Even taking the Flexagon Committee's work as a starting point, it doesn't seem that they ever wrote down much, and it is an interesting challenge to figure out exactly how much or how little they knew about flexagons. There are two main ways to regard flexagons - either as friezes or as spiral polygon stacks. The latter is more insightful, the former more historical. That is because flexagons originated from folding leftover strips of paper when someone was trimming notebook paper and got to playing with the debris. It is also true that rolls of adding machine tape or similar objects are a good source of material to play with.

However, using straight strips becomes a limitation when joining polygons into a strip seems to be a better vastage point than making polygons out of a predetermined strip. Also, there comes a moment when using them to make spirals in three dimensions rather than sticking to one or two seems to give a better perspective. The main thing about the spirals is that when they consist of two or more turns, edges where they can be folded become parallel rather than blocking each other, and the spiral can be laid out into two pieces alongside each other to make a plane figure. With triangles, three turns are required; the reason for this can be seen in the requirement for the sum of angles around a common center must sum to 360 degrees if the polygons are all to fit together in a nice rosette. So two of the basic rules of flexagons are:

- any polygon can be flexagonned
- the angles must add to 360 degrees.

To make the basic square flexagon, make a strip of five squares, so that the fifth could be laid over the first and pasted. That is one turn of the spiral, which is never enough. So take two such friezes, join the second to the first to get nine squares, then fanfold the frieze, open it out in the middle, and join the ninth to the first. Faces can now be painted, the flexagon flexed, more faces painted, and one has the basic tetraflexagon.

Besides squares, rhombi could be used, with angles of 60 - 120 degrees rather than just 90 degrees. Two turns will not be enough because of the 60 degree angle but three will work just as it does for the original triangle flexagon. Eventually it turns out that there are far more polygons that can be turned into flexagons than anyone would care to work with. Still, when first learning about flexagons, choosing a reasonable variety serves to illustrate the possibilities when one is still trying to formulate a theory and understand them. In that spirit, trying out some near, but not quite, divisors of 360 degrees helps to show why that choice is a good, if not necessary, one. Which is why building a flexagon out of pentagons is so educational.