
MAY/JUNE 2006 (Vol. 26, No. 3) pp. 45
02721716/06/$31.00 © 2006 IEEE
Published by the IEEE Computer Society
Published by the IEEE Computer Society
A Fractal Voyage
Article Contents  
Mathematical heresy  
Iterated function systems  
Download Citation  
Download Content  
PDFs Require Adobe Acrobat  
David Makin currently toils by day as a freelance programmer at Parys Technographix in North Wales, and he says he has no formal background in fractals or fractalrelated math whatsoever. He also says that he never produced any fine art before discovering fractals. He came to programming at age 21 when he discovered Basic while taking a chemistry course. "Going through the entire Basic manual in about 45 minutes made me realize that chemistry wasn't the best choice," he explained. "In the end I became a programmer by teaching myself machine code (6809) on a Dragon32. Since this time I've produced commercial software for the Dragon, the Atari ST, the Amiga, (DOS/Windows) PCs, and Windows mobile devices."
I always have to ask just what is it about fractals that artists find so alluring. Is it nature? The simplicity? God? "Personally, I've found fractals and fractal art to be very addictive," Makin explained. "I think this is due to the paradox between the (apparent) simplicity of the basic math and the amazing complexity of structure that results. Of course, the fact that many fractal forms are found in nature is another attraction (I'm thinking here in particular of iterated function systems [IFS]/flame fractals). I think one of the things that attracts so many people to fractals is the discovery that fractals are such a fundamental part of Creation...."
Mathematical heresy
Makin says that Disturbed Tiles (the cover image) is representative of how he messes around with math in an endless quest to create images that are out of the ordinary as far as fractal art is concerned. "I am constantly reinspired by the images that result from different mathematical constructions," he said. Makin first investigated fractal formulas using Frederik Slijkerman's Ultra Fractal software, and began to experiment with the Newton fractal. "One of the things I added to the basic Newton (and other formulas I've written for Ultra Fractal) was the option to modify the function that gets iterated, specifically allowing you to use z _{1} = f( z _{0}), z _{2} = f( z _{0} * z _{1}), z _{3} = f( z _{0} * z _{1} * z _{2}), and so on. Or z _{1} = f( z _{0}), z _{2} = f( z _{0} + z _{1}), z _{3} = f( z _{0} + z _{1} + z _{2}), and so on, instead of z _{1} = f( z _{0}), z _{2} = f( z _{1}), z _{3} = f( z _{2}), and so on."
He explained that Disturbed Tiles uses a Newton formula extended to a general system of two variables rather than "plain" complex values and uses the previously described process of iterating the product. "It's complete heresy mathematically speaking," said Makin, "since the Newton part treats the x and y variables as a twofunction system such that new x = f( x, y) and new y = g( x, y), but the iteratetheproduct part treats the values as a complex number by precalculating the continued multiple of the complex value (where z = x + I * y) prior to the Newton calculation. In Disturbed Tiles the two functions that the true Newton would be solving involve some normal powers and multiples of x and y but also have some trig functions in the mix and this is what gives Disturbed Tiles the repeated tiling. When I found the basic image I immediately had the idea for the title, and set about coloring it in what I hoped brought out the 'disturbing' aspect of the image." The formulas Makin used for Disturbed Tiles are all available in the public Ultra Fractal formula database (see http://formulas.ultrafractal.com). "In the end [ Disturbed Tiles] has five layers, each the same base formula/parameters, one colored using standard smooth iteration coloring, two using standard orbit trapping, and two using an alternative method of orbit trapping."
Anemonies of Vega 5 (see Figure 1 ) is another of Makin's many fractal images. Colony and Rose (see Figures 2 and 3 ) are from Makin's Renderosity gallery (see http://www.renderosity.com/ gallery.ez?ByArtist=Yes&Artist=MakinMagic); he says these images also come from playing around with algorithms. He explained that like Disturbed Tiles, Colony has a somewhat esoteric main formula and that Rose, all puns aside, stems from a new coloring algorithm: " Colony's main formula is in part plain z ^{2} + c but it's being mixed with a version of the formula for the Chebyshev polynomials and applied in such a way that c is the power rather than an additive constant—that is, the Chebyshev iteration alone would be z = cos( c * acos( z)). I see Colony as a space colony but others see it as something under a microscope."
Iterated function systems
Makin sells his prints on a regular basis and he's had his work published on a few other magazine covers in addition to IEEE Computer Graphics and Applications. The sales magazine for Swarovski crystals also recently published his work and he's hoping to devote more time to working on his art.
Basically, it all boils down to processor speed and IFS. "At the moment I'm revisiting creating 3D fractals using either complex fractals extended to 3D by mixing z and caxes in the Julibrot style or using quaternions or hypercomplex numbers," he explained. "I originally did some solid3D formulas for Ultra Fractal five years ago but didn't stay with them long due to the rendering times involved. I've gotten back to them now due to the improvements in computing power (my main machine is a 3GHz P4). In the immediate future I'm aiming to write a much optimized version of the solid 3D formulas. I'm hoping for around a tenfold speed increase, and then I'm going to get back to [other] IFS formulas that I've worked on ... hopefully adding 3D IFS."
Makin wants to get the word out and justify fractals as a compelling creative product. "In fractal art terms, rather than fractal math programming, I simply hope to spread the message that fractal art is a fully valid fine art medium," he said. "[And] to continue to find the unusual and different, to improve my coloring techniques, and to create images that are more interesting, thought provoking, and beautiful."
When asked if he's an artist, a mathematician, or both, Makin had this to offer: "Overall I'd say I'm a professional programmer, a budding artist (hoping to be fully professional), and an amateur mathematician (I need more study time)."
 x  