Flexagon

[1] The discovery of the first flexagon, a trihexaflexagon, is credited to the British mathematician Arthur H. Stone, while a student at Princeton University in the United States in 1939.

Stone's colleagues Bryant Tuckerman, Richard Feynman, and John Tukey became interested in the idea and formed the Princeton Flexagon Committee.

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 cyclic hexatetraflexagon does not have any "dead ends", but the person making it can keep folding it until they reach the starting position.

Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure.

(Note that the word hexaflexagons [with no prefixes] can sometimes refer to an ordinary hexahexaflexagon, with six sides instead of other numbers.)

To assemble, the strip is folded every third triangle, connecting back to itself after three inversions in the manner of the international recycling symbol.

(The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)

One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.

[9] In its flat state, the pentaflexagon looks much like the Chrysler logo: a regular pentagon divided from the center into five isosceles triangles, with angles 72–54–54.

[11] The pentaflexagon is one of an infinite sequence of flexagons based on dividing a regular n-gon into n isosceles triangles.

[17] A high-order hexaflexagon was used as a plot element in Piers Anthony's novel 0X, in which a flex was analogous to the travel between alternate universes.

[18] Vi Hart, a well-known recreational mathematician and public educator, gained attention for her video on hexaflexagons.

A hexaflexagon, shown with the same face in two configurations
A hexaflexagon, shown with the same face in two configurations
Diagram for folding a tritetraflexagon
A tritetraflexagon can be folded from a strip of paper as shown.
Sides of a tritetraflexagon
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.
Tritetraflexagon traverse
Hexatetraflexagon traverse
This trihexaflexagon template shows 3 colors of 9 triangles, printed on one side, and folded to be colored on both sides. The two yellow triangles on the ends will end up taped together. The red and blue arcs are seen as full circles on the inside of one side or the other when folded.
A strip of paper, divided into triangles, which can be folded into a hexaflexagon.
A series of photos detailing construction and "flexing" of a hexaflexagon
Figures 1-6 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 colours; 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. 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.