Here's how to visualize Enneper Surface looks like from far away.
Enneper Surface with Higher Order Symmetry
Generalizations to Ennerper with higher symmetry were found 100 years later.
Enneper Surface Parametric Equations
x = u - u * u * u / 3 + u * v * v
y = v - v * v * v / 3 + v * u * u
z = u * u - v * v
Enneper Surface is one of the finite total curvature immersions discovered in the 19th century. The generalizations to higher symmetry came in the 1980th.
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.
pure 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
For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see:
[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, 1993
[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab, Minimal Surfaces I, Grundlehren der math. Wiss. v. 295 Springer-Verlag, 1991