A pseudosphere is a saddle-shaped surface that is produced by rotating a tractrix about its asymptote. The name "pseudosphere," which means "false sphere,"
is misleading because it suggests something that is sphere-like; however,
a pseudosphere is almost exactly the opposite of a sphere. Whereas a sphere has a constant positive curvature (equal
to +1/r, where r is the radius) at every point on its surface,
a pseudosphere has a constant negative curvature (equal to -1/r)
everywhere. As a result, a sphere has a closed surface and a finite area,
while a pseudosphere has an open surface and an infinite area. In fact,
although both the two-dimensional plane and a pseudosphere are infinite,
the pseudosphere manages to have more room! One way to think of this is
that a pseudosphere is more intensely infinite then the plane. Another result
of the pseudosphere's negative curvature is that the angles of a triangle
drawn on its surface add up to less than 180°.
The geometry on the surfaces
of both the sphere and the pseudosphere is a two-dimensional non-Euclidean
geometry – spherical (or elliptical) geometry in the case of the
sphere and hyperbolic geometry in the case of the pseudosphere. Astronomers currently suspect that the
universe we live in may have a hyperbolic geometry and thus have properties
analogous to those of a pseudosphere.