11/3/2023 0 Comments Kandasoft callnote reviewI first heard how interesting this puzzle is in 1985, from a report by ShinichiIkeno appearing in Mathematical Science. In fact the puzzle is not new toJapan, and resembles an old toy known as a byoubugai. The puzzle is made ofpaper and has a hexagonal shape. 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. 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 twist may be to the left or right, and yields a connectedsurface for which an inside and outside cannot be distinguished. 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 3 face fold used for the hexaflexagon isa basic among basics, so Id like for the reader to master it completely. Ten equilateral triangles with sides of 6 cm are lined up sideways as shownin Figure 3(1). It should be possible to draw a diagram of this complexity witha ruler and compasses. The right hand edge of the 10 triangles is for gluing, soin fact 9 triangles are involved in the puzzle. The triangles each have inner andouter faces, so there are a total of 9 2 = 18 triangles. The hexaflexagon onthe other hand, is composed of 6 triangles 18 6 = 3, so mathematically, it isnatural that it constitutes a 3 face folding. While it may tally mathematically however, the appropriate arrangementof the triangles is key, and is explained below. Lets focus on the correct foldingtechnique first. 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)). This involves 3 valleyfolds which is a twist of 180 3 = 540 degrees. Squeezing two adjacent triangles of the glued hexaflexagon in the way shownin Figure 1 causes a new face to appear naturally from the center. If you would like to learn more or talk to us, please email STARTEAECSTRINGNTJlMGExNjI2NTVkNTc4ZThiODgxNjk2M2YxYjM4YzNjb250YWN0QGthbmRhc29mdC5jb201MmUwYTE2MjY1NWQ1NzhlOGI4ODE2OTYzZjFiMzhjMw=ENDEAECSTRING or call 61.If it doesntappear, try sliding back one of the triangles (at 60 degrees from the centralangle) without pulling too hard. In case you are wondering, we are open for new business. Be it relocation, extra security precautions or improved business continuity procedures, appropriate and immediate steps continue to be taken. In order to keep up with the new world, we are doing a lot of things in unison, with urgency. Presently, all of the projects that we have been working on are on pace and continuing as expected. We are executing rapidly to ensure that all of our commitments are met – and all of Kanda’s partners remain happy with us. We are grateful to our clients and their employees who have been providing support and assistance to us and, more importantly, to folks in Ukraine. We condemn this aggression and pray for Ukraine’s unconditional victory.įor many of us this is personal – we have roots and many friends in the area. At Kanda we are horrified by the unprovoked attack on Ukraine.
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