Flexagon

In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front.

Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon.

In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats.

Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation.

History

Discovery and introduction

The discovery of the first flexagon, a trihexaflexagon, is credited to the British student Arthur H. Stone who was studying at Princeton University in the USA in 1939, allegedly while he was playing with the strips he had cut off his foolscap paper to convert it to letter size. Stone's colleagues Bryant Tuckerman, Richard P. Feynman and John W. Tukey became interested in the idea and formed the Princeton Flexagon Committee. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon.

Flexagons were introduced to the general public by the recreational mathematician Martin Gardner, writing in 1956 in his inaugural "Mathematical Games" column for Scientific American magazine.

Attempted commercial development

In 1955, Russell Rogers and Leonard D'Andrea of Homestead Park, Pennsylvania applied for, and in 1959 were granted, U.S. Patent #2,883,195 for the hexahexaflexagon, under the title "Changeable Amusement Devices and the Like." The patent imagined possible applications of the device "as a toy, as an advertising display device, or as an educational geometric device." A few such novelties were produced by the Herbick & Held Printing Company, the Pittsburgh printing firm where Rogers worked. But the device, marketed as the "Hexmo", failed to catch on commercially.

Varieties

Tetraflexagons

Tritetraflexagon

The tritetraflexagon is the simplest tetraflexagon (flexagon with square sides). The "tri" in the name means it has three faces, two of which are visible at any given time if the flexagon is pressed flat.

It is folded from a strip of six squares of paper like this:

To fold this shape into a tritetraflexagon, first crease each line between two squares. Then fold the mountain fold away from you and the valley fold towards you, and add a small piece of tape like this

This figure has two faces visible, built of squares marked with "A"s and "B"s. The face of "C"s is hidden inside the flexagon. To reveal it, fold the flexagon flat and then unfold it, like this

The construction of the tritetraflexagon is similar to the mechanism used in the traditional Jacob's Ladder children's toy, in Rubik's Magic and in the magic wallet trick or the Himber wallet.

Cyclic hexa-tetraflexagon

A more complicated cyclic hexatetraflexagon doesn't require any gluing. To make it, take a square piece of paper and fold it twice vertically and horizontally to measure it into a 4 by 4 grid. Cut a square hole in the middle by removing 2 by 2 cell area. Make sure all edges are straight. Fold the sides to the middle.

Cyclic hexatetraflexagon doesn't have "dead ends" but you can keep folding it until you reach the starting position. If your colour the sides as you go, you can see the states more clearly.

Hexaflexagons

Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure.

Trihexaflexagon

A hexaflexagon with three faces.

While this is the simplest of the hexaflexagons to make and to manage, it is a very satisfying place to begin. It is made from a single strip of paper, divided into ten equilateral triangles. Patterns are available at The Flexagon Portal.

It is possible to automatically section and correctly place photographs (or drawings) of your own selection onto Trihexaflexagons using the simple program Foto-TriHexaFlexagon.

Hexahexaflexagon

This hexaflexagon has six faces.

Make a mountain fold between the first 2 and the first 3. Continue folding in a spiral fashion, for a total of nine folds. You now have a straight strip with ten triangles on each side. There are two places where 3's are next to each other; fold in both these places so as to hide the 3's, forming a hexagon with a triangular tab sticking out. Lift one end of the hexagon around the other so that the 3's near the ends are touching each other. Fold the tab over to cover the blank triangle on the other side, and glue it to the blank triangle. One side of the hexagon should be all 1's, one side should be all 2's, and all the 3's should hidden.

Photos 1-6 below show the construction of a hexaflexagon made out of cardboard triangles on a backing made from a strip of cloth. It has been decorated in six colors; orange, blue, and red in figure 1 correspond to 1, 2, and 3 in the diagram above. The opposite side, figure 2, is decorated with purple, gray, and yellow. Note the different patterns used for the colors on the two sides. Figure 3 shows the first fold, and figure 4 the result of the first nine folds, which form a spiral. Figures 5-6 show the final folding of the spiral to make a hexagon; in 5, two red faces have been hidden by a valley fold, and in 6, two red faces on the bottom side have been hidden by a mountain fold. After figure 6, the final loose triangle is folded over and attached to the other end of the original strip so that one side is all blue, and the other all orange.

Photos 7 and 8 show the process of everting the hexaflexagon to show the formerly hidden red triangles. By further manipulations, all six colors can be exposed. Faces 1, 2, and 3 are easier to find while faces 4, 5, and 6 are more difficult to find. An easy way to expose all six faces is using the Tuckerman traverse. It's named after Bryant Tuckerman, one of the first to investigate the properties of hexaflexagons. The Tuckerman traverse involves the repeated flexing by pinching one corner and flex from exactly the same corner every time. If the corner refuses to open, move to an adjacent corner and keep flexing. This procedure brings you to a 12-face cycle. During this procedure, however, 1, 2, and 3 show up three times as frequently as 4, 5, and 6. The cycle proceeds as follows:

1-3-6-1-3-2-4-3-2-1-5-2

And then back to 1 again.

Each color/face can also be exposed in more than one way. In figure 6, for example, each blue triangle has at the center its corner decorated with a wedge, but it is also possible, for example, to make the ones decorated with Y's come to the center. There are 18 such possible configurations for triangles with different colors, and they can be seen by flexing the hexahexaflexagon in all possible ways in theory, but only 15 can be flexed by the ordinary hexahexaflexagon. The 3 extra configurations are impossible due to the arrangement of the 4, 5, and 6 tiles at the back flap. (The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)

The one shown is not the only hexahexaflexagon. Others can be constructed from different shaped nets of eighteen equilateral triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.

Other hexaflexagons

While the most commonly seen hexaflexagons have either three or six faces, variations exist with four, five, and seven faces. Nets for these can be found at the flexagon.net website.

Higher order flexagons

Right octaflexagon and right dodecaflexagon

In these more recently discovered flexagons, each square or equilateral triangular face of a conventional flexagon is further divided into two right triangles, permitting additional flexing modes ([1]). The division of the square faces of tetraflexagons into right isosceles triangles yields the octaflexagons ([2]), and the division of the triangular faces of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons ([3]).

Pentaflexagon and right decaflexagon

In its flat state, the pentaflexagon looks much like the Chrysler logo: a regular pentagon divided from the center into five isosceles triangles, with angles 72-54-54. Because of its fivefold symmetry, the pentaflexagon cannot be folded in half. However, a complex series of flexes results in its transformation from displaying sides 1 and 2 on the front and back, to displaying its previously hidden sides 3 and 4. [4]

By further dividing the 72-54-54 triangles of the pentaflexagon into 36-54-90 right triangles produces one variation of the 10-sided decaflexagon ([5]).

Generalized isosceles n-flexagon

The pentaflexagon (described above) is one of an infinite sequence of flexagons based on dividing a regular n-gon into n isosceles triangles. There is the heptaflexagon ([6]), the isosceles octaflexagon ([7]), enneaflexagon ([8]), and on and on...

Nonplanar pentaflexagon and nonplanar heptaflexagon

Harold V. McIntosh also describes "nonplanar" (i.e., they can't be flexed so they lie flat) flexagons; ones folded from pentagons called pentaflexagons [9], and from heptagons called heptaflexagons [10]. These should be distinguished from the "ordinary" penta- and heptaflexagons described above, which are made out of isosceles triangles, and can be made to lie flat.

Bibliography

• Mitchell, David (2000). The Magic of Flexagons - Folding curiosities to cut out and make. Tarquin. ISBN 1899618287. 
• Pook, Les (2009). Serious Fun with Flexagons, A Compendium and Guide. Springer. ISBN 9048125022. 
• Pook, Les, Flexagons Inside Out, Cambridge University Press (2006), ISBN 0-521-81970-9 [11]
• Martin Gardner has written an excellent introduction to hexaflexagons in one of his Mathematical Games column in Scientific American. It also appears in:
  • The "Scientific American" Book of Mathematical Puzzles and Diversions (Simon & Schuster, 1959).
  • Hexaflexagons and Other Mathematical Diversions: The First "Scientific American" Book of Puzzles and Games (University of Chicago Press, 1988; ISBN 0226282546)
  • The Colossal Book of Mathematics (W.W. Norton & co., 2001; ISBN 0393020231)
  • Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games (Cambridge University Press, 2008; ISBN 0521735254)

See also

• Geometric group theory
• Cayley tree
• Octahedron: two identically formed nonplanar flexagons: one octahedron

External links

Flexagons:

• My Flexagon Experiences by Harold V. McIntosh — contains valuable historical information and theory; the author's site has several flexagon related papers listed in [12] and even boasts some flexagon videos in [13].
• The Flexagon Portal — Robin Moseley's site has patterns for a large variety of flexagons.
• Flexagons is a good introduction, including a large number of links.
• Flexagons — Scott Sherman's site, with a bewildering array of flexagons of different shapes.

Tetraflexagons:

• MathWorld's page on tetraflexagons, including three nets
• Folding User Interfaces - A mobile phone design concept based on a tetraflexagon; Folding the design gives access to different user interfaces.
• Flexifier - a simple online tetraflexagon generator
• Instructions for making cyclic hexa-tetraflexagon from just one piece of paper.

Hexaflexagons:

• Flexagons — 1962 paper by Antony S. Conrad and Daniel K. Hartline (RIAS)
• MathWorld entry on Hexaflexagons
• Hexaflexagon Toolkit software for printing flexagons from your own pictures
• Hexaflexagons — a catalog compiled by Antonio Carlos M. de Queiroz (c.1973).
  Includes a program named HexaFind that finds all the possible Tuckerman traverses for given orders of hexaflexagons.
• Crochet hexaflexagon cushion