(1979). V-Flexing the Hexahexaflexagon. The American Mathematical Monthly: Vol. 86, No. 6, pp. 457-466.

Flexagons are hinged polygons that have the intriguing property of displaying different pairs of faces when they are flexed. Workable paper models of flexagons are easy to make and entertaining to manipulate. Flexagons have a surprisingly complex mathematical structure and just how a flexagon works is not obvious on casual examination of a paper model. Flexagons may be appreciated at three different levels. Firstly as toys or puzzles, secondly as a recreational mathematics topic and finally as the subject of serious mathematical study. This book is written for anyone interested in puzzles or recreational maths. No previous knowledge of flexagons is assumed, and the only pre-requisite is some knowledge of elementary geometry. An attractive feature of the book is a collection of nets, with assembly instructions, for a wide range of paper models of flexagons. These are printed full size and laid out so they can be photocopied.

V-Flexing the Hexahexaflexagon

Flexagons Inside Out

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 mobius 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. The Java applets provided only work for the traditional integer data type.

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The combinatorics of all regular flexagons

This paper gives a well designed math class lesson plan covering five class periods in which creation skill and higher order thinking are set to be the main objectives. Based on the theory of Bloom taxonomic and Polya's how to solve it, the activities designed in the lesson plan focus on the exploring unknown structure in a certain origami and problem solving issue. The thinking level of the students is supposed to reach the highest creating level by this curriculum design. We come to the conclusion that origami as well as good questions motivate students' higher level thinking. As students' scaffolds, video and animation technology are essential in helping the learners understanding origami in right perspectives.

Computer enhanced instruction: a case study of a series of creative math activities design

Regular hexaflexagons mysteriously change faces as you pinch flex them. This paper describes a different flex, the V-flex, which allows the hexahexaflexagon (with only 9 faces under the pinch flex)...

https://www.jstor.org/stable/pdfplus/10.4169/college.math.j.43.1.006.pdf

The V-Flex, Triangle Orientation and Catalan Numbers in Hexaflexagons

This paper reports an experience of the author teaching high school 3D geometry with origami. In this case, students were originally expected to construct a 3D-hexaflexagon by imitating video tape so that they can understand some related math concepts better. However students demonstrated strong willing of exploring the concerned enchanting paper craft. They did not want to just memorize the folding procedures or be limited within verifying some math concepts. The teacher had then to extend the class time to 2 class periods, so that a series of creative activities are invited in. After the primary goals have been reached, two extra activities were launched. First, by exploring a torus forming animation, they explored a different algorithm of forming a hexaflexagon. Secondly, when another animation of Hexaflexagon made by the teacher was presented, the students can then explore the function of the slope of the creases. Some smart students found hexaflexagons with different shape or even different type. So the thinking level of the students reached the highest creating level by curriculum re-design. This case study comes to the conclusions that origami as well as good questions motivate students’ higher level thinking. As students’ scaffolds, video and animation technology are essential in helping the learners understanding origami in right perspectives.

V-Flexing the Hexahexaflexagon

Citations

Flexagons Inside Out

The combinatorics of all regular flexagons

Computer enhanced instruction: a case study of a series of creative math activities design

The V-Flex, Triangle Orientation and Catalan Numbers in Hexaflexagons

Origami in Education Enhanced by Computer Technology: A Case Study of Teaching Hexaflexagon in Math Class

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