Pretzel as a curve

The pretzel as a curve.

Equation of pretzel curve

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Mac OSX viewers can open Zettler's pretzel animation in a freeware program.  (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.


Jeffrey R. Weeks illustrates the topological equivalence of a pretzel
in The Shape of Space, 2nd ed.,
Marcel Dekker, 2002, p. 28.


Earlier in the past two centuries the leaders in investigating surfaces were Riemann, Klein and Poincaré. . . .

"If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living."
Henri Poincaré  (1854-1912)


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Mathematicians at the Bakery
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Surfaces:
Pretzel and Croissant

"A discussion with a friend led to the finding that there was no appropriate mathematical description of baker's ware available.  This had to be fixed.  The results named the Pretzel and the Croissant are shown here."
Dr. Thomas Zettler 
Munich, Germany  

This section . . . .
AnimatedPretzel
Replay the Animation

The three dimensional Pretzel is basically an ellipsoid of the formulas:

Equations
 and to plot,
set the parameters as . . .
Parametric equations
The Croissant is a limiting case of the Pretzel for beta approaching 0.
The transition is visible in the two animated files below:
Croissant animation
 Croissant equations
Replay the Animation
 Replay the Animation
Croissant Equations

A Brief Listing of references that should be in most university libraries.
Adams, C. C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots.  W. H. Freeman, 1994, p. 48.
Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 1998.
Weeks, Jeffrey R., The Shape of Space, 2nd ed., Marcel Dekker, 2002.
Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 1762 - 64.  See Surfaces.

Note:  According to an August 21, 2007 JEOPARDY® question, the common ordinary pretzel dates to ancient times.
The National Curve Bank thanks Dr. Thomas Zettler of Munich, Germany for Deposit # 79.
Dr. Zettler created these animations using GRAPHER running on Macintosh MAC-OS X.
thomas.zettler@arcor.de
 Zettler signature