construction of flexagons of any desired type. Recipes are described later in the book.

There is a fundamental difficulty in mapping the dynamic behaviour of flexagons. This is that in general their structure and dynamic behaviour are so complicated that there has to be a compromise between including relevant detail and keeping a map reasonably easy to follow. The terminology used in the analysis of the structure of flexagons is mostly based on that used by previous authors, but to keep the text concise it was found that some new terms are needed. […]