Square flexagons were the second variety of flexagon to be discovered. They are less well understood than hexaflexagons, partly because their dynamic behaviour is more complex. In particular relatively little is known about numbers of distinct types with a given number of faces. The leaves of a square flexagon are squares. In appearance a main position of a square flexagon is flat and consists of four leaves each with a vertex at the centre so there are four pats and two sectors. The outline is a square. Some square flexagons are twisted bands and hence exist as enantiomorphic (mirror image) pairs. Enantiomorphs are not usually regarded as distinct types. The handedness of a square flexagon is only mentioned when this is needed for clarity. Some square flexagons are untwisted bands so there is only one form.

Hexaflexagons are relatively simple because there are only one type of cycle and only one type of link between cycles. By contrast square flexagons have three different types of cycle and two types of link between cycles. There are three distinct types of single cycle square flexagon. Two of these have four faces and will traverse a complete 4-cycle. The third has three faces, and is incomplete in that it will only traverse an incomplete 3-cycle.

A ‘main position link’ between two cycles in a multicycle square flexagon is analogous to the type of link which occurs in multicycle hexaflexagons. Some statistics on numbers of distinct types of square flexagon with main position links are given. […]