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ANALECTA 

FRACTALS 
Amanda Schermer 

Fractals are everywhere-in the flowing of a stream; in the inter­
twining branches of a tree; in the rise and fall of voices in conversation. 
The language of fractal geometry describes the underlying patterns that 
we intuitively sense in each of these. Fractals are complex designs which 
are created by repeating a simple mathematical rule many times. They 
allow us to examine the subtle structures of our world and show us how 
the repetition of simple patterns results in the infinite variety of objects 
that we see around us. Fractals are exciting because they offer us a better 
visual description of the way natural objects change and grow than do 
simple shapes like circles and squares. For this reason, scientists often use 
them as models to describe natural processes. 

The best way to understand fractals is to see how one develops. 



Philofopher ' s Stone 

Although the formal study of fractals began in the late 1970s, philoso­
phers and artists have been using the ideas behind fractal geometry for 
centuries. Since the time of the ancient Greeks, philosophers have sug­
gested that we can train our minds to think abstractly by learning to see 
increasingly subtle patterns in the world. The study of fractal geometry 
teaches us to see the subtlest of patterns in the chaos of our 
experience. 

The human mind has an innate tendency to search for patterns. 
Many artists use this tendency in their visual portrayal of structure. 
Natural objects are so complex that it is not possible to communicate 
every detail in a picture. A cluttered image results from the attempt to 
draw everything we see. Artists often solve this problem by outlining the 
form of an object and allowing the viewer's mind to infer the rest of the 
structure. Ansel Adams discussed such a technique in his Basic Photo 
Series Book, Natural Light Photography: 

Consider photographing rocky landscapes, another 
instance of textural rendition. Beyond a certain 
distance, a great field of granite boulders will appear 
as perfectly smooth stones, the natural textures being 
beyond the resolving power of the lens and/ or the 
emulsion. In order to suggest the substance of these 
stones it is necessary to include in the very near 
foreground a boulder in which the texture is adequately 
revealed. You can then say that the photograph 'reads 
well.' While you cannot see the texture in the distant 
boulders, you can see it in the near boulder, and you 
assume that all the boulders are the same material. 

Adams reminds us that in art, as in everyday life, we need structure AND 
detail to form a coherent and meaningful view of the world. 

Our minds crave complexity. If we lived in a completely ordered 
environment, everything would be identical and our senses would not be 
useful. Everywhere we turned, we would see and hear exactly the same 
thing. After a few minutes of uniformity, our minds would cease to 
respond to sensory messages. Rigid order satisfies us aesthetically only 
when our senses are overloaded and need a break. 

Complete randomness does not meet our aesthetic needs, either. 
If we flip a coin a large number of times, we will not find any pattern in 
the resulting string of heads and tails. Because there is no structure in 
random occurrences, they are not complex. Our minds are not able to 
make any more sense out of completely random experiences than com­
pletely ordered ones. 

Great music is neither monotone nor noise. A great painting is 
neither a blank canvas nor completely random splotches of color. Artists 

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ANALECTA 

incorporate the random and the ordered into their work, but it is the way 
in which they do this, the structure they impose, which is beautiful and 
meaningful. Those objects and processes which are most complex and 
beautiful involve both an underlying form and the potential for 
infinite variation. In other words, they are fractals. 

To generate a fractal, we choose a simple "replacement rule" and 
then repeat it many times. Computers are usually used to create pictures 
of fractals because it is difficult to perform the rule enough times by hand 
to get a feel for the end result. The replacement rule might be geometric, 
such as the one used to create the bush that we saw earlier. In this fractal, 
each line segment is divided into five pieces, two of which remain aligned 
and three of which are duplicated and angled. Each of the new line seg­
ments is replaced using the same procedure. 

Another type of replacement rule involves algebraic equations. 
Classic Julia sets, for example, are created by using the equation 
(New Z)=Z2+C. Here is a picture of a Julia set. 

The original Z in the equation is a point in the complex plane 

and C is a complex number which stays constant. To generate this image, 
the computer starts with the point at the upper left corner of the monitor. 



Philofopher's Stone 

It takes the complex number Z corresponding to that point, multiplies Z 
by itself, and adds C to it. It then replaces the original Z with the result, 
squaring it and adding C to get the next iterate. The computer repeats 
this process until the iterates become "large," which means they are 
"escaping" to infinity, or until the operation has been performed a certain 
predetermined number of times. If the iterates do not escape to infinity, 
we say that the original point remained "bounded" under iteration, and 
the computer colors that point. Otherwise, it leaves the point uncolored. 
The computer then proceeds to the next point on the screen and repeats 
the entire process. 

Sometimes, instead ofleaving the background blank, the com­
puter assigns color to a point to indicate how many iterations it took for 
the results to become "large." Those points which escape first form the 
background color for the fractal. Those points which escape after one 
iteration form the outside color band. Each successive iteration produces 
a new, smaller escape band. Those points which never escape are colored 
black. The fractal on plate 31 was colored in this way. Because this 
image is a zoom from deep within the fractal, we cannot see the back­
ground or the outside color bands, but we can see which points escaped 
during each of the last few iterations. 

Using different values for C creates different pictures. The fol­
lowing images demonstrate how very small changes in C affect the result­
ing fractal. 

It is interesting to notice the similarities and differences between 

C=-0.736 + 0.097i C=-0. 7 43 + 0.097i 

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ANALECTA 

C=-0. 7 44 + 0.097i C=-0. 756 + 0.097i 

fractals created with different replacement rules. The first Julia set is 
more filled in than the others. What causes this? Some of these fractals 
"blow away" as we increase the number of iterations. There are other val­
ues for C which do not produce an image at all. Which values for C 
produce images that will not blow away? We notice that these four 
images are all symmetric around the center point. Is that true of all Julia 
sets? Both the Julia sets and the bush fractal have features which are 
repeated on different scales throughout the image. Do all fractals have 
this property? These are the types of questions that fractal geometry 
explores. 

The study of mathematics has long been used to develop our intu­
itive ability to recognize patterns and structure. Math students first learn 
to recognize simple patterns, and gradually learn to think about more 
abstract structures. Plato argued that this training would ultimately enable 
the mind to contemplate the subtlest concepts: the Beautiful, the Good, 
and the other ideal Forms. Fractal geometry is an extension of this ancient 
search for the essential forms in our world. 

To see a world in a grain of sand 
And a heaven in a wild flower 
Hold infinity in the palm of your hand 
And eternity in an hour 

Blake described what it means to understand the fractal structure 
in the world around us. We are everywhere surrounded by profound beau­
ty. By learning to see subtle patterns, we develop a deeper understanding 
and appreciation of the wonders around us. 



Philofopher's Stone 

FOR MORE INFORMATION 

An Eye For Fractals, Michael McGuire, Addison-Wesley 
Publishing Company, Redwood City, California, 1991. This beautiful book 
is an excellent introduction to fractal geometry and its use in art. 

Nick's Fractal Page, http:/ /www.bush.edu/-nick/nick.html, gives 
more detail on how Mandelbrot and Julia sets are generated. Nick offers 
copies of the freeware program Fractint, which is one of the most user­
friendly fractal programs around. This page also has links to fractal gal­
leries and other related sites. 

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