Deposit # 83
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"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
Dr. Thomas Zettler
|This section . . . .
The shell surface is
where is the
radius growth function.
plotting the shell surface . . .
|The second term (G) forms the
grooves of the shell.
while gamma ( γ )
segments, one with and the other without
In the K
portion of the equation, the number of grooves is defined by n
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 .
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
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."
der Messung mit dem Zirkel und Richtscheyt, 1525
Painters' Manual, Abaris Books, NY, 1977)
A brief list of printed
be in most university libraries.
Alfred, Modern Differential Geometry of
Curves and Surfaces with MATHEMATICA®, CRC Press,
1998, p. 72.
E. H., A Book of CURVES,
Cambridge Unniversity Press, 1961, pp. 127-129.
E. W., CRC Concise Encyclopedia of Mathematics, CRC Press,
1999, pp. 499-500. See Dürer's Conchoid etc.
R. C., Curves and Their Properties. NCTM, 1952,
pp. 31-33, etc.
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.