What Benoit found was a graph with symmetry, hierarchical self-similarity and infinite detail. No graph of this kind had ever been seen before. He had to invent a new branch of mathematics - fractal mathematics - to describe what he found. In 1982, Alan Norton - also of IBM - extended the mathematics of the Julia set (a "complimentary" graph to the Mandelbrot set) to Quaternion complex 4-space, which can be viewed in 3-space. [10]

The Seventeenth-Century French mathematician Pierre de Fermat wrote in the margin of his copy of Arithmetica by Diophantus, near the section on the Pythagorean Theorem (a squared plus b squared equals c squared),

"x ^ n + y ^ n = z ^ n - it cannot be solved with non-zero integers x, y, z for any exponent n greater than 2. I have found a truly marvelous proof, which this margin is too small to contain."

Andrew Hanson [11] has made some pictures, and I have in turn made sculpture (Figure 4), of a system analogous to Fermat's last theorem - a superquadric surface parameterized in complex four-space. We think that the mathematics of the n=3 case are similar to Fermat's own proof of the n=3 special case. Our pictures (Figure 2) have lent some visual concreteness to the recent news of Andrew Wiles' proof of the Taniyama-Weil conjecture, which implies the proof of Fermat.

Fig. 2 -- x^5 + y^5 = z^5
Image by Andrew Hanson and Stewart Dickson

By the 1890's the study of minimal surfaces was thought to be exhausted - no new surfaces could be described mathematically which were non-self-intersecting in three-space and which had vanishing mean curvature. However, in 1983 a graduate student in Rio de Janeiro named Celsoe Costa wrote down an equation for what he thought might be a new minimal surface, but the equations were so complex, they obscured the underlying geometry.

David Hoffman [12] at the University of Massachusetts at Amherst enlisted James Hoffman to make computer-generated pictures of Costa's surface (Figure 3). The pictures they made suggested first, that the surface was probably embedded - which gave them definite clues as to the approach they should take toward proving this assertion mathematically - and second, that the surface contained straight lines, hence symmetry by reflection through the lines.

Fig. 3 -- Costa's Minimal Surface
Image by David Hoffman and James Hoffman

The symmetry led Hoffman and William Meeks, III [13] to extrapolate that the surface was radially periodic and that new surfaces of the same class could be achieved by increasing the periodicity. They did so by altering the mathematical description of the surface to be the solution to a boundary-value problem constrained by the behavior of a minimal surface at the periodic lines of symmetry. The result: Hoffman and Meeks proved that Costa's surface was the first example of an infinitely large class of new minimal surfaces which are embedded in three-space.

The technique Hoffman and Meeks used was to make a picture which caused them to modify their mathematical theory and discover something totally unexpected about that theory. [14] They later extended their techniques to find minimal surfaces of more complex geometry and they also created pictures of them. This is a new kind of experimental mathematics and a procedure not far from creative visual art.

However, the pictures which were made in many cases were merely two-dimensional renderings of three-(or higher)-dimensional objects. The real discoveries were objects in three-(or higher)-dimensional spaces, but because of conventions in computer imaging technology (primarily the cathode-ray-tube) the observations of the discoveries were made through two-dimensional "windows" into cyberspace.

The real discoveries remained in the vacuum of abstract spaces, cyberspace and the cathode-ray tube.

I sent a copy of my exhibit proposal to Stephen Wolfram, author of the Mathematica System for Doing Mathematics by Computer. [15]

Mathematica implements a large subset of the "natural" language of mathematics in a machine-executable form. When making a graph of a mathematical expression, the machine can process symbolic language into a three-dimensional representation.

When the three-dimensional representation of a mathematical expression is cast into physical materials -- a sculpture in-the-round (Figure 4) -- via automated fabrication technologies, [16] I term this operation the direct concretization of mathematical language. [17]

Fig. 4 -- x^5 + y^5 = z^5
Stereolithograph by Stewart Dickson

The power of visualization has been proven to a fairly convincing degree by the history I have cited, but can similar arguments be made for tactilization? Clearly the segment of our population who cannot use computer pictures because their eyes do not work well enough to see the pictures have a real need for concrete artifacts from cyberspace.

I argue further that the physical presence provides a degree of immediacy missing from computer graphic displays, which in turn supplies more information than a two-dimensional picture. However, for the human observer, the meaning of the mathematical language which originates the form can become disconnected from the form when it becomes physical. I propose that improvement can be made by mapping descriptive text onto the three-dimensional object (See Figure 5).

For the purpose of discussion of the Internet, I believe mathematical objects may be of less importance to general human communication than are other types of three-dimensional shape information.

My on-line portfolio, in its current implementation, is no more than a traditional catalogue -- a published representation of artworks which might be purchased by a collector, or exhibited by a curator. Unlike a traditional catalogue, a Hypertext document on the Internet may be located by search engines and autonomous "Web Crawler" information-gathering agents. [18] Regardless of this feature, nothing about an Internet catalogue is intrinsically new, but the immediacy of information over distance. In general, publication on the Internet can be much more than this.

A portion of a hyperbolic paraboloid has been mapped with text information indicating special features of the surface -- in particular, the curves which result when one of the parameters in the equation is held to a constant: The Parabolas at X=0 and Y=0, the Hyperbolas parallel to the X-Y plane, achieved by setting Z to a non-zero constant, and the degenerate hyperbola, or two straight lines which lie in the X-Y plane at Z=0.

The regions of the surface in Figure 5 mapped with text could be active regions in a VRML application. Activating ("touching" with a computer cursor) one of these regions would supply the viewer with auxiliary information about that region, or could transport the user to another VRML "world".

Note that a similar integration of information could be achieved by translating the text into Braille on a physical model of the Hyperbolic Paraboloid.

Hoffman and Hoffman did in fact encode the Gaussian curvature of the surface as a color map in the images they made, of the type shown in Figure 3.

Image meaning has a level of detail in its context which can far outstrip verbal precision. I believe that there is much imprecision evident in current stock photography catalogues, if one views them strictly as image "Dictionaries". And the same is true of three-dimensional objects and worlds.

Given this difficulty in assigning an image to a verbal symbol, I determined that the dictionary system should be evolutionary -- dynamically re-assigning images to verbal contexts according to their frequency of use.

Furthermore, whereas MIME types are self-identifying objects to the extent that communications applications can decode and display them, any means for identifying the philosophical content of the objects is not standardized.

Identifying an image on the Internet per se, is probably limited at this point to the URL of the MIME object. This is a fundamental problem, of particular interest to those wishing to retain copyright over images published on the Internet. The ownership of an image itself is problematic in the age of post-mechanical reproduction.

At the present stage of the Internet, pictures of sufficient resolution to be valuable tend to occupy more data than can be economically transmitted for casual viewing. Therefore, the images which are routinely transmitted for informational viewing tend to be of a resolution sufficiently low to be of little commercial value in themselves.

But, bandwidth will not be a limiting factor for long. Owners of copyrighted material are beginning to use so-called "digital watermarking" [42][43] to identify images.

Similar issues also exist for 3-D object data, being often viewed as equivalent to the set of moulds for an original sculpture, over which the author typically wishes to keep edition control. I know of no method for similarly "watermarking" a 3-D object file, outside of "signing" it with geometrical text, embedded in the object's surface.

In the testing of the prototype Visual Dictionary server system, I have developed a HTML-based user interface from which anyone can add word and image definitions to the dictionary. Thus, I can potentially harness the collaborative power of the Internet to build the dictionary. The system is intended to eventually use a combination of natural language processing and evolutionary programming to automatically update word and image definitions from recorded frequencies of contextual word usage.

Clearly, in this age of electronic communication, the language of our civilization is evolving faster than published dictionaries can keep pace. Given, however, than a dictionary can produce information on the way people think about the ideas words describe -- I believe that an automatic, evolutionary dictionary can produce unexpected new information on the culture which engages in discourse over the Internet.

The system is designed to be extensible to support any MIME type, including VRML objects. I believe such a system could serve as a tool for studying ideas regarding the philosophical content of images and three-dimensional shapes.

4. Conclusion: Future Extensions

I have shown how the Internet can be a searchable catalog and repository of three-dimensional shapes. I have also presented an overview of the current technology of the World-Wide Web -- the active Hypertext document or VRML world which combines words, pictures and three-dimensional objects -- with special emphasis on the content as philosophical symbols in a logical universe of discourse.

I have demonstrated that a VRML entity can be a multi-dimensional, self-describing, integrated information object. If translated into Braille on a physical model, in-the-round of a mathematical surface, something interesting in addition occurs: As the reader's hand passes over the text, describing the surface at that region, the hand is constrained to the surface under all six degrees of freedom -- that is, the hand not only occupies a three-space position determined by the mathematics of the surface, but it is also oriented according to the normal vector to the surface and the line along which the text lies in the surface. I believe there occurs in such a case an element of corporeal awareness of the object which should not be discounted.

Where can we go from here? There are a host of improvements and developments which can and will be made to the visual dictionary system.

In the present form of the visual dictionary, verbal contexts could be translated into rules of two-dimensional image composition, which could in turn produce HTML 4.0 extensions to add further expression to the existing "automatic poetry" system. Similarly, rules of three-dimensional composition could be applied to generate 3-D VRML constructions from text input.

I have proposed a two-way, person-to-person application, which utilizes the visual dictionary server, to act as a semiotic amplifier: To multiply "by 1000" the meaning of words in a typed conversation, by automatically in-lining image icons into the conversational stream. In the virtual 3-D world of cyberspace, it is not unreasonable to speak of similar "conversational" modes in three dimensions as well.

What could three-dimensional conversation mean? There exist formal languages which can describe three-dimensional form and gesture. American Sign Language is a commonly used language which is expressed in the three-dimensional form of the hands.

Fig. 7 -- The Laban notation of the classical ballet