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# Flexagons

In 1939 Arthur Stone moved from the UK to the US to study at Princeton University. Unfortunately, his new American paper was too big to fit into his British binder so he cut off the excess. This gave him strips of paper to play with during classes. At one point he folded it into a hexagon and tried to collapse it into three triangles when it seemed to open up from the inside. This was the first flexagon. Soon he found a design involving a coiled strip that looked similar but was more complex. He showed it to his friends Bryant Tuckerman and Richard Feynman. Tuckerman managed to find a quick way of finding all the faces. This method is now known as the Tuckerman traverse. Richard Feynman drew a diagram known as the Feynman diagram. (These have nothing to do with quantum mechanics which involve other diagrams called Feynman diagrams. Feynman drew a lot of diagrams). Those three and also John Tukey found more flexagons with different Feynman diagrams.

The first flexagon was shaped like a hexagon so it was called a hexaflexagon.It also had three faces so it was called a trihexaflexagon. The more complex variant was a hexahexaflexagon. To flex one, pinch two sides then push in the third.(This is easier said than done). However, not all flexagons are hexagonal. There are also square flexagons known as tetraflexagons. To flex these you need to fold it in half then pull it apart. Incidentally, I first got introduced to flexagons through the simplest tetraflexagon, known as the tritetraflexagon.

This simple flex (analogous in both versions) is known as the pinch flex. In complex flexagons, other flexes such as the V flex are possible. In a hexaflexagon, this mixes the sides. Also, other shapes exist such as pentaflexagons, heptaflexagons, octaflexagons, and more. Each type has its own unique methods of flexing and Feynman diagrams. Details for each flexagon will be given below.

Trihexaflexagon
Tetrahexaflexagon
Pentahexaflexagon
Hexahexaflexagon (Stone's version)
Tritetraflexagon
Tetratetraflexagon

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