Wavy Enneper Surface

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Wavy Enneper Minimal Surface. Note that it has 3-fold rotational symmetry.
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Wavy Enneper Minimal Surface. Note that it has 3-fold rotational symmetry.

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The Enneper surfaces can be modified to develop waves around their boundary.
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These boundary waves can travel around without visibly disturbing the center part.
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                       About Wavy Enneper

                             H. Karcher

   The surfaces Wavy Enneper, Catenoid Enneper,  Planar Enneper,
and Double Enneper are finite total curvature minimal immersions
of the once or twice punctured sphere---shown with standard
polar coordinates. These surfaces illustrate how the different
types of ends can be combined in a simple way.

   Morphing Wavy Enneper rotates a high order Enneper perturbation
(of amplitude = aa, frequency=ff) over the (ee+2) tongues of a lower
order Enneper Surface. aa=0,  ee=0 gives the standard Enneper Surface.

   The pure Enneper surfaces (Gauss(z)=z^k , k=ee+1) and the Planar
Enneper Surfaces have been re-discovered many times, because
the members of the associate family are CONGRUENT surfaces
(as can be seen in an associate family morphing) and the
Weierstrass integrals integrate to polynomial (respectively)
rational immersions.  Double_Enneper was one of the early
examples in which I joined two classical surfaces by a handle;
we suggest to morph the size of the handle or the rotational
position of the top Enneper Surface against the bottom Enneper
surface. Formulas are taken from:

     H. Karcher, Construction of minimal surfaces, in "Surveys in
     Geometry", Univ. of Tokyo, 1989, and Lecture Notes No. 12,
     SFB 256, Bonn, 1989, pp. 1--96.

  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in "Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday", K. Uhlenbeck Editor, Publish or Perish Press,

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
          Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991