Graphics Reference
In-Depth Information
The cell structure of a manifold gives us all the information that one needs to
define homology groups and any derived algebraic invariants. One could also define
discrete geometry concepts such as the discrete curvature at edges and vertices from
the attaching maps, although this has not yet been implemented.
Current Available Queries.
One can ask for the homology groups.
16.5
Where to from Here?
The SPACE program is obviously only a start on a general purpose three-dimensional
manifold program. For one thing, there are a lot more interesting and important
invariants associated to three-dimensional manifolds other than homology groups.
Some of these and algorithms for computing them can be found in [Matv03].
Although mathematicians at various university research centers have developed
software over the years to study aspects of the intrinsic geometry of spaces along the
lines that we have been discussing here, the programs tended to be more specialized
than is the intent of the SPACE program and restricted to predefined classes of man-
ifolds. There is no room to go into all these interesting programs here, but we mention
several below.
Higher Dimensions.
Four dimensions have been of particular interest. A simple
example of a four-dimensional space is the
hypercube
[0,1]
4
. Getting a good visual
understanding of such a space boils down to using good projections of it to
R
3
(and
then to
R
2
). Thomas Banchoff and others pioneered the visualization of surfaces in
R
4
in the 1970s at Brown University. Showing the surfaces as they rotated helped give
the viewer a feeling of depth. See [Banc95], [HanH92], and [HaMF94].
Special Task Programs.
One research center (shut down in 1998) that seems to
have produced an extraordinary number of modeling programs was the Geometry
Center, a National Science Foundation Science and Technology center at the Univer-
sity of Minnesota. Gunn's MANIVIEW program ([Gunn93]) is probably the program
that comes closest to what the SPACE program is trying to do. Here the viewer is also
inside a three-dimensional manifold (one defined by a discrete group in this case).
The program was an external module for the well-known surface visualization
program GEOMVIEW. Another program, also from the Geometry Center, is Jeff
Weeks' SNAPPEA program for studying hyperbolic 3-manifolds. (A
hyperbolic mani-
fold
is a Riemannian manifold with constant negative sectional curvature. In a hyper-
bolic plane, given a line and a point not on the line, there are an infinite number of
lines through the point that are parallel to the line. The sum of the angles of a trian-
gle is less than p.). Such manifolds are important to the classification of 3-manifolds.
The program is also an important tool for studying knot theory. See [Week85].
For a discussion of some further visualization programs see [HaMF94].
