Surfaces of constant negative Gaussian curvature
Gerard Westendorp
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
Andrew D. Hwang
> 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.
>
Surfaces of rotation are much easier to describe than general
surfaces... :)
> 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?
>
A theorem of Hilbert guarantees no *complete* surface of constant
negative curvature can be immersed in R^3, so there's no hope of
"slightly" deforming the triply-periodic Schwarz surface (the right-
hand image in the URL you posted) to have constant Gaussian curvature.
Offhand, I don't know about deforming one "unit".
Your (nifty) model seems to have "constant polyhedral
curvature" (three regular Euclidean heptagons meeting at each vertex),
but it doesn't look as if two copies can be joined like units of the
Schwarz surface so that three faces meet at each vertex. There's an
associated tiling of the hyperbolic plane, though. Don Hatch has a
large gallery of this and other tilings:
http://www.plunk.org/~hatch/HyperbolicTesselations/
Yours is in the fifth row:
http://www.plunk.org/~hatch/HyperbolicTesselations/7_3_trunc0_512x512.gif
(My own favorite is the 5,5 tiling, which covers Kepler's great
dodecahedron.)
Best,
Andy
P.S. Regarding the Klein quartic, do you know Silvio Levy's book The
Eightfold Way and Helaman Ferguson's sculpture of the same name?
Andrew D. Hwang
Dept of Math and CS
College of the Holy Cross
Worcester, MA, 01610-2395, USA
Oscar Lanzi III
the faces of a cube. Now draw any body diagonal of the cube and
identify its intersections with the P-surface. Because of the threefold
symmetry and the continuity of the curvature, the Gauss curvature must
vanish at these points. To get constant negative Gauss curvature you'd
have to puncture the surface at the body diagonals of the cube.
Unfortunately, your model does not satisfy a periodic bonndary
condition. To fit the protrusions of a like version of your model with
the indentations in the original, you'd have to rotate one copy by 45
degrees. Thus you can't get a Schwarz-P like surface.
--OL
Narasimham
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 sheet 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
Gerard Westendorp
[..]
> A theorem of Hilbert guarantees no *complete* surface of constant
> negative curvature can be immersed in R^3, so there's no hope of
> "slightly" deforming the triply-periodic Schwarz surface (the right-
> hand image in the URL you posted) to have constant Gaussian curvature.
> Offhand, I don't know about deforming one "unit".
Hilbert's theorem appears to forbid my surface, but I'm not quite sure
if the (Schwarz) surface qualifies as "complete regular". The surface
has no boundary, it extends to infinity in 3D space. Maybe it is not
"complete" in that sense.
> Your (nifty) model seems to have "constant polyhedral
> curvature" (three regular Euclidean heptagons meeting at each vertex),
> but it doesn't look as if two copies can be joined like units of the
> Schwarz surface so that three faces meet at each vertex.
[..]
A slightly simpler model of "constant polyhedral curvature" is one
formed by 3 sets of orthogonal intersecting bars
http://www.uwgb.edu/dutchs/graphic1/polyhedr/hyperbol/skew6x4.gif
The surface has 6 squares at each vertex.
I also found this one, a (3,7) infinite polyhedron:
http://www.superliminal.com/geometry/infinite/3_7a.htm
It appears to be the dual of the (7,3) case. So I think (7,3) *can* fill
space in a "zeolite" fashion, rather like other examples such as (4,6).
I might make a 3D model.
> P.S. Regarding the Klein quartic, do you know Silvio Levy's book The
> Eightfold Way and Helaman Ferguson's sculpture of the same name?
Yes, we had quite a lot of discussion on this following one of the "This
weeks's Finds" by John Baez:
http://www.math.ucr.edu/home/baez/klein.html
[..]
Oscar Lanzi III wrote:
> In the Schwarz P-surface, take a "unit cell" and cap its boundaries
> with the faces of a cube. Now draw any body diagonal of the cube and
> identify its intersections with the P-surface. Because of the
> threefold symmetry and the continuity of the curvature, the Gauss
> curvature must vanish at these points.
> To get constant negative Gauss curvature you'd
> have to puncture the surface at the body diagonals of the cube.
I can see the Schwarz P-surface has zero Gaussian curvature at these
points, but for example, the 4^6 tiling described above has 6 squares
meeting at the corresponding points, so I' m not sure symmetry alone
prohibits negative curvature there.
[..]
Narasimham wrote:
[..]
> Just as the popular method of making soap films is by soap-films. A
> similar method has not been suggested in literature to best of my
> knowledge. 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
A method I am working on myself is based on discrete Ricci Flow on
Circle packings. This is really beautiful stuff, for example, look at this:
http://www.cs.sunysb.edu/~vislab/papers/RicciFlow.pdf
I'm trying to implement this in a suitable computer program, so I can
make pictures myself. Should have it working soon.
Gerard
Narasimham
Beautiful indeed. Just for my take,is it so that line expansions/
contractions are conformally defined,or may be even intrinsically,
making mapping always possible from a surface of arbitrary Gauss
curvature to a flat plane?
So for example,any face can be made to resemble in its physical
features with any other face by a morphing that can be calculated by a
Laplace-Beltrami patch mapping matrix ?
If so,is there a Law that connects strains and Gauss curvatures that
can be stated at least approximately? Von-Karman's relations are also
relevant here.I had studied the subject to a limited extent.The first
and second forms of surface theory could be connected in elegant ways.
Narasimham
</x-flowed>
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Gerard Westendorp
curvature surfaces, covered by hyperbolic tilings:
http://www.xs4all.nl/~westy31/Geometry/Geometry.html#constant
The reason this does not violate Hlibert's theorem, is I believe related
to what I call "Ripple points".
Consider a surface that is locally described in polar coordinates
(r,theta) by
z = r^2 cos(N*theta)
If N is a positive integer, then along any line through the origin
(r=0), we have that
d2r.ds2 = 2.
So along any line, the coordinates of the surface are smooth functions.
But at the origin we cannot express z as
z= a x^2 + b y^2
so we do not have any way to define extrinsic curvature in the
conventional way. The intrinsic curvature is perfectly OK of course.
I believe ripple points are not regular in the sense of Hilbert's
theorem, so my surfaces do not violate Hlbert's theorem because they
contain Ripple points.
Allowing ripple points seems to me to open up quite an interesting range
of surfaces, that are worth studying.
Narasimham wrote:
[..]
> Beautiful indeed. Just for my take,is it so that line expansions/
> contractions are conformally defined,or may be even intrinsically,
> making mapping always possible from a surface of arbitrary Gauss
> curvature to a flat plane?
I think the answer is yes. Anything you can do with a conformal mapping,
you can do with a circle packing.
>
> So for example,any face can be made to resemble in its physical
> features with any other face by a morphing that can be calculated by a
> Laplace-Beltrami patch mapping matrix ?
I believe so.
>
> If so,is there a Law that connects strains and Gauss curvatures that
> can be stated at least approximately? Von-Karman's relations are also
> relevant here.I had studied the subject to a limited extent.The first
> and second forms of surface theory could be connected in elegant ways.
I think you are looking for some mechanical way to produce surfaces of
constant negative curvature, just like soap bubbles are positively
curved. Actually, if they are not pressurized from the inside as in a
spherical bubble, soap bubbles produce zero mean curvature, or a
"minimal surface".
I can think of a couple of ways to generate positive curvature. For
example, a surface of "finite" thickness that is heat on one side, will
curve due to thermal expansion. I feel that a mechanism for generating
negative curvature will involve some symmetry breaking effect, because a
saddle point has 2 definite spatial directions. Something with magnets...
I have thought about it, but I can't think of anything yet. But I feel
there must be something possible.
Gerard