# Information VR and Hyperbolic space

Paul Burchard (burchard@horizon.gw.umn.edu)
Fri, 10 Jun 94 12:04:03 -0500

One of the advantages of living in a virtual world is that you can
change its geometry to something more convenient than the essentially
flat, Euclidean universe in which we happen to live.

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

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:

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Paul Burchard <burchard@geom.umn.edu>
``I'm still learning how to count backwards from infinity...''
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