Sea Shell Maths

October 16, 2012 by  

This post is one of several exploring the research and creative processes Giles and I have undertaken for our project Lifestreams, an Art+Tech collaboration with industry partner, Philips R&D in Cambridge as part of Anglia Ruskin University’s Visualise programme.

Our interest in using sea shells as the basis for making tangible lifecharms meant that I had to dive deeper into the maths, biology and development of shells to get a handle on how they grow and also to understand a bit more about what actually goes on at a physiological level. Fortunately there is a long history of the study of shell shapes and morphologies by different disciplines, biologists, mathematicians, artists et al to draw upon.

In my research I have come across many descriptions and models of shells, ranging from mathematical descriptions to those exploring the more complex biological processes involved in their genesis: these for example incorporate the growth of so called cellular templates which then undergo bio-mineralisation solidifying a soft scaffold of tissue into a rigid structural extension of the shell.

Essentially shells represent a geometric pattern that nature embraces and uses repeatedly in many biological structural systems such as the cochlea in our ears. sunflower blossoms and pine tree kernels. It is the the logarithmic helico-spiral. Imagine a flat logarithmic spiral that is then dropped from its centre on to the top of a cone and the spiral path then successively drops and drapes itself onto the surface of the cone.

This results in the 3-dimensional spiral that provides the growth direction for a generating curve which deposits different types of cellular tissue in sequence. The generating curve moves in three dimensions twisting, turning and changing dimension and rotation, at times even (depending on the shell  species)  changing its edge shape along the route. This then creates the intermittent bulges and outgrowths and sometimes sinusoidal waves along the shell edge.

Different types of shells come to being from a variety of generating curves and shapes that expand along the length of this spiral path as the shell grows. The height or flatness of the cone determines the compression of the helico-spiral on its central axis. As the shell grows the leading edge can vary in shape following rhythmic patterns or sporadic outgrowths. This has equivalents in natural growth phenomena in plants and becomes visible for example as  growth rings in trees.

As a shell grows in volume it simultaneously adds variable patterning on the exterior surface of the shell affecting growth based both on environmental and health factors. The surface colouring of the shell is patterned through a diffusion reaction process taking place just at the outward facing shell edge. The mollusc itself is never in direct control of this external pattern as it grows and even within the same species these patterns can vary dramatically.

What has been interesting in taking the formulas as a departure point into a series of parametric and other model variants is that the math evidently only is an approximation of the sea shell form. Some nuances are missed in the pure formula  generated shell approach and this became evident when I changed the way I was modelling my shells in different systems and moved away from using straight functional geometric models to more iterative and generative types.

Further reading:

D’Arcy Thompson: On Growth and Form @ InternetArchive

Seashells: the plainness and beauty of their mathematical description

Seahsells @ Wolfram MathWorld

D. Fowler, H. Meinhardt, and P. Prusinkiew, Modeling Seashells



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