#Hexaflex flexture how to#
#Hexaflex flexture series#

The hexagon is constructed from six triangles,and by squeezing two adjacent triangles between the thumb and index finger asshown in Figure 1, a new face can be revealed from the center.įigure 2 illustrates a face which from a topological perspective has no insideor outside.

The puzzle is made ofpaper and has a hexagonal shape. In fact the puzzle is not new toJapan, and resembles an old toy known as a byoubugai. I first heard how interesting this puzzle is in 1985, from a report by ShinichiIkeno appearing in Mathematical Science. There are flexagonsin shapes other than hexagons, such as tetraflexagons, which are square, butthe most interesting from both a theoretical and practical perspective, is thehexaflexagon. The hexa in hexaflexagon means six, and flexagon indicates somethingthat is flexible, easy to bend, and can take many shapes. Sounds unfamiliar it is often called an origami hexagon or pleated origami inJapanese, which all refer to the same thing. Received: Decemc 2010 Academic Publications The puzzle was devised in 1939 by the English mathematician Arthur H.Stone, and is known as a hexaflexagon. I would like to introducethis puzzle to those readers who do not know it, and explain its close relationto mathematics. The theoretical workrelated this puzzle has advanced significantly, and the puzzle is now understood.A new folding technique has in fact been developed. In the December 1990 edition of the Basic mathematics magazine, I introduceda handmade puzzle known as a hexaflexagon under the title Folding PaperHexaflexagons. Referring to thearticles of Gardner and Madachy the author discovered a general solution formultiple foldings of hexaflexagons, which is described.ĪMS Subject Classification: 00A08, 00A09, 97A20Key Words: hexaflexagon, Mobius strip, topology, paper folding Hexaflexagons are known to besurfaces with no inside or outside, similar to Mobius strips.

#Hexaflex flexture how to#

Osaka University of Economics2, Osumi Higashiyodogawa, Osaka, 533-8533, JAPANĮ-mail: This article explains hexaflexagons: how to make them, how tooperate them, and their mathematical theory. 1 2010, 113-124ĭepartment of Business InformationFaculty of Information Management

#Hexaflex flexture series#

This work was commissioned by the NASH ensemble and performed as part of their concert series at Wigmore Hall on the 22nd of March, 2022.International Journal of Pure and Applied MathematicsVolume 58 No. Programme note written by the composer Dedication, Awards & Acknowledges Hexaflex offers a relaxing opportunity to forgo the torment of progression to some sort of future achievement, and instead, focuses on the joy of presenting what can be reached right now, through the simple processes of flapping, pressing, and pinching of its musical fabric. The form is ultimately cyclical, as these chords do not necessarily function as points of arrival or departure, just as how there is no starting or ending face of any flexagon.

These “flexagations” often require the musicians on stage to play more than two notes at a time, with overlapping chords seemingly not possible in a traditional trio setup, representing the phasing between two sides of a flexagon. The music is constructed around various singular, distinct chords, that can all be reached from one another through different “flexagations” of the musical texture. Mathematical mumbo jumbo aside, Hexaflex aims to invite the listener to experience a sonic equivalent of the movements inherent in flexagons. These shapes can be folded using a strip of paper and act similar to the origami paper fortune teller (also known as a “chatterbox”, “salt cellar” or “cootie catcher”) except, when flexed, flexagons will open up entirely to reveal one of its many sides. A “hexaflexagon” is the hexagonal variant of flexagons that can possess either three sides (“trihexaflexagon”) or six sides (“hexahexaflexagon”). However, when these models are manipulated (flexed), they reveal that they contain more sides in addition to its front and back face. In geometry, “flexagons” are a set of models that at first appear to be flat.