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Constructing π Via Origami

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origami, create a circular crease and measure part of its circumference with a. straight crease, you’ll have constructed a multiple of π. ...

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Constructing Via Origami Thomas C. Hull Merrimack College May 5, 2007 Abstract We present an argument for the constructibility of the transcendental number by paper folding, provided that curved creases are allowed. 1 Introduction Determining which numbers are constructible by paper folding has been the heart of research in origami geometry. H. Abe s angle trisection method 1, 4 demonstrates that origami is a more powerful construction method than straightedge and compass, and as far back as the 1930s it was known, as shown by Beloch 2 , that paper folding can solve general cubic equations. In 2003 R. Lang proved that the standard set of origami axioms people had been using for such constructions could not be made more powerful 5 , which proved that quintic equations were not solvable by traditional origami. However, by traditional we mean only using straight creases making one fold at a time. And so, in due course R. Lang came up with an angle quintisection method 6 that employed the simultaneous creation of two creases, thus demonstrating that by using multifolds (aligning parts of the paper so as to produce more than one crease line simultaneously) the solutions of equations of higher degree were constructible. It now seems very plausible that, in theory, if an arbitrary number of simultaneous creases are allowed in origami constructions, then polynomials of arbitrary degree are solvable. The in theory part of that last sentence must be emphasized, however. In practice it seems that making more than two simultaneous creases is extremely difficult, if not impossible, even under the best of conditions. 1 Figure 1: Crease pattern for constructing from a strip of paper. Despite all this activity, two techniques in paper folding which have yet to be employed in origami geometric construction explorations are non-flat folding and curved creases. (However, curved creases are non-flat by their very nature, so perhaps these two ...

Source: www.merrimack.edu

 

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