Ordinary Euclidean space gets cluttered very easily, and doesn't
allow for comfortable layout of the typically tree-like data
structures that you're trying to navigate in info-space. The amount
of room at a distance R from your current location only increases
linearly with R in two dimensions, and quadratically in three
dimensions. Spherical space -- a positively curved universe -- is
even worse, since the total amount of room available is finite!
However, hyperbolic space -- a negatively curved universe -- has an
exponential amount room at distance R from your current location.
This allows you, for example, to lay out a city of identical blocks
having *five* instead or four sides. All the streets would still be
straight and meet at right angles -- it's just that as you went
around each block, you'd come to five intersections instead of four
(see HREFs below for pictures). In such a city, the number of
different locations you can arrive at by travelling N blocks actually
increases *exponentially* with N (think of the sociological
consequences!).
The benefit for info-navigation is that *any* tree of bounded
branching will fit nicely inside a hyperbolic universe, without any
crowding or distortion. If you need more room to lay out a data
structure, you just have to make all the links uniformly a little
longer (note that hyperbolic space is not scale-invariant like
Euclidean space!).
You can see a 3D analog of the 5-sided city-block layout in the image
http://www.geom.umn.edu/pix/archive/homepages/not_knot_HSpace.html
from the Geometry Center's movie "Not Knot"; this view shows what it
would look like to a person living in the hyperbolic universe. You
can also move around in hyperbolic space interactively using the
Geomview program, which runs on most UNIX-based platforms:
http://www.geom.umn.edu/docs/software/viz/geomview/geomview.html
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Paul Burchard <burchard@geom.umn.edu>
``I'm still learning how to count backwards from infinity...''
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