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Understanding Hexagons and Mathematics

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Afew years ago l learned of a neat math trick using a strip of paper folded into small equilateral triangles. When folded correctly and glued together, the strip of paper transforms into a hexagon. When the sides are pinched, the hexagon can fold out into a new hexagon. l have always wondered why this anomaly happens and would like to understand the full reason of why the hexaflexagon can do this. In 1939, a man named Arthur Stone moved to the United States from England. He came to the United States to go to school at Princeton University. When he moved here he discovered that his notebook paper would not fit into his English style folder. He then cut off the extra strips of paper to make it fit into his folder. He began experimenting with the strips of paper and folding it into what we now call a flexagon. Aflexagon is a flexible polygon that can be folded inside out to reveal hidden faces. The fiexagon no inside or outside. If the flexagon is twisted 180 degrees a new side of the flexagon appears. Repeating this‘3 times shows each of the sides of the trihexafiexagon. it has three faces and six sides to it. The trihexaflexagon is made up of ten sixty degree triangles. The last triangle is used for gluing so the whole puzzle is made up of nine equilateral triangles. Each triangle has an inner and outer side so there are eighteen sides in all. The hexaflexagon is therefore made up of six triangles so there are three sides of the hexaflexagon. in the diagram below, it shows

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