I've seen no presentations of what hyperbolic space would look like to
an Earth-based observer. Hyperbolic space models an additional large
dimension which causes the "shells of space" to have larger volumes
per del-radius than would flat 3D space. Since we don't see this
large extra dimension locally, it follows that (in this model) our 3D
space is embedded in a 4D space such that the 4th dimension manifests
only over large scales; the Randall-Sundrum model is like this. An
extension of this is that the 4D space is itself bounded within a 5D
sphere, as a boundary point of 4D space can be thought of as an
asymptote to the hyperbola, i.e. the line in the cone
x(0)^2-x(1)^2-x(2)^2-x(3)^2-x(4)^2-x(5)^2 = phi. It follows that the
boundary del-H(5) is a sphere, being just the 5D case of a general
rule across n dimensions.

In the absence of redshift we wouldn't be able to visually distinguish
between such a space and flat space, as e.g. if the shell of space at
radius R were twice the volume expected in flat space and thus
contains twice the objects, we would see those objects at just half
their expected angular size and so would conclude that they were
further away than they really are, using a perceptual correction which
mapped it into flat space.

However, if we now model that redshift is linear to distance, then we
have a ruler against which to measure the visible shells of space.
Objects at e.g. z=0.5 would be seen to be smaller in size and more
numerous than provided by flat space. This is in fact observed, see
e.g. astro-ph/0407216 where Sne II (core collapse stars) are found to
be 3x as numerous at z=0.3 as the local rate. Some cosmologists would
explain this with "evolution", but hyperbolic space explains both the
higher rate and the lower apparent magnitude, as the smaller angular
size naturally leads to lower flux received. Bye-bye accelerating
universe.

An additional refinement comes when redshift is modelled not just as
linear with distance, but as a function of distance modified by the
cosmological model. We know that the BB model is characterized by the
Hubble constant H(0) which however seems to manifest as different
values depending on how far out we are measuring. In a static model
we can model redshift as polarization in the large extra dimension,
i.e. z=tan^2(A) where A is the angle on the 5D sphere, and so redshift
goes to infinity at theta(5D)=pi/2; thus we can see half the universe
in such a model. Combining such a redshift dependency on distance
with hyperbolic space makes a volumetric map which accomodates current
observations nicely.

Of course, then there is the CMB which altho tractable is a topic for
another day.

Eric

P.S. to the moderator: feel free to boot this article; I can post it
to sci.astro no problem. Just looking for signal sans noise.