In a systematic treatment flexagons can be classified into two main infinite families. The first is the convex polygon flexagon family and the second is the star flexagon family. A principal main position of a star flexagon is flat, and has the appearance of an even number of regular polygons arranged about its centre, each with a vertex at the centre. The first two varieties of star flexagons are square flexagons and hexaflexagons. Square flexagons are the first variety of the convex polygon flexagon family. The first two varieties are not typical of star flexagons. Typical star flexagons have at least eight polygons arranged about the centre of a principal main position, and the constituent polygons are regular star polygons. The dynamic behaviour of a typical star polygon flexagon is similar to that of the corresponding convex polygon flexagon. Interpenetration of the stellations during flexing doesn't matter in the ideal situation but does make the construction of paper models impossible.

Typical star flexagons are precursors to ring flexagons. If all the stellations are removed from the constituent polygons of a star polygon flexagon then it becomes a ring flexagon. Hence for any given star polygon flexagon there is always a corresponding ring flexagon with similar dynamic behaviour, and there is an infinite family of ring flexagons. A principal main position of a ring flexagon has the appearance of a flat ring of an even number of regular convex polygons. The rings are regular in that each polygon is the same distance from the centre of the ring. […]