Flexagon

Hexaflexágono showing its two main faces

A flexagon is a flat object in the shape of a polygon (square, rectangle or hexagon) created by folding a sheet of paper (or other sufficiently flexible and thin material), whose main characteristic is It resides in that, through its correct bending, it allows showing more faces than the only two that initially have a flat polygon. This has meant that flexagons have become a fun hobby since their creation, although they have also been studied in the field of geometry. Specifically, the study of its properties is carried out by topology, a branch of mathematics that is in charge of studying the properties of surfaces.

Discovered by Arthur Stone in 1939, flexagons belong to the group of geometric bodies called kaleidocycles and their name comes from the words flexible and hexagon, since the first of They had six sides, although later models with four sides, square or rectangular, have been created. In addition, Harold V. McIntosh describes two types of non-planar flexagons formed from pentagons and heptagons, which he calls, respectively, pentaflexagons and heptaflexagons .

In hexaflexagon theory (i.e., relative to flexagons with six sides), flexagons are generally defined in terms of portions.

Two flexagons are equivalent if one can be transformed into the other by a series of pinches and rotations. The equivalence between flexagons is an equivalence relation.

History

Flexagons were discovered in 1939, when he was 23 years old, by Arthur Stone, an English student studying mathematics on scholarship at Princeton University (United States). The discovery was accidental: one day, after cutting some sheets of paper to fit his folder brought from England, Stone absently began to fold the excess strips in various ways, thus achieving an interesting flat figure in the shape of a regular hexagon that had a curious feature, since by joining three of its alternate corners, it could be opened again, similar to a flower, but showing a new face that was not visible before and hiding one that was.

This first flexagon, which in current nomenclature is called a trihexaflexagon, had only three faces, two visible and one hidden, but Stone himself managed to build a new type of flexagon the next day, this time with six faces, two visible and four hidden, that is, a hexahexaflexagon. Fully convinced that he had something interesting on his hands, he showed them to some of his classmates and friends and in a short time, those already dubbed "flexagons", became the most popular pastime among them, creating in company from his fellow and later famed scientists, the mathematician Bryant Tuckerman, the physicist Richard P. Feynman, and the statistician John W. Tukey, the Princeton Flexagon Committee ("Princeton Flexagon Committee" 3. 4;). The first of them would develop a topological method, called "Tuckerman traverse", to discover all the faces of a flexagon.

The flexagons were made known to the general public since the mid-50s thanks to the work of the mathematician and science popularizer Martin Gardner, with the publication of his articles on mathematical puzzles in the magazine Scientific American. The first of his columns, published in 1956, was called Flexagons. Later he would publish the book Hexaflexagons and other mathematical hobbies .

Nomenclature

The word flexagon comes from the terms flexible and hexagon, since the first models created by Stone had six sides. Currently, two prefixes followed by the ending -flexagon are used to name them; the three terms are sometimes separated by a hyphen. The first prefix indicates the number of faces, while the second depends on the authors: for some it indicates the number of sides of the flexagon while for others it indicates the number of polygons that each face forms. For example, the tetraoctaflexagon does not have eight sides, but only four, but it is called that because it is made up of eight triangles. But for all the authors the original flexagon created by Stone is a tri-hexa-flexagon (three faces, six sides), while the second model is a hexa-hexa-flexagon (six faces, six sides).

The four-sided models, since they are not shaped like hexagons, should not be called, strictly speaking, nor "tetraflexagons" nor "square flexagons", but, for example, "square kaleidocycles", although in practice those terms are used.

Types

Tetraflexagons

Tetraflexagons are four-sided flexagons, with four or six squares, or rectangles, on each face. They flex, closing and opening them on the opposite side as if they were a book. Examples:

• Tritetraflexágono. Three faces.
• Tetratetraflexágono. Four faces.
• Hexatetraflexágono. Six faces.

Octaflexagons

Octaflexagons are a particular type of tetraflexagons. Each one of the four squares that form each face is in turn divided into two isosceles right triangles; therefore, since there are eight polygons around the center of each face, the octo- prefix is used. This division into triangles means that, despite being square, their bending is like that of hexaflexagons, through the union of alternate points, and can be of three different types. This causes figures with different shapes to appear during handling. Examples:

• Tetraoctaflexágono. Four faces.
• Octaoctaflexágono. Eight faces.

Hexaflexagons

Hexaflexagons are flexagons with six sides. It is the family of flexagons with the most types:

• Trihexaflexágono. Three faces.
• Tetrahexaflexágono. Four faces.
• Pentahexaflexágono. Five faces.
• Hexahexaflexágono. Six faces. There are three different models.
• Heptahexaflexágono. Seven faces. There are four different models.
• Octahexaflexágono. Twelve faces. There are twelve different models.
• Eneahexaflexágono. Nine faces. There are 27 different models.
• Decahexaflexágono. Ten faces. There are 82 different models.
• Dodecahexaflexágono. Twelve faces. It is a special type of hexahexaflexágono created using a double paper tape. Following this method, Tuckerman came to make a 48-sided model.

Dodecaflexagon

The dodecaflexagon, described by Ann Schwartz, is a special type of hexaflexagon, just as the octaflexagon is a special type of tetraflexagon. In this case, each of the equilateral triangles that form it is subdivided in turn into two right triangles, which produces faces in which triangles with several different faces can be combined and also non-hexagonal figures are formed during handling.