were discovered in 1939 by Arthur
Stone, then a graduate student at Princeton University who later became
a lecturer in mathematics at the University of Manchester. An Englishman,
Stone was trimming American notebook sheets to fit in his English binder
when it occurred to him to fold the strips of paper to make figures, one
of which was the trihexaflexagon (so called because it has six sides and
three faces). He elaborated on this model to create the hexahexaflexagon
(six sides, six faces).
Stone showed his models to his friends: Bryant Tuckerman, a graduate student in mathematics who became a mathematician at IBMs research center; Richard Feynman, a graduate student in physics who went on to win the Nobel Prize in 1965; and John W. Tukey, a math student and later a professor of mathematics at Princeton who made significant contributions to topology and statistical theory. They formed the Flexagon Committee, and were later joined by Tuckermans father, the physicist Louis B. Tuckerman.
The Committee devised ways to make flexagons with 9, 12, 15 or 48 faces, as well as square flexagons. Tukey and Feynman worked out a complete mathematical theory of hexaflexagons in 1940; however, it was never published. Feynman also did diagrams of flexagons which were precursors of his Feynman Diagrams describing the behavior of subatomic particles.
Flexagons began catching on with the general public in 1956, when author and Renaissance man Martin Gardner wrote an article on them that was published in Scientific American magazine (the article became the first of Gardners regular columns on recreational mathematics).
Flexagon fans have devised variations, created new flexagons with various shapes and properties, and even contemplated flexahedra, which are four-dimensional analogues of flexagons.