About general order regular flexagons

The three and six face constructions of flexagons first appeared in the literature in 1956 and 1957 by Gardner and Oakely - Wisner respectively, and have been of great interest to mathematicians ever since. A recent article by Anderson et. al. at the European Journal of Combinatorics conducted a systematic study of the mathematics involved in regular flexagons made from straight strips of equilateral triangles. In particular, the authors defined equivalent pats and faces mathematically and then used recursion to count the pats and faces. For the number of pat classes with a given degree, we first make an observation that arranges "sub-pat numbers" at leaves of a binary tree. This observation, together with the previously achieved recursion, provides an elementary approach to show the explicit formula for the number of equivalent pat classes. We point out that this formula can also be achieved using classic but technical combinatorial approaches as the binomial theorem and the Lagrange inversion formula. Based on the compositions of numbers, we obtain the explicit formula for the number of faces of a regular flexagon of any order. Then our focus shifts to achieving an even better understanding of how the faces/pats change under flexes. A novel labeling of the triangles in the construction of a Hexagon is proposed. As a result, a detailed state diagram is shown to explain the transition between one face to another. The theoretical results of this flavor lead to algorithmic procedures to construct a regular hexaflexagon and reproduce the orientation of the top triangles of a Hexagon. Consequently, a Java applet was produced for those who are interested in playing flexagons themselves.