If it doesntappear, try sliding back one of the triangles (at 60 degrees from the centralangle) without pulling too hard. Squeezing two adjacent triangles of the glued hexaflexagon in the way shownin Figure 1 causes a new face to appear naturally from the center.
This involves 3 valleyfolds which is a twist of 180 3 = 540 degrees. Make a valley fold along line a b (Figure 3(2)), a valley foldalong line c d (Figure 3(3)), then without restricting the glue part, make avalley fold along line e f and glue (see Figure 3 (4)). Lets focus on the correct foldingtechnique first. While it may tally mathematically however, the appropriate arrangementof the triangles is key, and is explained below. The hexaflexagon onthe other hand, is composed of 6 triangles 18 6 = 3, so mathematically, it isnatural that it constitutes a 3 face folding.
The triangles each have inner andouter faces, so there are a total of 9 2 = 18 triangles. The right hand edge of the 10 triangles is for gluing, soin fact 9 triangles are involved in the puzzle.
It should be possible to draw a diagram of this complexity witha ruler and compasses. Ten equilateral triangles with sides of 6 cm are lined up sideways as shownin Figure 3(1). The 3 face fold used for the hexaflexagon isa basic among basics, so Id like for the reader to master it completely. In general, gluing together a strip with anodd multiple of 180 degree twists yields a surface with no inside or outside,while an even multiple yields a face with an inside and an outside.įigure 2: Surfaces with no inside or outside The Mobiusstrip involves a 180 twist but the hexaflexagon is made with a 540 degree twist 540 degrees is 3 times 180 degrees. The twist may be to the left or right, and yields a connectedsurface for which an inside and outside cannot be distinguished. Known as a Mobius strip, it is a normal loop glued together witha 180 degree twist, and was devised by the German astronomer A.F. 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 International Journal of Pure and Applied MathematicsVolume 58 No.