E-mail address: Ak@KelleyArt.com (A. Kelley).1http://www.KelleyArt.com

Computers & Graphics 24 (2000) 611}616

Chaos and Graphics

Layering techniques in fractal art

Alice Kelley1

23 East Maynard Avenue, Columbus, OH 43202, USA

Abstract

A fractal program released in 1999, Ultra Fractal, was the "rst publicly available fractal software package to includeconvenient layering methods previously limited to image editing programs. The artistic e!ects of layering within a fractalprogram are demonstrated. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Fractal art; Fractals; Computer art; Ultra fractal; Lapsing techniques

Computer-generated fractal imagery, originally in therealm of physicists and mathematicians, has been appear-ing with increasing frequency as popular art. Galleriesdisplay high-quality prints of fractal images, and storeso!er fractal merchandise such as posters and calendars[1]. `Fractala geometry, from the Latin `fractusa, mean-ing `brokena, was introduced in 1975 by mathematicianBenoit Mandelbrot to describe irregular and intricatenatural phenomena such as coastlines, plant branching,and mountains that cannot be described by Euclideangeometry. Fractal shapes, like coastlines, exhibit self-similarity* similar details at di!erent size scales. Thesecharacteristics continue for many magni"cations, bothwith natural phenomena and digital fractal art. Com-puters, with their ability to quickly perform thethousands of iterations necessary to graphically rendera fractal mapping, have made it possible to create andexplore abstract fractal geometric shapes. As fractalgenerating programs become easier to use and have moreoptions, dazzling fractals are being created with increas-ing ease.

Most computer-generated digital fractal images startout as a single layer. The concept of an image `layera iswell known to users of image-editing tools such as AdobePhotoshop. Generally speaking, layers allow di!erentimages to be superimposed, or merged, with a variety ofoptions. A fractal generating program called Ultra Frac-

tal, written by Frederik Slijkerman (http://www.ultra-fractal.com, Win 95/98/NT), allows additional fractals tobe rendered within the same boundaries as the "rst frac-tal. With each added fractal consisting of a new layerwith its own set of properties, and with a merge modethat dictates how the added fractal layer interacts vis-ually with the previous fractal's layer. Individual layerscan be manipulated without a!ecting the other layers.Users may combine as many layers as they wish in oneimage, either using variations of the original layer, oradding entirely new layers that use di!erent fractalgenerating formulas. There are 19 di!erent merge modes,including screen, overlay, di!erence, and hue, with eachhaving an adjustable opacity from 0 to 100%. By com-bining di!erent layers and altering the merge modesbetween them, it is possible to create several di!erentfractals from one original single-layer image, which canallow for more artistic interpretation of traditional frac-tal forms.

Moonscape (Fig. 1), a spiral derived from the well-known Mandelbrot set, `z"z2#ca [2], is an exampleof a single-layer fractal that was originally renderedusing the DOS, 256 color program Fractint (The StoneSoup Group, http://spanky.triumf.ca/www/fractint/fractint.html) [3]. Ultra Fractal has the ability to readFractint parameter "les, which are the text "le `recipesathat contain all the fractal's information, allowing thefractal to be recreated when Fractint, or in this case UltraFractal, reads the "le. Moonscape was re-rendered inUltra Fractal, and the color banding within the imagethat results from Fractint's 256 color limitation wasautomatically smoothed by Ultra Fractal's true color

0097-8493/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 0 6 2 - 5

Fig. 1. Moonscape.

Fig. 2. Corona.

(224 colors) capability. A stark image such as this doesnot bene"t from the addition of layers.

Corona (Fig. 2), a spiral created using the Phoenixformula `z(n#1)"z(n)a#cHz(n)b#p*z(n!1)a [4], is

another image that was "rst rendered in Fractint. Shownwith it are two examples of how adding a single layer canalter the original fractal. In the "rst example, EtchedCorona (Fig. 3), a new layer was added that was identical

612 A. Kelley / Computers & Graphics 24 (2000) 611}616

Fig. 3. Etched Corona.

Fig. 4. Beta Lyrae.

to the "rst. The coloring algorithm, a function that trans-forms the color index value of each point, was switched inthe new layer from the original `outside color"iteraalgorithm, to a `hearta orbit trap; all coloring algorithmsmentioned are documented and available in Ultra Frac-

tal. The `di!erencea merge method was selected, whichreturns the absolute di!erence between the blend colorand the base color [5], and which typically is used atopacity 100%. In the second example, Beta Lyrae (Fig. 4),which the author "nds even more artistically e!ective,

A. Kelley / Computers & Graphics 24 (2000) 611}616 613

Fig. 6. Tropical Fish/Sea.

Fig. 5. Tropical Fish.

again a new layer was added that was identical to the"rst. This time a Gaussian integer coloring algorithm wasapplied to the new layer, and a `Hard lighta mergemethod, at opacity 100%, was chosen, which multipliesor screens the colors, depending on the blend color [5],and "nally the coloring gradient editor, which controls

the colors in each layer, was shifted slightly in this layerto give the "nal image a more pleasing appearance. UltraFractal's graphical user interface allows the user to seethe e!ects these changes have as they are being made,and to quickly adjust variables in order to attain a vis-ually pleasing image. A fractal like Corona may also be

614 A. Kelley / Computers & Graphics 24 (2000) 611}616

Fig. 8. Hatchling.

Fig. 7. Rainforest.

duplicated several times within the program's window,allowing exploration of di!erent variables and layers foreach duplication to occur nearly simultaneously.

Tropical Fish (Fig. 5), a fractal made with a formulacalled Gallet-5-08, `z"F(real(z), imag(z))#iHF(imag(z),real(z))a [6], was also "rst rendered in Fractint. Once its

parameter "le was read by Ultra Fractal, the image couldbe altered any number of di!erent ways by adding layers.Tropical Fish/Sea (Fig. 6) is one example out of the "vefractals that ultimately evolved from the original, whichincorporates a total of four layers, all using the Overlaymerge method, which multiplies or screens the colors,

A. Kelley / Computers & Graphics 24 (2000) 611}616 615

Fig. 9. Terraria.

depending on the base color [5], at between 44 and 50%opacity. The "rst new layer uses the same formula, withthe coloring algorithm change to `outside"reala, whichcreates a very simple fractal that provides the diagonalwhite swirl that runs across the image. The second newlayer uses the formula used in the Corona example, thePhoenix Julia [4], to provide the `seaweeda in the imagewith the bene"t of a `linesa orbit trap coloring algorithm.The last new layer uses the formula `Newton's methodfor exp(z)"log(z)a [7] with a `Shapes 2"ellipsea color-ing algorithm to provide the green and blue swirlingshapes that look like water. This version of tropical "sh isalso notable because two of the layers make use of UltraFractal's `alpha channela feature, which allows the userto specify an opacity value for each point in the gradient[5]. Each layer has its own alpha channel which may betoggled on or o!.

Examples of fractals that did not start out as single-layer Fractint images and which exist because of thecomplexity and interaction of multiple layers includeRainforest (Fig. 7), and Hatchling (Fig. 8), which areboth three-layer images, and Terraria (Fig. 9), a four-layer image. Though these images are complex, artistsroutinely mix 15 or more layers to create images that inthe past would have been very di$cult to create andmanipulate. The approaches described in this paper ap-pear to further expand the artistic possibilities of fractalcreation. To see more examples of both single andmulti-layer fractal art, visit the author's web gallery:http://www.KelleyArt.com.

Acknowledgements

The author thanks Cli!ord Pickover for his encour-agement and comments.

Appendix A. Computer details

All "gures were rendered on a 233 MHx Pentium IIrunning Windows 95 with 64 megs RAM and a1024]768 screen display.

References

[1] Kelley A. Fractal Cosmos Calendar. Amber Lotus,http://www.amberlotus.com/fractal.html, P.O. Box 31538,San Francisco, CA, 94131, 2000, 2001.

[2] Mandelbrot B. The fractal geometry of nature. New York:Freeman, 1982.

[3] Wegner T, Tyler B. Fractal creations. 2nd ed. Corte Mad-era, CA: The Waite Group, Inc., 1993.

[4] Jones DM. DMJ formula "le for Ultra Fractal,http://www.fractalus.com/ultrafractal/.

[5] Slijkerman F. Ultra fractal help "les, http://www.ultrafrac-tal.com.

[6] Gallet S. Gallet formula "les. http://ourworld.compu-serve.com/homepages/Sylvie}Gallet/linke.htm.

[7] Mitchell, L. Kerry LKM formula "le for Ultra Fractal.http://www.primenet.com/&lkmitch/uf.htm.

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