Published in the journal Computers and Graphics
Article © Alice Kelley
No portion of the contents of this article may be reproduced or transmitted in any form or by any means without express written permission of Alice Kelley

Abstract: A fractal program released in 1999, Ultra Fractal, was the first publicly available fractal software package to include convenient layering methods previously limited to image editing programs. The artistic effects of layering within a fractal program are demonstrated.

Computer-generated fractal imagery, originally in the realm of physicists and mathematicians, has been appearing with increasing frequency as popular art. Galleries display high quality prints of fractal images, and stores offer fractal merchandise such as posters and calendars[1]. "Fractal" geometry, from the Latin "fractus," meaning "broken," was introduced in 1975 by mathematician Benoit Mandelbrot to describe irregular and intricate natural phenomena such as coastlines, plant branching, and mountains that cannot be described by Euclidean geometry. Fractal shapes, like coastlines, exhibit self-similarity -- similar details at different size scales. These characteristics continue for many magnifications, both with natural phenomena and digital fractal art. Computers, with their ability to quickly perform the thousands of iterations necessary to graphically render a fractal mapping, have made it possible to create and explore abstract fractal geometric shapes. As fractal generating programs become easier to use and have more options, dazzling fractals are being created with increasing ease.

Most computer-generated digital fractal images start out as a single layer. The concept of an image "layer" is well known to users of image-editing tools such as Adobe Photoshop. Generally speaking, layers allow different images to be superimposed, or merged, with a variety of options. A fractal generating program called Ultra Fractal, written by Frederik Slijkerman, (http://www.ultrafractal.com, Win 95/98/NT), allows additional fractals to be rendered within the same boundaries as the first fractal, with each added fractal consisting of a new layer with its own set of properties, and with a merge mode that dictates how the added fractal layer interacts visually with the previous fractal's layer. Individual layers can be manipulated without affecting the other layers. Users may combine as many layers as they wish in one image, either using variations of the original layer, or adding entirely new layers that use different fractal generating formulas. There are 19 different merge modes, including screen, overlay, difference, and hue, with each having an adjustable opacity from 0-100%. By combining different layers and altering the merge modes between them, it is possible to create several different fractals from one original single-layer image, which can allow for more artistic interpretation of traditional fractal forms.

Moonscape (Fig.1), a spiral derived from the well-known Mandelbrot set, "z = z^2 + c" [2], is an example of a single-layer fractal that was originally rendered using the DOS, 256 color program Fractint, (The Stone Soup Group, http://spanky.triumf.ca/www/fractint/fractint.html)[3]. Ultra Fractal has the ability to read Fractint parameter files, which are the text file "recipes" that contain all the fractal's information, allowing the fractal to be recreated when Fractint, or in this case Ultra Fractal, reads the file. Moonscape was re-rendered in Ultra Fractal, and the color banding within the image that results from Fractint's 256 color limitation was automatically smoothed by Ultra Fractal's true color (2^24 colors) capability. A stark image such as this does not benefit from the addition of layers.

Corona (Fig.2), a spiral created using the Phoenix formula "z(n+1) = z(n)^a + c*z(n)^b + p*z(n-1)"[4], is another image that was first rendered in Fractint. Shown with it are two examples of how adding a single layer can alter the original fractal. In the first example, Etched Corona (Fig.3), a new layer was added that was identical to the first. The coloring algorithm, a function that transforms the color index value of each point, was switched in the new layer from the original "outside color = iter" algorithm, to a "heart" orbit trap; all coloring algorithms mentioned are documented and available in Ultra Fractal. The "difference" merge method was selected, which returns the absolute difference between the blend color and the base color[5], and which typically is used at opacity 100%. In the second example, Beta Lyrae (Fig.4), which the author finds even more artistically effective, again a new layer was added that was identical to the first. This time a Gaussian integer coloring algorithm was applied to the new layer, and a "Hard light" merge method, at opacity 100%, was chosen, which multiplies or screens the colors, depending on the blend color[5], and finally the coloring gradient editor, which controls the colors in each layer, was shifted slightly in this layer to give the final image a more pleasing appearance. Ultra Fractal's graphical user interface allows the user to see the effects these changes have as they are being made, and to quickly adjust variables in order to attain a visually pleasing image. A fractal like Corona may also be duplicated several times within the program's window, allowing exploration of different variables and layers for each duplication to occur nearly simultaneously.

Tropical Fish (Fig.5), a fractal made with a formula called Gallet-5-08, "z = F(real(z),imag(z)) + i x F(imag(z),real(z))"[6], was also first rendered in Fractint. Once its parameter file was read by Ultra Fractal, the image could be altered any number of different ways by adding layers. Tropical Fish/Sea (Fig.6) is one example out of the five fractals that ultimately evolved from the original, which incorporates a total of four layers, all using the Overlay merge method, which multiplies or screens the colors, depending on the base color[5], at between 44% and 50% opacity. The first new layer uses the same formula, with the coloring algorithm change to "outside = real," which creates a very simple fractal that provides the diagonal white swirl that runs across the image. The second new layer uses the formula used in the Corona example, the Phoenix Julia[4], to provide the "seaweed" in the image with the benefit of a "lines" orbit trap coloring algorithm. The last new layer uses the formula "Newton's method for exp(z) = log(z)"[7] with a "Shapes 2 = ellipse" coloring algorithm to provide the green and blue swirling shapes that look like water. This version of tropical fish is also notable because two of the layers make use of Ultra Fractal's "alpha channel" feature, which allows the user to specify an opacity value for each point in the gradient[5]. Each layer has its own alpha channel which may be toggled on or off.

Examples of fractals that did not start out as single-layer Fractint images and which exist because of the complexity and interaction of multiple layers include Rainforest (Fig.7), and Hatchling (Fig.8), which are both three-layer images, and Terraria (Fig.9), a four-layer image. Though these images are complex, artists routinely mix fifteen or more layers to create images that in the past would have been very difficult to create and manipulate. The approaches described in this paper appear to further expand the artistic possibilities of fractal creation. To see more examples of both single and multi-layer fractal art, visit the author's web gallery: http://www.AliceKelley.com.

All images © 1999 Alice Kelley

REFERENCES

1. Kelley, Alice, Fractal Cosmos Calendar, Amber Lotus, http://www.amberlotus.com/fractal.html, P.O. Box 31528, San Francisco, CA, 94131, 1999, 2000, 2001.
2. Mandelbrot, Benoit, The Fractal Geometry of Nature, Freedman, New York, 1982.
3. Wegner, Tim, and Tyler, Bert, Fractal Creations, Second Edition, The Waite Group, Inc., Corte Madera, CA, 1993.
4. Jones, Damien M., DMJ formula file for Ultra Fractal, http://www.fractalus.com/ultrafractal/.
5. Slijkerman, Frederik, Ultra Fractal Help Files, http://www.ultrafractal.com.
6. Gallet, Sylvie, Gallet formula files, http://ourworld.compuserve.com/homepages/Sylvie_Gallet/linke.htm.
7. Mitchell, L. Kerry, LKM formula file for Ultra Fractal, http://www.primenet.com/~lkmitch/uf.htm.
8. Pickover, Clifford, Chaos and Fractals: A Computer Graphical Journey, Elsvier Science B.V., New York, 1998.

ACKNOWLEDGMENTS

The author thanks Clifford Pickover for his encouragement and comments.

APPENDIX
Computer details
All figures were rendered on a 233 MHx Pentium II running Windows 95 with 64 megs RAM and a 1024x768 screen display.