Folding $π$

TL;DR

The paper investigates how origami can construct or approximate $\pi$ and other transcendental numbers, contrasting straight-crease origami with curved creases. It presents convergent straight-crease methods (Osler's product adaptation, Fujimoto-style continued fraction convergents, and trig approximants) and introduces an exact construction of $\pi$ using a parabola-based curved crease, with generalizations to constants such as $\Gamma\left(\tfrac{1}{4}\right)$. The results show that while straight-crease origami only yields convergents, curved creases enable exact values, suggesting a broader framework for realizing transcendental constants through carefully chosen crease patterns and curves. The work highlights practical limitations, such as measurement precision and sheet size, and situates these constructions among prior methods in origami mathematics.

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 $π$ can only be approximated using a finite number of straight line folds. Using these methods we give a convergent sequence for folding $π$ as well as other methods to approximate $π$. Folding along curved creases, however, allows for the construction of transcendental numbers. We here give a method to construct $π$ exactly by folding along a parabola, and we discuss generalizations for folding other transcendental numbers such as $Γ(1/4)$.

Figures (8)

• Figure 0: Convergent folding sequence for $\pi$.
• Figure 1: Folding Fujimoto's exponentially convergent approximation for $\frac{1}{5}$ from an initial guess of $\frac{1}{5}$ with error $\epsilon$. The steps taken can be represented as RRLL.
• Figure 2: Rational angle approximations can be found for the angle $\theta\approx 72.34321^{\circ}$ in the top left corner, for which $a/b=\pi$.
• Figure 3: Lill's method for finding the real root $3.1413\ldots$ of the cubic $x^3-x^2-8x+4=0$. A single fold is made by aligning point a to line D while at the same time aligning point b to line L, making a crease that crosses line M. The measurement in cm from O to M will give the $\pi$ approximant when the length from b to O is 1 cm.
• Figure 4: Annotated cropped windows of Jun Mitani's Ori-Revo software showing the silhouette (top left), folded model (top right), and crease pattern (bottom).