A formal textbook on flexagons, after some preliminaries, would probably start with a chapter with this somewhat daunting title, or something similar. In this book the chapter has been deferred to this point because convex polygon flexagons are nothing more than generalisations of the square flexagons and triangle flexagons described earlier. Understanding of convex polygon flexagons in general is incomplete.

There is an infinite family of convex polygon flexagons. Varieties are named after the constituent polygons. A feature of some varieties of convex polygon flexagon is that there may be more than one type of main position and more than one type of complete cycle. It then becomes necessary to refer to principal and subsidiary main positions and cycles. The distinction is only made when needed. In a principal main position a convex polygon flexagon has the appearance of four leaves each with a vertex at the centre so there are four pats and two sectors. Some are twisted bands and so exist as enantiomorphic (mirror image) pairs.

If a flexagon is regarded as a linkage then bending the leaves during flexing is not permissible. However, allowing bending during flexing does makes it easier to rationalise dynamic behaviours of the convex polygon flexagon family, and also makes the manipulation of some types of convex polygon flexagon more interesting.

The first variety of the convex polygon flexagon family, the digon flexagon, can only be flexed with the leaves truncated and then only by bending the leaves of a paper model: the ‘push through’ flex. […]