The Sum43 Hexagon

The Sum43 Hexagon

The sum43 hexagon is a polyhedron made of nineteen triangles folded from a single strip.sum43 hexagon It can be decorated in six colors and it flexes to reveal different sides, as shown below. This is a variation of the classic hexaflexagon, which has three or six faces and was first patented in 1939 by Arthur Stone. Its invention has spawned many variations of the design, including hexahexaflexagons and octahexaflexagons, with any number of faces.

The hexaflexagon is a special case of the flexagon, a class of polyhedra that can be constructed from a straight strip of paper or other material.sum43 hexagon A flexagon can be twisted and flipped to show different faces, and the shapes of its vertices can also change. Some flexagons have no symmetry and can be described as flat, while others have cyclic or noncyclic symmetries.

A regular hexagon has a special property: any pair of adjacent sides shares one vertex, and the longest diagonals connecting diametrically opposite vertices are twice the length of one side.sum43 hexagon This enables the regular hexagon to be partitioned into six equilateral triangles, which can be stellated into a regular hexagonal tiling with two types (colors) of edges.

In addition, a regular hexagon is bicentric, meaning that it has both a circumscribed circle and an inscribed circle.sum43 hexagon The smallest normal magic hexagon, of order 3 displaystyle 2n - 1, can be constructed using a compass and straightedge, because all the vertices of the circle are equilateral. The earliest known proof of this fact is by Ernst von Haselberg in 1887.

The first hexaflexagon was patented in the United States by Arthur Stone, a British mathematician who worked at Princeton University in New Jersey. Stone began by cutting a long piece of American paper into small squares and then folding them into different shapes. Hexaflexagons are now widely distributed as toys and science-fair exhibits, and are also used in mathematical education.

There are other hexaflexagon designs with any number of faces, and some can be built from straight strips that produce hexaflexagons of the same size. Other hexaflexagons are based on dividing a regular n-gon into n isosceles triangles, which can be stellated or truncated into different shaped hexaflexagons.

The hexaflexagon has been used to explore a number of other mathematical ideas, such as the Hexagonal Number Theory and the Hard Hexagon model in particle physics. The hexaflexagon has also been studied in relation to the kinetics of liquid crystals and fractal geometry. The hexaflexagon is also important in the field of cryptography because it has properties that make it resistant to certain types of attacks. It is often used as a secure symbol, such as the letter 'E' in the graphical password encryption system RSA. This is because it cannot be copied by computer programs. It is therefore difficult to spoof, and it can be adapted to other cryptographic protocols as well. The hexaflexagon also has applications in computational biology and in the field of pharmacology, because it can be used to identify potential drug side effects.

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