The pretzel as a curve.
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. . . .
nature were not
beautiful, it would not be worth knowing, and if nature were not worth
knowing, life would not be worth living."
Mathematicians at the
|Back to . . .
"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."
Pretzel and Croissant
|This section . . . .
Replay the Animation
dimensional Pretzel is basically an ellipsoid of the formulas:
|and to plot,
set the parameters as . . .
Croissant is a limiting case of the Pretzel for beta approaching 0.
The transition is visible in the two animated files below:
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.
Alfred, Modern Differential Geometry of
Curves and Surfaces with MATHEMATICA®, CRC Press,
Jeffrey R., The Shape of Space, 2nd ed., Marcel Dekker, 2002.
E. W., CRC Concise Encyclopedia of Mathematics, CRC Press,
1999, pp. 1762 - 64. See Surfaces.
to an August 21, 2007 JEOPARDY® question, the common ordinary
pretzel dates to ancient times.