FEBRUARY 2011 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS 5

One of the seminal lessons thatcame with the emergence ofchaotic dynamics and fractal

mathematics was the realization thateven the most seemingly complex and disorderly systems in nature might be followingsimple rules. Thisdiscovery broughtabout a major shiftin perspective inapplied fields committed tounderstanding such unruly things as weathersystems, animalpopulations, and financial markets. Ithas also brought about a significantchange of perspective in art.

Over the years, many artists haveturned to the tools of fractal geometryto create tableaus that are most oftenrepresentations of fractals arising frommathematical formulas. The resultingimages are necessarily abstractbecause this is a quality of the fractalsthemselves. To be sure, abstractfractal-based art has exploited thecharacteristic complexity and self-simi-larity of fractals to great effect, with

suggestive images that create a senseof wonder and expectation for theviewer. These images, however, arerarely, if ever, representational. This isodd because, generally speaking,chaotic dynamics is not an abstract

endeavor. The techniques of chaos theory are routinely put to use to modelthe turbulent and unpredictable realworld; therefore, it should also be possible to use fractal images in thecreation of natural, realistic phenomenaand landscapes.

We are going to briefly explain a way of using an abstract fractal image tocreate an absolutely realistic painting ofa natural landscape called Sea at theSunset. (See figure 1.) To create arealistic illustration, it is not essential toobtain a fractal similar to our subject.

Fractals do often have some primaryshape, and it is possible to makechanges to this shape by altering theunderlying formulas and parameters ofthe fractal, but this is a limited tool thatwe will largely avoid.

The mostimportantobservation isthat fractalimages sharemany propertieswith nature: self-similarity,fractaldimension,

complexity, chaos, simultaneous orderand disorder. In a very real sense,fractals are a complex and mixedillustration of natural behavior. The idea,then, is to capture these qualities ofnature contained in our fractal andapply them in the image-makingprocess. It should be noted that therewere a few artists, such as KatsushikaHokusai and Jackson Pollock, whoinstinctively recognized self-similarityand fractal dimension as intrinsicproperties of nature and applied themin their paintings.

Mehrdad Garousi

A Mandelbrot Sunset

Figure 1. Sea at the Sunset, created by Mehrdad Garousi (2008).

The intricate structure of mathematical fractals issuggestive of the complexity of nature, but rarely does a fractal on its own communicate the beauty ofnature. It is the creative artist who, by recognizingand applying these properties, can create an aesthetic work of art.

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Layering the Mandelbrot SetThe software package we use for our painting is called Ultra Fractal(http://www.ultrafractal.com), butmany others would work just as well.We never use any distortion, erasing, or image-processing software. Thefractal we will use is the famous Mandelbrot set, which lives in the complex plane and is created using the following algorithm:

Initial Condition: z = 0Loop: z = z2 + cBailout Criterion: � z � ≥ 2.

The value of c in the loop equation is acomplex number that corresponds tothe pixel location. For a given pixellocation c, the software iterates theloop, starting with z = 0, and checks tosee if the resulting sequence staysbounded (by two.) If it does, the pixel iscolored black and is in the Mandelbrotset; if the iterated sequence heads off to infinity, the point c is not in theMandelbrot set and is given a (nonblack) color determined by thespeed of divergence. Figure 2a showsthe results of this process.

The complexity in the Mandelbrot setlies at the boundary of the set (theblack points) and its complement. Tocreate our painting, we successivelyzoom in on a piece of the boundary toattain an image with an approximatemagnification of 5.8 � 107. (See figure2.) Ultra Fractal, like most fractalsoftware, has an option for rotating the flat page on which the fractal is produced, which effectively changesthe viewing angle but otherwise leaves

the fractal unchanged. Using this feature, we adjust the viewing angle tothe lower, more horizontal perspective

shown in figure 3. (Note that the central, thick black line in figure 2dbecomes the slim, black line that starts at the center of the right side of figure 3.)

Next, we generate five other versions ofthis image, each with a slight differencein zoom and transparency from the previous ones. The result of overlayingthese five images on the first layer isshown in figure 4. The combination is arich, textured image suggestive of thesea and clouds, and the strategy at this

6 FEBRUARY 2011 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS

Figure 2. Process of magnification of the border of the Mandelbrot fractal. Image (a)is the entire set. The magnification factor of (b) is 5.8 � 103, (c) is 5.8 � 105, and (d)is 5.8 � 107.

Figure 3. The piece of the Mandelbrot boundary from figure 2d, viewed from apoint near the surface.

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point is to continue layering even moreimages to perfect this effect. Addingcopies of the same five layers as beforewith their colors modified and a littlerelocation from their previous positionsyields figure 5.

Now we need to control the dispersionof the painting’s colors with an eyetoward differentiating the sea and sky.This time, we add five more layers ofthe Mandelbrot set with more blue, red,yellow, and white in the appropriateplaces. In most stages of this process,we don’t need to change the color ofthe Mandelbrot fractal; rather, we justzoom in on the color that we need. Forexample, to make the image moreyellow, we magnify a mostly yellow areaof the Mandelbrot set and add it in as anew layer. The image in figure 6consists of 16 layers of complex and

disordered views taken from theMandelbrot fractal, but by managingthe superposition of different layers and by controlling the color dispersionin the whole image, we have created an evidently ordered paintingrepresenting a blue sea with a red and yellow sunset.

In contrast to the chaotic sea and sky,the round shape of the setting sun ishighly regular. To create a circular formusing Ultra Fractal, we can change theloop equation to

Initial Condition: z = 0Loop: z = cBailout Criterion: � z � ≥ 2,

and the resulting set of pixel valueswhose orbits stay bounded by two isprecisely the disk of radius 2. Afteradding five more layers that balance

the colors and display the sun in thepainting, we obtain the completedlandscape shown back in figure 1.

In the end, we have created a 21-layerlandscape in which every part of thecanvas is covered with detail andcomplexities at the finest scale. If welook at the space between two waves,we observe other, more irregular waves,and between these irregular waves,there are still more. The clouds of thepainting also have this infinite property,making them visually similar to realones in the sky.

The whole of this image is made fromthe Mandelbrot fractal and thus takeson these fractal properties of nature.But unlike the Mandelbrot fractal, wehave an unmistakably representativeimage—a unique sunset painting thatcould not have been created on anycanvas or with any camera. Its creationdepends on computer assistance andon exploiting fractal geometry. It alsodepends on the eye of the artist. Theintricate structure of mathematicalfractals reflects the behavior andproperties of nature, but rarely does afractal on its own communicate thebeauty of nature. It is the creative artistwho, by recognizing and applying theseproperties, can create an aestheticwork of art.

DOI: 10.4169/194762111X12954578042777

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About the author: Mehrdad Garousi isa painter and photographer who hasrecently devoted his artistic energy tomathematical and fractal art entirely.For more information, please seewww.mehrdadart.deviantart.com.

email: mehrdad_fractal@yahoo.com

Special thanks to Mehdi Damali Amirifor his unsparing help in this project.A much shorter version of this articleappeared in the Bridges 2009Conference Proceedings onmathematical connections in art,music, and science.

Figure 4. Result of layering six slightly modified copies of the image from figure 3.

Figure 5. The painting with 11 separated layers of the Mandelbrot set.

Figure 6. This stage of the painting, showing the sea and sunset, contains 16layered images.

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