In all the flexagons with which we have dealt thus far, the notion of ``class'', relating to the structural components of a flexagon, has been both clear and useful. We have been able, using the three general concepts of class, cycle, and order, to give a general idea of the shape, operation, and size, respectively, of any given flexagon. Several relationships have arisen among these three quantities: In complete flexagons, the cycle is found to be equal to the class, and the order is then of the form , where is an integer; the cycle is always less than the class and less than or equal to the order; and so forth.
However, in this section and those following, we will begin to come upon flexagons in regard to which the concept of ``class'' becomes vague and the rules regarding it no longer hold. These flexagons will reveal that the class, as a description of a flexagon, actually has far less significance than the cycle. As the distinctions between classes is lessened more and more, we will find that the very concept of ``class'' will change. Before proceeding with more radical changes in the flexagon structure, we can treat a number of variations that grow directly out of flexagons we have already discussed. Each of these deserves individual observation.