# A Physical Model for Self-Similar Seashells

## TL;DR

This paper introduces a coordinate-free physical model for self-similar seashell growth, using differential equations to describe the growth dynamics from both local and global perspectives, with practical implementation in Mathematica.

## Contribution

It presents a novel coordinate-free differential equation model for seashell growth that links local biological parameters to global shell measurements.

## Key findings

- Model accurately describes self-similar seashell growth.
- Provides transformations between local and global shell representations.
- Includes Mathematica code for simulation and visualization.

## Abstract

This paper presents a simple physical model for self-similar (gnomonic, or first-order) seashell growth which is expressed in coordinate-free terms. The shell is expressed as the solution of a differential equation which expresses the growth dynamics, and may be used to investigate shell growth from both the local viewpoint of the organism building it and moving with the shell opening (aperture), as well as that of a researcher making global measurements upon a complete motionless shell. Coordinate systems needed to express the global and local descriptions of the shell are chosen. The parameters of growth, or their information equivalent, remain constant in the local system, and are used by the organism to build the shell, and are likely mirrored in the DNA of the organism building it. The transformations between local and global representations are provided. The global model of Cortie, which is very similar to the present model, is expressed in terms of the present model, and the global parameters provided by Cortie for various species of mollusk may be used to calculate the equivalent local parameters.Mathematica code is provided to implement these transformations, as well as to plot the shells using both global and local parameters.