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Paper-folding and cutting activities to demonstrate five-fold symmetry

Published online by Cambridge University Press:  17 October 2018

King-Shun Leung*
Affiliation:
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong e-mail: ksleung@eduhk.hk

Extract

We can obtain a two-fold symmetric figure by folding a square sheet of paper in the middle and then cutting along some curves drawn on the paper. By making two perpendicular folds through the centre of the paper and then cutting, we can obtain a four-fold symmetric figure. We can also get an eight-fold symmetric figure by making a fold bisecting an angle made by the two perpendicular folds before cutting. But it is not possible to obtain a three-fold, five-fold or six-fold symmetric figure in this way; we need to make more folds before cutting. Making a three-fold (respectively five-fold and six-fold) figure involves the division of the angle at the centre (360°) of a square sheet of a paper into six (respectively ten and twelve) equal parts. In other words, we need to construct the angles 60°, 36° and 30°. But these angles cannot be obtained by repeated bisections of 180° by simple folding as in the making of two-fold, four-fold and eight-fold figures. In [1], we see that each of the constructions of 60° and 30° applies the fact that sin 30° = ½ and takes only a few simple folding steps. The construction of 36° is more tedious (see, for example, [2] and [3]) as sin 36° is not a simple fraction but an irrational number. In this Article, we show how to make, by paper-folding and cutting a regular pentagon, a five-pointed star and create any five-fold figure as we want. The construction obtained by dividing the angle at the centre of a square paper into ten equal parts is called a pentagon base. We gained much insight from [2] and [3] when developing the method for making the pentagon base to be presented below.

Type
Articles
Copyright
Copyright © Mathematical Association 2018 

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References

1. Harrison, Ian, Folding angles of 30 and 60 degrees, British Origami Society, accessed April 2018 at http://www.britishorigami.info/\\academic/3060.phpGoogle Scholar
2. Haga, Kazuo, Fold Paper and Enjoy Math: ORIGAMICS, 3OSME in Asilomar Conference Center, Monterey, California, USA. Plenary Talk 1 (2001).Google Scholar
3. Walser, Hans, The Golden Section, The Mathematical Association of America (2001).Google Scholar