The Combinatorics of All Regular Flexagons

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European Journal of Combinatorics




Flexagons were discovered in 1939 by topologist Arthur Stone. A regular flexagon is one that contains 9n equilateral triangular regions on a straight strip of paper. This paper is then rolled into smaller strips of paper and finally into a hexagon with 6 triangular regions called pats, producing one mathematical face. The pinch flex removes the uppermost triangular regions and replaces them with a new set producing a new face. The flexagon is said to have order 3n because you can color 3n of the faces with 3n different colors. It is well known that when only the pinch flex is used, a flexagon of order 3n is a möbius band with 3(3n−2) half-twists, and has 6n−3 different mathematical faces. Even though a colored face appears more than once, the uppermost triangles might be rotated producing a different mathematical face. When T. Bruce McLean described the V-flex on the flexagon of order 6 in 1979, he showed that it now had 3420 mathematical faces and provided a graph that demonstrated how to reach all of the different faces. This flex scrambles the colors similar to the way the Rubik’s cube does except that a flexagon is flat. It is the purpose of this paper to provide an algorithm that counts the number of mathematical faces for flexagons of order 3n for all n, once the V-flex is included.

A theorem in this paper gives a recursive formula that counts the number of different pats of a given thickness (or degree). To start the count for the number of mathematical faces of a regular flexagon of order 3n an ordered set of 6 degrees that add to 9n is considered. The adjacent degrees must add to a multiple of three according to the axioms of a flexagon using both flexes. Two sets are equivalent if you can rotate one six-tuplet into the other. Then for each case, the number of pats given by this Theorem that have degrees of those 6 numbers can be multiplied by the fundamental theorem of counting and these are called initial faces after rotations are removed. The last step in the count is to allow for translations of the flexagon and when there is no symmetry, you can multiply the number of initial faces by 9n. You can only multiply by 3n when there is complete symmetry.