Chapter 1 is an introductory chapter. Nets and assembly instructions are given for a simple hexaflexagon, the trihexaflexagon, and for a simple square flexagon. The pinch flex used to manipulate them is described. Nets for other types of flexagon are given later in the book to illustrate various points made. General assembly instructions are given for these nets.

Flexagons are a twentieth century discovery. Their early history is given in Chapter 2. In 1940 two members of a Flexagon Committee at Princeton University worked out a mathematical theory of flexagons but this was never published. The subject can be said to have reached maturity with the issue in 1962 of a comprehensive report on flexagons, but it was not published in a form which reached a wide audience.

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.