Folding $\pi$

Abstract

It is well known that the set of origami constructible numbers is larger than the classical straight-edge and compass constructible numbers. However, the Huzita-Justin-Hatori origami constructible numbers remain algebraic so that the transcendental number $\pi$ can only be approximated using a finite number of straight line folds. Using these methods we give a convergent sequence for folding $\pi$ as well as other methods to approximate $\pi$. Folding along curved creases, however, allows for the construction of transcendental numbers. We here give a method to construct $\pi$ exactly by folding along a parabola, and we discuss generalizations for folding other transcendental numbers such as $\Gamma(1/4)$.