Herbert W. Franke – historic text from 1998 (published in Franke’s classic Webpage)
An Indeterminate Border between two Worlds
It began with mathematicians using their computer graphics systems to visualize what their formulas expressed. Of course, algebraic curves and geometric figures had already been put on paper before, but for understandable reasons they were content with the simplest shapes – straight lines, circles, rectangles… Anything else would have been too laborious to create. The obvious ease with which the computer coped with laborious arithmetic work led to it being given ever more complicated tasks, simply out of curiosity as to what would emerge. And a lot came to light.
Mathematical relationships become visible, statements from formulas that are difficult to tease out analytically are laid bare before your eyes. That is one side, the cognitive side. A lot could be said about this, for example about the pedagogical aspects, about better understanding: the “royal road to mathematics” that this opens up. For some mathematical disciplines, this meant a breakthrough, for example with fractals. But there is another side, that of art.
I am already getting ahead of myself, because it is by no means clear whether what I am suggesting can be considered art. And I’m not really interested in making a claim to art, which, as always in such cases, would lead to endless discussions. It’s more about pointing out something highly peculiar and noteworthy. In general, it is quite difficult to get a large number of people to start a new activity, to do something that takes a lot of time and effort – and without sponsorship or a fee. The fascination of mathematical images is so great that this hurdle is overcome, and the pursuit of mathematical-aesthetic experiments has become a worldwide movement. This case is also a good example of the positive side of the Internet, namely the opportunity it offers for the smallest interest groups with a very specific hobby to communicate across all borders. If you search the web, you will come across a flood of addresses. By far the largest group is dedicated to fractals, but you also discover other subject areas that you would hardly see under a common aspect at first, but which undoubtedly belong together. During my search, for example, I became aware of a group of people who deal with Arabic ornaments and try to describe them using mathematical means, and another one who dedicated themselves to the game of “slimming” and bring up the artillery of knot theory in order to grasp it theoretically. The protagonists include university professors as well as interested laypeople, scientific institutes as well as schoolchildren and students. Some of them issue publications, but the high costs of production and shipping make things difficult. There are certainly several of them, but they are not very well known. I will mention two privately published journals of this kind, namely “Fraktalia” from Romania and “The Fractal Translight Newsletter” from the USA. In both cases, the focus is on computer programs – because what is being created here would not be possible without computers.
The main purpose of my article is to draw attention to this phenomenon and to make it easier for lone fighters who are driven by similar interests to make contact with like-minded people. I have therefore included a few Internet addresses in the appendix, although I must point out that I have not been able to check all the information. This area is constantly changing, so that some things disappear and others reappear within a very short space of time. But once you start researching, you will quickly find many ways to get to the center of your wishes.
And with that I could actually leave it at that, but the problem of which category the activity I have described should actually be categorized in does not let me go, and so I would like to add a few thoughts for those who want to venture into the slippery ground of art discussions with me.
The debate actually began much earlier – long before there were computers. You could start with the ancient Greeks, with harmonic relationships and the golden ratio. This is about aesthetic orders, and these can certainly also be found in many mathematical visualizations. The rational theory of art based on the concept of information even offers explanations for the aesthetic impression of such structures. So that is not the problem. The question reminds me more of the discussion about Ernst Haeckel’s “Art Forms of Nature” from 1904. When the biologist, who was already widely respected at the time, made drawings of the shells of diatoms and stinging animals that he had seen under the microscope – microphotography was still unknown at the time – people doubted his fidelity to the original and considered the forms to be figments of his imagination. However, it was only when he referred to the forms he had discovered as “art forms” that heated discussions arose. If one limits the term art to man-made objects, then the title of his famous illustrated book is a provocation (today one would say: a gag intended to arouse interest). But the creator of the drawings, Ernst Haeckel, obviously cannot be described as an artist either, as he did not invent these forms, but merely reproduced them. A similar discussion broke out later about photography, because the photographer (with a few notable exceptions) only captures existing things in pictures and does not create them himself.
There is obviously a correspondence here to the current situation – to the images from the computer. But the comparison is flawed, because those who deal with the visualization of mathematics do not reproduce natural forms, but those of mathematics, i.e. something originating from humans. Is that even correct? – Opinions differ, because the mathematical relationships in question are determined by overriding logical laws and therefore humans cannot be their creators. On the other hand, creativity is certainly involved, namely in recognizing the mathematical problem and in the method for solving it – but this requires cognitive creativity and not creative creativity.
When a group of mathematicians from Bremen brought back the first fractals created with modern computer graphics systems from a study visit to the University of Utah, the head of the team, Heinz-Otto Peitgen, visited me and asked about my experiences with such images. We agreed that they could not be visualized without aesthetic decisions by the mathematician, because he determines the way they are displayed, the detail, the colors and so on. This is similar to photography, and I predicted that the discussion would begin anew using fractals as an example. And that is what happened: The Bremen group brought their works to the public under the title MapArt, triggering not only approval but also contradiction.
Heinz-Otto Peitgen makes use of all the possibilities of impressive representation offered by computer graphics, but the main purpose remains the reproduction of mathematical relationships. But there is another way – as computer art, which has existed since 1965, has proven. Those who practice it usually use the mathematical description as a kind of graphic notation. It helps them to compose images according to their own ideas, albeit in an unconventional way, without the use of manual means. This no longer has anything to do with the graphic recording of mathematical formulas, and what emerges does not have to make mathematical sense, but it is still remarkable – from an aesthetic point of view. If it had been produced by hand, even the opponents would have no problem calling the result a work of art.
This concludes my digression, which was merely intended as a brief stimulus to reflect on the background to what we find today in the form of mathematically described images. But no one is forced to dwell on such considerations – it is certainly far more enjoyable to start straight away with mathematical composition experiments, regardless of what you call what emerges.
Journals
Roger L. Bagula: The Fractal Translight Newsletter, USA, 11759 Waterhill Road, Lakeside, CA 92040
Marius-F. Danca: Fractalia, P.O.Box 524, Cluj 9, Rumänien, e-mail: mdanca@bavaria.utcluj.ro
Web addresses
Art Baker: Fractal Images – http://www.rainorg:80~ayb/
Alan Beck: Virtuel Mirror – http://sashimi.wwa.com:80/mirror/gallerie/fracgali/fg941101.html
Andy Burbanks: Lyapunov Pictures – http://info.Iboro.ac.uk/departments/ma/Gallery/lyap/index.html
Robert W. Ghrist: http://www.math.utexas.edu/MatGal9.html
Stewart Dickson: 3d Fractals – http://www.wri.com/~mathart/portfolio/SPD_Frac_portfolio.html
Geometry Center at University of Minnesota – http://www.geom.umn.edu/pix/archive/subjects/fractals.html
Herbert W. Franke: http://www.zi.biologie.uni-muenchen.de/ ~franke/kunst3.html
Javier S. Gonzales: Islamic Rectilinear Interlaced Lattices – httw://www.anglia.ac.uk/troshford/puertra4/index.html
Ryan Grant: Favourite Fractals – http://ncsa.uiuc.edu/SDG/People/rgrant/fav_pics.html
Noel Giffin: The Spanky Fractal Database – http://spanky.triumf.ca/www/spanky.html
Brian N. Hershey: SineArt Offering – http://members.aol.com/BNHershey/sineart.html
International String Figure Association: http://members.iquest.net/webweavers/isfa.html
David. E. Joyce: Mandelbrot and Julia sets – http://aleph0.clarku.edu/~djoyce/home.html
Pascal Massimino: Quaternion Julia Set – http://acacia.ens.fr8080/home/massimin/quat/quat.ang.html
o.N.: ANU Images – http://acat.anu.edu.au/works/gallery.html
o.N.: Computer Graphics Gallery – http://www.mathe.tcd.ie/pub/images/images.html
o.N.: Contours of the Mind – http://acat.anu.edu.au/contous.html
o.N.: Dalhousie University Fractal Gallery – http://is.dal.ca:3400/~adiggins/fractal/
o.N.: Fractal Commander Gallery – http://www.geocities.com/SoHo/Lofts/5601/gallery.html
o.N.: Fractal Gallery – http://irc.umbc.edu/gallery/Fractals/grindex.html
o.N.: Fractal Microscope – http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
o.N.: Gratuitous Fractals – http://www.vanderbilt.edu/VUCC/Misc/Art1/fractals.html
o.N.: San Francisco Fractal Factory – http://awa.com/sfff/sfff.html
o.N.: SHIKA Fractal Image Library – http://wwfs.aist-nara.ac.jp/shika/library/fractal/
o.N.: Soft Source – http://www.softsource.com/softsource/fractal.html
o.N.: Stanford University Pointers – http://akebono.stanford.edu/yahoo/Art/Computer_Generated/Fractals/
o.N.: Xmorphia – http://www.ccsf.caltech.edu/ismap/image.html
J. C. Sprott: Fractal Gallery – http:/sprott.physics.wisc.edu/fractals.htm
Fa. Wolfram Research: Computergraphics-Gallery – http://www.graphica.com/artists
Frank Rousell: Fractal Gallery – http://www.cnam.fr/fractals.html
Giuseppe Zito: Algorithmic Image Gallery – http://www.ba.Infn.it/~zito/dagal/dagal.html
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As web addresses are often changed, the topicality of this list cannot be guaranteed.
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Copyright H.W. Franke 1998