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:

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, and in the magic wallet trick .

External links

Tetraflexagons:

• MathWorld's page on tetraflexagons, including three nets

Hexaflexagons:

• MathWorld entry on Hexaflexagons
• How to make a Hexaflexagon
• How to make a hexa-hexa-flexagon by Magnus Enarsson