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. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...

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] (http://www.eighthsquare.com/12-gon.html)). 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. Princeton University, located in Princeton, New Jersey, is the fourth-oldest institution of higher education in the United States. ... 1939 was a common year starting on Sunday (link will take you to calendar). ... A4 is a standard paper size, defined by the international standard ISO 216 as 210×297 mm (roughly 8. ... Bryant Tuckerman (November 28, 1915 _ May 19, 2002) was an American mathematician born in Lincoln, Nebraska. ... Richard Phillips Feynman (May 11, 1918–February 15, 1988) (surname pronounced FINE-man; in IPA) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ... John Wilder Tukey (June 16, 1915 - July 26, 2000) was a statistician. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...

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). Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ... Martin Gardner (born October 21, 1914) is an American recreational mathematician and author of the long-running but now discontinued Mathematical Games column in Scientific American. ... Scientific American is one of the oldest and most serious popular-science magazines. ... 1959 was a common year starting on Thursday (link will take you to calendar). ... 1961 (As MAD Magazine pointed out on its first cover for the year) was the first upside-down year—i. ...

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. A square as a geometric shape is described and illustrated at square (geometry). ...

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. A homemade Jacobs ladder with an electric arc. ...

External links

Tetraflexagons:

• MathWorld's page on tetraflexagons (http://mathworld.wolfram.com/Tetraflexagon.html), including three nets

Hexaflexagons: MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

• MathWorld entry on Hexaflexagons (http://mathworld.wolfram.com/Hexaflexagon.html)
• How to make a Hexaflexagon (http://home.xnet.com/~aak/hexahexa.html)
• How to make a hexa-hexa-flexagon (http://www.enarsson.nu/Flexagon/) by Magnus Enarsson