# Seashells: the plainness and beauty of their mathematical description

5. The origins of limpets

The limpets (Patella, Helcion) have small angles alpha (smaller than 30º) which make them ideally adapted to live in rocky environments exposed to the action of sea waves. It is believed that their ancestors are gastropods with angle alpha larger than 80º, because of the similarities between the animal that lives inside the limpet and those gastropods [6]. The result of that evolution process is a flat shell with alpha smaller than 30º. There exist only a few shells (fossils or of living animals) with alpha between 30º and 60º. Below we present a simulation of that evolution made with the mathematical model previously presented. We assume that the original shell is a Turban Shell (modern member of the family of the Trochoidea) and that the overall shape is similar to a Patella, the common prototype of a limpet.

(Click with the mouse over the pictures to rotate them)

## 2:

alpha=83, beta=20, phi=-50, mu=1, Omega=-10 alpha=73, beta=20, phi=-40, mu=1, Omega=-10 alpha=63, beta=30, phi=-30, mu=1, Omega=-20
A=9, a=8, b=7 A=9, a=8, b=7 A=9, a=8, b=7

## 5:

alpha=53, beta=40, phi=-20, mu=1, Omega=-25 alpha=43, beta=50, phi=-10, mu=1, Omega=-30 alpha=33, beta=70, phi=1, mu=1, Omega=-35
A=9, a=8, b=7 A=9, a=8, b=7 A=9, a=8, b=7

## 7: Common Limpet (Patella [2, p. 33])

alpha=23, beta=89, phi=1, mu=1, Omega=-40 alpha=3, beta=89, phi=1, mu=1, Omega=-80
A=9, a=8, b=7 A=5, a=8, b=7

[2] S. Peter Dance, Shells, Dorling Kindersley, 2002.

[6] E.R. Truemann and M.R. Clarke, The Molluscs. Vol 10: Evolution, Academic Press, 1985.