TY - JOUR

T1 - The combinatorics of all regular flexagons

AU - Anderson, Thomas

AU - McLean, T. Bruce

AU - Pajoohesh, Homeira

AU - Smith, Chasen

PY - 2010/1

Y1 - 2010/1

N2 - Flexagons were discovered in 1939 by topologist Arthur Stone. A regular flexagon is one that contains 9 n 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 3 n because you can color 3 n of the faces with 3 n different colors. It is well known that when only the pinch flex is used, a flexagon of order 3 n is a möbius band with 3 (3 n - 2) half-twists, and has 6 n - 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 3 n 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 3 n an ordered set of 6 degrees that add to 9 n 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 9 n. You can only multiply by 3 n when there is complete symmetry. The Java applets provided only work for the traditional integer data type.

AB - Flexagons were discovered in 1939 by topologist Arthur Stone. A regular flexagon is one that contains 9 n 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 3 n because you can color 3 n of the faces with 3 n different colors. It is well known that when only the pinch flex is used, a flexagon of order 3 n is a möbius band with 3 (3 n - 2) half-twists, and has 6 n - 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 3 n 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 3 n an ordered set of 6 degrees that add to 9 n 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 9 n. You can only multiply by 3 n when there is complete symmetry. The Java applets provided only work for the traditional integer data type.

UR - http://www.scopus.com/inward/record.url?scp=70350101304&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2009.01.005

DO - 10.1016/j.ejc.2009.01.005

M3 - Article

SN - 0195-6698

VL - 31

SP - 72

EP - 80

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 1

ER -