3-manifolds

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### 3-manifolds

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.

Tamara Munzner
Thu Sep 21 19:17:33 CDT 1995