Local growth laws determine global shape of molluscan shells

TL;DR

This paper demonstrates that molluscan shell shapes can be mathematically modeled using local growth laws and Lie group actions, revealing that most shells are characterized by just three parameters.

Contribution

It introduces a novel geometric framework linking local growth rules to global shell shapes, simplifying their description to three key parameters.

Findings

Shell shapes are governed by fixed local growth laws.

Most molluscan shells can be described by three parameters.

The model relates shell shape parameters to phylogenetic relationships.

Abstract

Molluscan shells come in various shapes and sizes. Despite this diversity, each species produces a shell with a characteristic shape that is independent of environmental conditions. We seek to understand this robust complexity. We are guided by two principles in the spirit of D'Arcy Thompson. First, the growth is governed by the repeated and continuous application of a fixed growth law, even as the shell evolves in overall shape, without any complex biological machinery to monitor and control the growth. Second, the growth law depends solely on local geometry at the shell's growing edge. The first principle naturally leads to the mathematical statement that the shape of the shell is generated by the action of a Lie group on a protoconch. The second naturally leads to a particular representation of the Lie group. We use this representation to show that the shapes of nearly all known molluscan shells can be described by essentially three parameters: a scalar (scaling), a vector (orientation), and a curve (edge of the protoconch). We relate these parameters to the phylogenetic tree. In addition to the morphogenetic insight, our results potentially point to a new approach to engineering complex structures.