The shell as a curve and a surface.

Graph of the equation


. . . and to plot,
set the parameters as . . .

Grapher icon OSX



All animations were created using the Grapher program in MacOS X. 

Mac OSX viewers can open Zettler's shell animation in a freeware program.  The surface can be rotated using the Grapher tool.  Simply click on the graph and move the mouse.  (In the Applications folder on the hard drive, find the Utilities folder.  Download the free Grapher to the desktop.)  Download the attached file to your desktop, unzip the file and then open in Grapher.   

Related Curves



Durer Cook Island stamp
Albrect Dürer's famous Melancholia on a Cook Islands stamp.  His magic square is in the upper right hand corner.

Durer's Magic Square

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Deposit # 83

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Sea Shells
"Recently I studied a shell collection.  The beautiful shapes fascinated me and made me search for a mathematical model.  I found a rich and time-honored history."

Dr. Thomas Zettler
Munich, Germany

This section . . . .

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The shell surface is given by

Shell surface equations

where    Shell function   is the radius growth function.
For plotting the shell surface . . .
The second term (G) forms the grooves of the shell.

while gamma ( γ ) separates two angular segments, one with and the other without grooves. 

In the K portion of the equation, the number of grooves is defined by n:

Shells have a venerable history.  Pappus ascribes their "invention" to Nicomedes (ca. 240 BC).  Later in the seventeenth century, the conchoid was a favorite specimen for the new methods of analytical geometry and calculus.  Today, we find shells, conchoids and limaçons are popular for those experimenting with computer graphing software and graphing calculators.

Note the first term in the first line of the initial equation is limacon equation.  This is a special case of Pascal's conchoid, better know as the limaçon (from the Latin word for "snail," limax, and was discovered by Etienne Pascal (1588-1640), father of the famous Blaise Pascal.

The limaçon was in fact constructed before Pascal by the famous artist Albrecht Dürer.  Due to the appearance of lines he used in the construction, Dürer called the image a "spider curve."
Albrect Dürer, Unterweysung der Messung mit dem Zirkel und Richtscheyt, 1525
(The Painters' Manual, Abaris Books, NY, 1977)

A brief list of printed sources that should be in most university libraries.
Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 1998, p. 72.
Lockwood, E. H., A Book of CURVES, Cambridge Unniversity Press, 1961, pp. 127-129.
Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 499-500.  See Dürer's Conchoid etc.
Yates, R. C., Curves and Their Properties.  NCTM, 1952, pp. 31-33, etc.

The National Curve Bank thanks Dr. Thomas Zettler of Munich, Germany for Deposit # 83.
Dr. Zettler created these animations using GRAPHER running on Macintosh MAC-OS X.

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