When we extend our domain from 2-manifolds to 3-manifolds, we are confronted with the problems of volume visualization. The traditional surface visualization approach is to embed the 2-manifold in 3-space and let the user fly around in the empty spaces, viewing the manifold from the outside. This is harder for 3-manifolds, but still feasible if one can do rapid volume rendering. In essence, one projects from 4D to 3D, treating space as a photo-sensitive medium that one can also fly through. In [4][5], Hanson, Pheng, and Cross introduce ``outside viewer'' techniques that allow interaction with 4D-lit, thickened 2-manifolds as well as moderately complex tessellated 3-manifolds.

Charlie Gunn's imaging system, Maniview [3],
an external module of Geomview,
takes an alternative
approach that dates to Bernard Riemann, the founder of manifold theory.
The viewer is placed *inside* the 3-manifold, with
no notion of an embedding in some ambient, higher-dimensional space. This
is an elegant, mathematical solution because it avoids artifacts of
any particular embedding. One can interact with an environment that is
3-dimensional, just like our familiar 3-space, but with surprises. For
example, the barbershop experience of sitting between two parallel
mirrors is similar to being inside certain 3-manifolds, provided you
ignore images of yourself that face you. What seems subjectively like
being inside a vast, repeating
volumetric tiling of space is objectively the
gluing of one wall to another to form a 3D cylinder.
Conceptually, the ``insider's view'' obtained by this
approach is an infinite tessellation of space. Of course, the tessellation
drawn by Maniview must be finite, but the combination of a large
tessellation radius with light attenuation yields a convincing picture. In
Figure 12, we see Maniview's representation of
life inside a 3-dimensional Klein bottle.

Thu Sep 21 19:17:33 CDT 1995