Computers And Society

David D. Thornburg, Associate Editor

The Computer As A Tool For Discovery

The notion that the development of low-cost computers is "revolutionary" is not a new one, but the word revolutionary is used so much that one is likely to dismiss it as pure advertising hype along with words like "new" and "improved." And yet those of us who have been involved with this industry since its inception are aware that the development of the personal computer is not, by itself, revolutionary just because it may bring computer technology into people's homes.

"Revolutionary" is a special word – it implies that a technology or tool causes far-reaching changes in many aspects of our lives.

The development of the steam engine was revolutionary; the development of steam cleaning for carpets was not. The development of the telephone was revolutionary; the development of the answering machine was not. The development of the airplane was revolutionary; the development of in-flight entertainment was not.

Computer technology has had an impact that reaches far beyond the world of the computer itself. Computer users in industry and academia have known this for many years. Now that the power of the computer has reached the home, can we expect that people will start thinking about their world differently?

I think so.

The computer will help people to explore ideas that they wouldn't begin to explore if the computer hadn't given them the leverage to start thinking about them.

Beauty And Practicality

As an example of this, let's explore the development of a new field of mathematics called "fractal geometry." I have touched on this branch of mathematics in the "Friends of the Turtle" column a few times. I am intrigued by it because it deals with topics of considerable beauty and practical interest. Its seeds were planted a hundred years ago, but it was only after the development of the computer that anyone was able to begin to advance this field beyond the crudest level.

I realize the risk of illustrating a computer application based on mathemetics, since it tends to reinforce the erroneous concept that computers are primarily mathematicians' tools. The only reason for pursuing this example is because it is an interesting story in its own right.

In the late 1800s mathematicians were exploring some questions that went to the very foundations of geometry. One question of interest was if one could construct a curve that would fill a plane. At first thought, the idea of filling a two-dimensional surface with a curve made from a one-dimensional line is as absurd as asking for a roll of optically flat steel, or asking how many angels can dance on the head of a pin.

To the Italian mathematician Guiseppe Peano, this was a most intriguing question. In 1890 he published a proof that space-filling curves were, in fact, possible – that one could construct a curve that has the dimension of a surface. While this proof attracted the attention of several other mathematicians, the bulk of the academic community abhorred the thought of such "ill-behaved" curves.

In 1904 Helge von Koch continued the pursuit of strange types of functions by publishing the discovery of the "snowflake" curve. This curve is created by preparing successive generations from a simple motif. The rule to be followed is that each new generation is made by replacing each straight line in the previous generation with a copy of the motif itself.

If this process is carried on to infinity, one gets a very strange curve indeed. First, the curve is everywhere bumpy – there are no smooth regions. Second, even though the curve has clearly defined boundaries, it has infinite length. Third, the curve has a "dimension" that is intermediate between that of a line and a surface. To mathematicians of the early twentieth century, this curve was monstrous. To the contemporary mathematician Benoit Mandelbrot, it represented the need for a new field of mathematics, to be called fractal geometry.

The history and development of this field is beautifully illustrated in Mandelbrot's new book, The Fractal Geometry of Nature (W. H. Freeman, San Francisco). Through the pages of this richly illustrated volume, the reader is treated to a new way of thinking about geometry and nature.

For example, if you want to model a coastline, you are far better off to use a fractal curve than a smooth approximation, simply because coastlines are not smooth. Coastal lengths depend on the ability of the measuring stick to follow the nooks and crannies along the way. A coarse measuring stick gives a result corresponding to an early generation of a fractal curve. As the length of the measuring stick gets smaller, the total measured length of a coastline grows ever larger. This is also true for fractal curves.

Where does the computer fit in all of this? The notion of defining a curve in terms of itself may challenge the imagination, but it has a simple implementation in computer programming called recursion. Furthermore, the speed and accuracy with which computer-driven plotters can graph the various stages of curves free the mathematician to study their properties without being bogged down in drafting.

Computer graphics plays another pivotal role in the practical application of fractal geometry as well, since it is the tool that allows the creation of the simulated landscapes seen in movies such as Star Trek II. This practical application of a branch of mathematics would not have been possible were it not for the computer.

Those of you who read "Friends of the Turtle" know that fractal curves can be created on home computer systems using turtle graphics. Their expression in languages such as Logo is quite simple, and Mandelbrot's book provides hundreds of challenges for the interested programmer.

It is important to keep the role of the computer in perspective. The reason that these curves were not explored in depth in the early 1900s is that there was no appropriate tool to aid in their exploration. Now that the computer has made the study of fractals accessible to millions of people, one can expect the field to advance rapidly.

I Call It Kring

I saw a T-shirt that carried the message: "Recursion is a way of expressing the infinite in finite guise." My friend Sam Savage (the computer scientist/mathematician that invented the jigsaw puzzle called "Shmuzzles") likes to play with the infinite recursively. While I have used Logo to tinker with the latest of his ideas, you may wish to implement them mechanically.

Consider The Kite

Your normal garden variety kite is on the end of a string that droops gracefully in an arc.

This is fine for garden variety kite fliers, but suppose you wanted to make the string straighter. One way to accomplish this would be to add a second kite in the middle of the string. Because each kite would carry less weight, they would each be smaller.

But we still have some droop in the string, so we can add two more kites.

And two more ...

And so on, reducing the area of each kite and placing them closer and closer together.

If we keep repeating this process, we will end up with a substance I call kring – a combination kite/string that rises straight up in the air as it is unfurled.

Now that's revolutionary!