Origami-based growing tube model for reproducing shell shapes

Mathematical models for shell morphology have been well-studied in theoretical morphology. The growing tube model proposed by Takashi Okamoto was developed to reproduce heteromorph ammonoids and can generate various shells by changing three parameters: curvature, torsion, and enlarging ratio. Previous studies in theoretical morphology have employed computers to visualize mathematical models. However, computers are not accessible to everyone, and merely observing the graphics of the reproduced shells does not deepen understanding of the mathematical model. Therefore, in this study, we considered using origami, which is inexpensive and readily available, to make users understand mathematical models through hands-on folding. First, we simplified the crease pattern of seashells, originally devised by an origami artist Tomoko Fuse, to create a basic crease pattern. We then identified the crease pattern elements corresponding to the three parameters in the growing tube model and the aperture-apex position determiner. Based on them, we reproduced various shell shapes, including a heteromorph-ammonoid-like shape, with origami. Finally, we investigated the constraints between parameters in the origami-growing tube model. The origami-growing tube model is expected to help disseminate mathematical models and promote theoretical biology.