Mathematical Modeling of the Origami Navel Shell

The well-known Nautilus shell has been modeled extensively both by mathematicians and origamists. However, there is wide disagreement on the best-fitting mathematical model — partly because there is significant variability across different Nautilus Shells found in nature, and no single model can describe all of them well. Origami structures, however, have precise repeatable folding instructions, and do not exhibit such variability. Ironically, no known mathematical models exist for these structures. In this research, we mathematically model a prominent origami design, the Navel Shell by Tomoko Fuse, believed to be based on the Nautilus. We use first-principles geometric and trigonometric constructs for developing a non-smooth Geometric Model of the ideal origami spiral. We then search for the best-fitting parametric smooth spiral approximation, by formulating the fitting problem as a minimization problem over four unknowns. We write a Python computer program for searching the space numerically. Our evaluations show that: (i) the Smooth spiral is an excellent fit for the Geometric Model; (ii) our models for Origami Navel Shell are different from prior mathematical models for the Nautilus shell, but they come close to a recent model for a rare species of Nautilus; (iii) the Geometric Model explains the outer edges of origami images quite well and helps identify construction errors in the inner edges; and (iv) the Smooth Model helps understand how well the ideal Navel Shell matches different variants of the Nautilus species. We hope our research lays the foundation for further mathematical modeling of origami structures.