Hexaflexagons were the first variety of flexagon to be discovered and they have been analysed in the most detail. The leaves of a hexaflexagon are equilateral triangles. In appearance a main position of a hexaflexagon is flat and consists of six leaves, each with a vertex at the centre so there are six pats and three sectors. The outline is a regular hexagon. All hexaflexagons are twisted bands and hence exist as enantiomorphic (mirror image) pairs. Enantiomorphs are not usually regarded as distinct types. The handedness of a hexaflexagon is only mentioned when this is needed for clarity.

In some ways hexaflexagons are the simplest type of flexagon. There are only one possible type of cycle and one possible type of link between cycles and normally only the pinch flex is used. On the other hand hexaflexagons do have a large number of degrees of freedom. In consequence with most types of hexaflexagons spurious transformations may occur during flexing, and it is possible to get some types badly tangled.

The trihexaflexagon has three faces and only one cycle. Since there are three main positions this is called a ‘3-cycle’. Its decorative possibilities are illustrated by a transformation from a happy face to a sad face. Multicycle hexaflexagons have been extensively analysed and their dynamic behaviour is well understood. A ‘Tuckerman traverse’ is a systematic method of traversing a multicycle hexaflexagon which guarantees that all possible main positions are visited. Numerous nets have been published, and some examples are given. […]