Re: Surfaces of constant negative Gaussian curvature

On Aug 27, 1:00 am, Gerard Westendorp <west...@xxxxxxxxx> wrote:
I'm looking for "nice" surfaces of constant negative Gaussian curvature.
There are quite a few pictures on the web of "minimal surfaces", which
have zero mean curvature. Their Gaussian curvature is generally
negative, but not necessarily constant.

As an example, I like this one, the "Schwarz P-surface":

http://www.indiana.edu/~minimal/archive/triply/schwarzp/schwarzp.html

The only surface of negative Gaussian curvature that seems to pop up
everywhere on the web is the pseudosphere. But would't it be possible to
deform the Schwarz P-surface so that not its mean curvature is zero,
but its Gaussian curvature constant?

The ultimate motivation for this is that I built a model of Klein's
quartic surfaces with 24 heptagons, that looks rather like the Schwarz
P-surface:

http://www.xs4all.nl/~westy31/Geometry/KleinHoles.jpg

It would be really cool if this tiling could be viewed as a tiling of a
constant curvature surface, just like polyhedra can be viewed as tilings
of the sphere.

Gerard


Breather surfaces in
http://3d-xplormath.org/

(Soliton solutions of Sine-Gordon equation) appear to be compositions
of several pseudospherical surfaces in

http://virtualmathmuseum.org/Surface/gallery_o.html#PseudosphericalSu...

which are spectacular both visually and in their mathematical
formulation.
Of the three pseudospheres (central, hyper and hypo) the central one
is
rotationally symmetric and has a simple closed form, others involve
elliptic integrals or Jacobi functions to describe 3D orientation.
Kuen surface
(e,g., pl. see Mathworld) and Dini's surface also have closed form
surface
parameterizations. Dini is made by cutting central a pseudosphere
along a meridian
and twisting it physically. Mathematically one introduces an extra z
term for
twist into central pseudosphere parameterization.

Had seen the pictures of Schwartz P surfaces earlier in "The Science
of Soap Films and Soap Bubbles" by Cyril Isenberg, H = 0 minimal soap
bubble/film surfaces. More recently holes are also appearing in Costa
and Hermann Karcher minimal surfaces.

Hilbert's theorem stipulates every surface of constant (negative)
curvature to have a cuspidal edge. It is intuitive to expect it also.
Since product K = k1*k2 is constant, when one curvature is infinite
there appears a sharp "edge" separating periodical segments in
a negatively curved semi-infinite surface. (The same logic applies
to positive Gauss curvature also). The asymptotic lines which are
edges
of regression proceed from one nappe to the other continuously, as
seen
in the central pseudosphere for z = th-tanh(th) at start of cuspidal
edge
th = 0.

To create a cuspidal edge in a physical model, each surface cell has
to be
deformed severely, from a condition of orthogonal asymptotes for
minimal surfaces H =0 , K <0, towards narrow rhombuses in isometric
deformation at first and then by dilatation into K< 0 constant
surfaces. Deformation proceeds from a variable square cell to
rhombic cells of Tschebychev Nets. This is difficult but not
impossible in physical models. It would be also an interesting subject
for research in general differential geometrical formulation. When a
helicoid
is to be so converted, an infinite *** has to becomes finite as an
isometric
equivalent of the hyper pseudosphere.

Just as the popular method of making minimal films is by soap-films,
a
similar method has not been suggested in literature to best of my
knowledge for K< 0 constant surfaces. My own proposal is to use
fishnet stockings
that make a Chebychev Net, filaments are twisted with constant torsion
as asymptotic
lines everywhere (by virtue of Enneper-Beltrami Theorem). We find the
such
construction in old style lobster/crab pots.

http://i7.ebayimg.com/02/i/000/99/88/7c48_1.JPG

To make physical model of Schwartz surface with const negative K, one
can start with a fishnet stocking, which is to be first draped on a
spherical mandrel employing three mutually perpendicular circle rings
inside.
They have to be drawn apart allowing the stocking to develop saddle
points everywhere.
For tiling I suggest such asymptotic lines as borders rather than the
lines of principal curvature. I studied these surfaces, finding the
topic to be of absorbing interest.

Regards,
Narasimham



.