flexagon

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In geometry, flexagons are flat models usually constructed by folding strips of paper that can flexed or folded in a certain way, 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.

History

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 A4 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.^[1]

Flexagons were introduced to the general public by the recreational mathematician Martin Gardner writing in Scientific American magazine.^[2]

Formal definition

This section requires expansion.

In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats.^[3][4]

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.^[3]

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.

There is also a method of creating a more complicated hexatetraflexagon. To make it, take a piece of square paper and cut a square hole in the middle. Make sure all edges are straight. Then from the left hand edge, make a valley fold towards the middle. From the top, make another valley fold towards the middle. Now make a valley fold from the right hand edge towards the middle. Finally, make a valley fold from the bottom towards the middle. You now have the hexatetraflexagon. Please note that you do not need any tape or paste to make this flexagon. The most exciting thing to do with this one, is to colour both faces, then keep on flexing and colour the faces as you find them, until you get back to your starting position.

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 that exact 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 is 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.

Cyclic hexa-tetraflexagon

• Here you can see how to make a cyclic hexa-tetraflexagon from just one piece of paper.

Octaflexagon and dodecaflexagon

This section requires expansion.

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, and the division of the triangular faces of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons.

Pentaflexagon and heptaflexagon

Harold V. McIntosh also describes nonplanar flexagons folded from pentagons called pentaflexagons [2] and heptagons called heptaflexagons [3].

Bibliography

• 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 [4]
• 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)
  • 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

References

1. ^ Gardner, Martin (1988). Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games. University of Chicago Press. ISBN 0226282546. 
2. ^ Jürgen Köller. "Flexagons". http://www.mathematische-basteleien.de/flexagons.htm. Retrieved 23 September 2009. 
3. ^ ^{a b} Oakley, C. O.; Wisner, R. J. (March 1957), "Flexagons", The American Mathematical Monthly (Mathematical Association of America) 64 (3): 143–154, http://www.jstor.org/stable/2310544 
4. ^ Anderson, Thomas; McLean, T. Bruce; Pajoohesh, Homeira; Smith, Chasen (January 2010), "The combinatorics of all regular flexagons", European Journal of Combinatorics 31 (1): 72–80, ISSN 0195-6698, http://www.sciencedirect.com/science/article/B6WDY-4W5VD4F-1/2/e1d94639a2f71f509b049f8ab6480cb7 

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 [5] and even boasts some flexagon videos in [6].
• Flexagons — David King's site is one of the more extensive resources on the subject; it includes a Java 3D Simulation of a hexahexaflexagon.
• 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.

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