Surfaces of constant negative Gaussian curvature

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

.

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