Flexagon
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In geometry, flexagons are flat models made from folded strips of paper that can be folded, or flexed, to reveal a number of hidden faces. They are amusing toys but have also caught the interest of mathematicians.
Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A flexagon whose hexagonal faces are each divided into twelve right triangles as opposed to six equilateral triangles, and which can consequently flex into nonhexagonal shapes, has recently been christened a dodecaflexagon ([1]). 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 flexagon with a total of six faces is called a hexahexaflexagon.
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. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon. Tukey and Feynman developed a complete mathematical theory that has not been published.
Flexagons were introduced to the general public by the recreational mathematician Martin Gardner writing in Scientific American magazine. The columns have been reprinted in, among other books, Mathematical Puzzles and Diversions (1959; Pelican, UK ISBN 0140207139) and More Mathematical Puzzles and Diversions (1961; Pelican, UK ISBN 0140207481).
The 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: Image:Tritetraflexagon-net.PNG
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 Image:Tritetraflexagon-making.PNG
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 Image:Tritetraflexagon-flexing.PNG
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.
The hexaflexagon
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.
Image:Hexaflexagon-construction-and-use.jpg
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. Each color 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.
External links
Tetraflexagons:
- MathWorld's page on tetraflexagons, including three nets
Hexaflexagons: