While writing up earlier chapters it became clear that a fairly systematic treatment of at least some types of flexagon made from distorted polygons was possible. This rather advanced chapter is the result. In order to illustrate some of the enormous range of possibilities nets for several types of distorted polygon flexagon are given, together with descriptions of their dynamic behaviour.

A ‘distorted polygon’ is a convex polygon derived from a regular convex polygon by changing the shape without changing the number of sides. The leaves of flexagons can be made from any convex polygon, but only a limited range of distorted convex polygons result in flexagons whose paper models are reasonably easy to handle. In practice paper models are most satisfactory when the distorted polygons are semiregular. Distorted polygon flexagons are usually named after the polygons from which they are made.

There are several ways in which the leaves of a flexagon made from regular polygons can be modified to produce distorted polygon flexagons. A distorted polygon can sometimes be regarded as a partially stellated version of a regular convex polygon with a different number of sides. Alternatively, a distorted polygon can sometimes be regarded as a star polygon from which some of the stellations have been removed. If the proportions of the leaves are changed without changing the angular relationships between their sides then, in general, the dynamic behaviour of the flexagon is not affected. Changing the angular relationships between leaf sides does change the dynamic behaviour. Most distorted polygon flexagons are best regarded as variants of either convex polygon flexagons or star flexagons.