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Pre-emptive strike #1

In order to hasten the inevitable triumph of Geomview as the  
cyberspace browser of choice, I sent the following message to WIRED's  
mailing list...

Begin forwarded message:

Date: Fri, 10 Jun 94 12:03:59 CDT
From: burchard (To: www-vrml at wired.com)
To: www-vrml at wired.com
Subject: Information VR and Hyperbolic space

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  

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:


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

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