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In geometry, flexagons are flat models usually constructed by folding strips of paper that can flexed or folded in a certain way, to reveal faces besides the two that were originally on the back and front.
Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons), but can be a wide variety of shapes. Two prefixes are usually added to denote a specific flexagon: the first prefix is the number of faces it has (including the front and back), and the second is the number of sides each face has. If only one prefix is used, it refers to a class of flexagons with the same number of sides each face has, denoted by that prefix. For example, a trihexaflexagon is a flexagon with three faces, each with six sides.
Formal Definition edit
In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats. The definition that follows is mathematical abstraction of a flexagon.
Pats edit
Pats are defined recursively. A pat of degree 1 is the permutation of a single element, corresponding to a single triangle in a real flexagon. A pat of degree m where m is a natural number greater than 1 is ...
Pinch edit
A pinch is a...
Rotation edit
A rotation is a...
Flexagons edit
Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation. [1]
History edit
The discovery of the first flexagon, a trihexaflexagon, is credited to the British student Arthur H. Stone who was studying at Princeton University in the USA in 1939, allegedly while he was playing with the strips he had cut off his A4 paper to convert it to letter size. Stone's colleagues Bryant Tuckerman, Richard P. Feynman and John W. Tukey became interested in the idea and formed the Princeton Flexagon Committee. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon.[3]
Flexagons were introduced to the general public by the recreational mathematician Martin Gardner writing in Scientific American magazine.[4]
Flexagon Mechanics edit
Steps 6-8 in the picture demonstrate a "flex" of a hexahexaflexagon.
Types of Flexagons edit
Tetraflexagon edit
A tetraflexagon has four sides on each face, and thus generally assumes a square or rectangular shape.
Hexaflexagon edit
A hexaflexagon has six sides on each face, causing it to form a hexagonal shape. Hexaflexagons are the most studied flexagons.
Higher Orders edit
References edit
- ^ a b Oakley, C. O.; Wisner, R. J., Flexagons, pp. 143–154
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- ^ Jürgen Köller. "Flexagons". Retrieved 23 September 2009.
See also edit
External links edit
Flexagons:
- My Flexagon Experiences by Harold V. McIntosh — contains valuable historical information and theory; the author's site has several flexagon related papers listed in [1] and even boasts some flexagon videos in [2].
- Flexagons — David King's site is one of the more extensive resources on the subject; it includes a Java 3D Simulation of a hexahexaflexagon.
- The Flexagon Portal — Robin Moseley's site has patterns for a large variety of flexagons.
- Flexagons is a good introduction, including a large number of links.
- Flexagons — Scott Sherman's site, with a bewildering array of flexagons of different shapes.
Tetraflexagons:
- MathWorld's page on tetraflexagons, including three nets
- Folding User Interfaces - A mobile phone design concept based on a tetraflexagon; Folding the design gives access to different user interfaces.
Hexaflexagons:
- Flexagons — 1962 paper by Antony S. Conrad and Daniel K. Hartline (RIAS)
- MathWorld entry on Hexaflexagons
- Hexaflexagon Toolkit software for printing flexagons from your own pictures
- Hexaflexagons — a catalog compiled by Antonio Carlos M. de Queiroz (c.1973).
Includes a program named HexaFind that finds all the possible Tuckerman traverses for given orders of hexaflexagons. - Crochet hexaflexagon cushion