# Seashell surface

Seashell surface with parametrization on left
Wheel-like Star Shell Astralium calcar, Diameter 3,5 cm; Originating from the Philippines

In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.

## Parametrization

The following is a parameterization of one seashell surface:

\begin{aligned}x&{}={\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\cos(2v)(1+\cos u)+\cos 2v\\\\y&{}={\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\sin(2v)(1+\cos u)+\sin 2v\\\\z&{}={\frac {10v}{2\pi }}+{\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\sin(u)+15\end{aligned}

where $0\leq u<2\pi$ and $-2\pi \leq v<2\pi$.

## References

• C. Illert (Feb. 1983), "the mathematics of Gnomonic seashells", Mathematical Biosciences 63(1): 21-56.
• C. Illert (1987), "Part 1, seashell geometry", Il Nuovo Cimento 9D(7): 702-813.
• C. Illert (1989), "Part 2, tubular 3D seashell surfaces", Il Nuovo Cimento 11D(5): 761-780.
• C. Illert (Oct 1990),"Nipponites mirabilis, a challenge to seashell theory?", Il Nuovo Cimento 12D(10): 1405-1421.
• C. Illert (Dec 1990), "elastic conoidal spires", Il Nuovo Cimento 12D(12): 1611-1632.
• C. Illert & C. Pickover (May 1992), "generating irregularly oscillating fossil seashells", IEE Computer Graphics & Applications 12(3):18-22.
• C. Illert (July 1995), "Australian supercomputer graphics exhibition", IEEE Computer Graphics & Applications 15(4):89-91.
• C. Illert (Editor 1995), "Proceedings of the First International Conchology Conference, 2-7 Jan 1995, Tweed Shire, Australia", publ. by Hadronic Press, Florida USA. 219 pages.
• C. Illert & R. Santilli (1995), "Foundations of Theoretical Conchology", publ. by Hadronic Press, Florida USA. 183 pages plus coloured plates.
• Deborah R. Fowler, Hans Meinhardt, and Przemyslaw Prusinkiewicz. Modeling seashells. Proceedings of SIGGRAPH '92 (Chicago, Illinois, July 26-31, 1992), In Computer Graphics, 26, 2, (July 1992), ACM SIGGRAPH, New York, pp. 379-387.[1]
• Callum Galbraith, Przemyslaw Prusinkiewicz, and Brian Wyvill. Modeling a Murex cabritii sea shell with a structured implicit surface modeler. The Visual Computer vol. 18, pp. 70-80. http://algorithmicbotany.org/papers/murex.tvc2002.html