Variations on a Theme in Paper Folding
Burkard Polster · American Mathematical Monthly · 2004
INTRODUCTION.Starting in 1971, Jean Pedersen published a number of articles in which she introduced and investigated an ingenious paper-folding construction for approximating certain rational angles and regular star polygons.Later on, she and Peter Hilton wrote a series of joint articles in which they generalized this construction, thereby allowing one to approximate any rational angle and any regular star polygon.Furthermore, they did an in-depth investigation of the number-theoretical ideas that their construction suggested.See [1], [2], [4], and [5] for excellent surveys of their work on this topic.When first encountered, the Hilton-Pedersen construction produces a magical "AHA!" effect similar to the one produced by a M6bius strip cut in half.Just like the Mibius strip trick, it uses only a simple strip of paper.It can be put to good use in the classroom, as a mathematical magic trick to impress nonmathematical friends and, last but not least, as a source of some very pretty nontrivial mathematics.Anyway, once the reader has understood what it is all about, he or she will never forget it.In this article, we summarize the construction in a way that enables us to describe other related paper folding constructions as alternative geometrical front-ends to the sound mathematical base created by Hilton and Pedersen.One such construction is Fujimoto's method for approximating rational subdivisions of arbitrary angles and line segments as described in [7].To really appreciate the beauty of the constructions presented in what follows, the reader is strongly encouraged to try them out with real paper.