March 7, 2000

Web Mind #1

by Rudy Rucker

Introduction

You might think of my Web Mind column as a clear-channel broadcast of a mad scientistís wild ideas. Like the good Dr. Frankenstein, one of my pet interests is the creation of life, and this will be a topic I return to over and over. For the first couple of columns Iím going to be talking about how (and why) you might go about making a computer copy of your mind.

The Medium is the Message

This summer I read a terrific book by Margaret Wertheim called The Pearly Gates of Cyberspace (W. W. Norton, 1999). Iíll probably get around to discussing the main point of the book in a later column, but for now I want to focus on a particular idea about the history of science which Wertheim raises.

The idea is this: the invention of pictorial perspective paved the way for Newtonian physics. This happened because perspective provides a tool for mapping unbounded three-dimensional space onto a finitely large two-dimensional canvas: the whole world in a square meter of cloth! Each object of the world gets assigned to one particular location upon the picture plane and, looking from the picture back out at the world, we can then see that the individual objects are contained in an all-encompassing world-space. Perspective teaches us to think of each objectís location as mapped into a mathematical (x, y, z) triple of coordinate numbers --- and this is the space of mathematical physics.

Itís fascinating to think that a new trick of artists made it possible to invent physics. Art matters! Accustomed as we are to seeing photographs, the perspective mapping of the world onto a square of paper seems obvious, even trivial, but it took people a long time to come up with it. And it was impossible for people to do modern physics until they had the idea of a unified underlying space. So, yes, maybe the invention of perspective really did lead to physics.

Might it be that the newborn Web provides a mapping tool which will lead to a mathematics of the human mind? As Marshall McCluhan taught, the effects of new media are wide-ranging and unpredictable.

In the most concise possible form, my idea is this.

Web : Mind :: Perspective : Space.

I have three reasons for thinking the Web is good for modeling the mind. First of all, the Web can display any type of media. Secondly, the Web has a hyperlinked structure reminiscent of mental associations. Thirdly, the Web and the mindís pattern of links are mathematical fractals of a similar kind.

Regarding the first point, the Web, a.ka. cyberspace, is a network containing all the kinds of data that one might conceivably access via a computer. In and of itself, the Web is not limited to any particular form of media. It can dole out printed words, sounds, images, movies, or active programs. Just like the mind.

The second point has to do with the fact that the Web pages by which we access Web data are written in hypertext (as in "HyperText Markup Language," a.k.a. HTML). One of the essential features of hypertext is that it contains hyperlinks: buttons you use to hyperjump to different locations in the hypertext. Later on, weíll look at how this compares to the mindís process of making associations.

And thirdly, I feel that the mind and Web are both fractals, specifically they are fractals of a similar kind of dimensionality. Before arguing this any further, Iíd like to give you some background on fractals.

Fractals

The word "fractal" was coined by Benoit Mandelbrot, Fractals: Form, Chance and Dimension (Freeman, 1977). It means a shape that has an exceedingly fragmented form, but which also has a certain kind of regularity. The regularity of a fractal lies in its self-similarity. If you select a small part of a fractal and magnify this part, then the magnified image will resemble the entire fractal shape itself.

Fractals can be either regular or random according to whether the small pieces of the fractal bear an exact or only a statistical resemblance to the whole form. Figure 1 shows a regular fractal called the Koch curve. We generate it by repeatedly replacing each line-segment by a little wiggle.

Rudy Rucker, Web Mind #1

Figure 1: The Helge von Koch curve, a fractal of Dimension 1.26. Click on the illustration to refresh the drawing.

There is a way to assign a numerical dimension to a fractal, but I wonít go into the details here. Suffice it to say that, firstly, the bumpier and gnarlier fractal, the higher its dimension and, secondly, the maximum dimensionality of a fractal is bounded by the space that it sits in. The Koch curve is an unruly line in two-dimensional plane, and itís thought of as having dimension 1.26. A mountain is a messy surface in three-dimensional space, and its dimensionality might be something like 2.1. If we had a sufficiently spiky fractal we might actually need a higher N-dimensional space to hold it without its part having to overlap.

Speaking of mountains, the parts of a mathematical fractal need not be perfect copies of the whole. Itís perfectly all right to have the patterns vary a bit from level to level. The idea is that a spur on a mountain looks quite a bit like the whole mountain, even though it isnít an exact replica. The outcroppings on the spur in turn resemble the spur, even though they arenít scale models of it. The outcroppings have mountainous little bumps on them, and the bumps have little jags, and if you get a magnifying glass youíll find zigs and zags upon the jags.

Among the physical forms that are commonly thought of as being like fractals are the following. Dimensions between 1 and 2: coastlines, trees, river drainage basins. Dimensions between 2 and 3: mountains, clouds, sponges. Fractal forms are found within the human body as well. Among these are the circulatory system, the nervous system, the texture of the skin, the eyeís iris, the convoluted surface of the brain, and the spongy masses of the internal organs.

A tree is a particular kind of fractal thatís particularly important for the present discussion. If you look closely at a tree, youíll readily notice that it has a trunk with big branches. There are subbranches coming off of the branches, and there are subsubbranches upon the subbranches, and so on through five to seven levels of branching.

I used to have the mistaken idea that a tree branched by splitting the tips of its branches, but this isnít really the way it works. The way a tree grows is that a new branch forms upon the smooth part of any sufficiently long piece. The basic move is like in Figure 2.



Figure 1: A branching tree fractal of dimension 1.46. Click on the illustration to refresh the drawing. Download the Java source for the Koch and tree fractals. The same code draws the two figures, using different parameters that are read off this HTML page. (Code by Jim Graham, copyright (C) 1996, Sun Microsystems, Inc.)

You might object to my calling the oak tree in your yard a fractal, because your oakís branching structure does not in fact have endlessly many levels of detail (as a true mathematical fractal would). When you get down to the twig level, the parts no longer resemble the whole. No matter. Even though an actual physical tree has a limited number of branching levels, it can be useful to think of it being a fractal. What weíre doing here is a special kind of idealization in which we approximates high complexity by infinite complexity. Oddly enough, this makes things easier. As the mathematician Stan Ulam once said about a particular problem, "The infinite case is easy. The finite case takes too long."

Alright, now Iím ready to state my point. Both the Web and the mental world of your ideas are N-dimensional fractal trees.

Tune in next month for more.