Extensive analysis has resulted in variations on the theme of hexaflexagons. Three are described in this chapter. These are: a different variety of flexagon, triangle flexagons; a different way of flexing hexaflexagons, the ‘V-flex’; and origami like recreations with hexaflexagons.

A triangle flexagon is a variant of a hexaflexagon in which the number of sectors is reduced from three to two, correspondingly the number of leaves visible on a face of a main position is reduced from six to four. The sum of the leaf vertex angles at the centres of main positions is less than 360° and this results in slant main positions. These have the appearance of a square pyramid without a base. The dynamic behaviour of a triangle flexagon is similar to that of the corresponding hexaflexagon except that it isn't possible to traverse a complete cycle without disconnecting a hinge, refolding the flexagon and reconnecting the hinge.

The V-flex is a complicated flex which exploits the large number of degrees of freedom of all but the simplest types of hexaflexagon. Using the V-flex results in the faces of a hexaflexagon becoming mixed up. How to operate the V-flex is described in detail. By using both the V-flex and the pinch flex the hexahexaflexagon can be flexed to display numerous different main position faces. Individual leaf faces have to be identified in order to characterise these main position faces. Whether or not the V-flex is regarded as legitimate is a matter of personal taste. In this book it is not regarded as legitimate. […]