Fractals in Nature
Fractals and Their Place in Nature
Mandelbrot and Fractal Beginnings Terminology Robert Brown Fractals and Plants Land Formations The Human Body Movies Works Cited
Fractals are one of the more recent discoveries in the field of mathematics. Uncovered in the late 1960s by Benoit Mandelbrot, fractals, simply put, are geometric figures that are made up of patterns and repeat themselves at smaller scales infinitely (Hastings 443). Mandelbrot and other mathematicians showed through their mathematics and computer programming that fractals are also common in everyday life, in nature, our bodies, and even in popular culture.
Benoit Mandelbrot contributed greatly to the study of fractals. Prior to his work, investigations into fractals began as early as the late 1800s, with German mathematicians George Cantor and Karl Weierstrass. There are two basic types of fractals, regular (geometric) and random. Regular fractals consist of large and small structures that are exact copies of each other, except in size. One of the more well known regular fractals is the Koch snowflake, which is made up of small triangles added to the sides of larger triangles to an infinite degree (Hastings 43). Random fractals are more apparent in nature as their small scale structures may differ in detail. It was this type of pattern that greatly influenced Mandelbrot, who gave these patterns the name “fractal,” from the Latin word fractus, which means a broken stone with an irregular surface (Hastings 43). In the late 1970s, Mandelbrot began to study an equation that later became known as the Mandelbrot Set, which under computer magnification, reveals an endless succession of repeating patterns (Hastings 43).
To understand the relation of fractals to nature, it is important to understand some of the math and math terminology behind them. Two important properties of fractals are self-similarity and dimension. Self-similarity is the property where a small portion of the object is similar in shape and structure to the shape and structure of the whole object. For example, picture the coastline of any beach from high up. It looks wiggly. However, as you continue to get closer and closer, the wiggles do not smooth out, but rather the closer you get to the coastline it remains wiggly (Liebovitch 8). There are two types of self-similarity, geometrical and statistical. Geometric self-similarity means that the smaller pieces of an object are exact copies of the larger piece (Liebovitch 12). A good example is the Koch snowflake. If you were to zoom in on any portion of the snowflake, it would be an exact copy of the original whole. Thus, usually only mathematically defined objects have geometrical self-similarity. Statistical self-similarity occurs when the statistical properties of the smaller piece can be geometrically similar to the statistical properties of the biggest piece (Liebovitch 12). Therefore, the smaller pieces of objects with statistical self-similarity are not exactly like the larger pieces, but are very close. I will refer to statistical self-similarity throughout the remainder of this essay whenever I refer to self-similarity, as I will be discussing real life objects, which are not perfectly described mathematically. A prime example are the arteries and veins in the retina, which have a branching pattern of the larger arteries that is repeated in the branching patterns of the veins and other smaller vessels (Liebovitch 14).
Fractal dimension is another important property of fractals. Simply put, the fractal dimension tells us the number of new pieces of the fractal we will see when we look at a higher resolution. There are three types of fractal dimensions: self-similarity dimension, which describes how an object fills space; topological dimension, which describes how points inside an object are connected; and embedding dimension, which describes the space containing an object (Liebovitch 47). Throughout this essay I will refer to self-similar dimension when I refer to fractal dimension. The fractal dimension is calculated from d in the equation N = Md, where N is the number of pieces left after an object is divided M times. For example, if we divide the sides of a square into thirds, we are left with 9 total pieces. Hence, 9 = 32, thus the fractal dimension is 2 (Liebovitch 48).
One of the interesting properties of fractals is that they can be used to model patterns in nature more accurately than convention geometry models. A good example of fractals in nature is Brownian motion. It was discovered by the Scottish biologist Robert Brown, as he observed the movements of small particles in liquid in his microscope. He noticed that the particles made small, erratic, and unpredictable movements, which he attributed to physical causes. Einstein later discovered it was due to irregular thermal changes (Lauwerier 112). We can use fractals to help us understand this motion, or more specifically, fractal curves. Mandelbrot describes fractal curves as “curves for which the fractal dimension exceeds the topological dimension 1” (Mandelbrot 31). The path of each of the particles moves in a fractal-like pattern, or if you were to picture a coastline, the particles’ movements resembled the shape of the coastline only in three dimensions. That means, like the coastline example, the particles have self-similarity; thus, if Brown were to have looked at the particles under a microscope that was 100 times more powerful, he would have observed the same thing. Therefore, Brown was really looking at a spatial fractal curve, which became known as Brownian motion (Lauwerier 113).
With the massive amount of calculations used for creating fractals, computer programs are employed to make fractal images, and one such program used to model real-world images, especially plants, is called an iterated function system (IFS, cerca 1988) (Barnsley 221). This program models two-dimensional objects based off the Collage Theorem, which can be used to create various foliage, such as ferns (Barnsley 239). Ferns are not the only image that can be created by the program. In Figure 1, notice how the IFS models (based off fractal math) can create an object resembling a tree.
Perhaps the best example of the technology behind IFS images and fractal’s close ties to nature can be seen in Figure 2a. This is known as the Black Spleenwort fern (Barnsley 242). Notice how each of the smaller “branches” of the image is a copy of the main image. Compare it to a real fern in Figure 2b. Although each of its “branches” is not an exact copy of the main plant, it can be seen how a fractal pattern is quite accurate in
describing these plants.
Consider the California oak tree in Figure
Figure
3: California Oak Tree (Barnsley 265)
Fractals are also very useful in describing natural formations on earth, such as land formations and rivers. It has been said that “ . . . mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line” (Barnsley 23). Coming back to Robert Brown, we now examine Brownian Landscapes. Actual landscapes can only be self-similar over a set distance, not infinitely. However, this distance can be quite large, and fractals are an excellent way of representing landscapes, especially mountains (Barnsley 30-1). Using computers, mathematicians can create images that greatly resemble earth formations, or could be the planet from a distant galaxy. In Figure 5, you will notice that a Brownian landscape bears a strong resemblance to a mountain range. If that particular landscape seems familiar for some reason, it would be because Brownian landscapes are often used in science fiction films (Lauwerier 104).
Fractal formations are not just limited to earth.
Lunar landscapes can be represented by fractals, where a few large
craters could be the main formation, and smaller craters represent smaller
scales and repetitions of the original formation (Lauwerier 114).
Figure
5: Brownian Mountain Range (Lauwerier 114)
Look at Figure 6 and Figure 7. At first glance, it is easy to mistake the computer generated images as pictures of the moon and earth. However, both of these images are generated by fractal programs. These figures illustrate the accuracy and realism of fractal images and how they can represent real life. Neil Armstrong himself might have problems differentiating between Figure 7 and the real moon!
Figure 6: Brownian moon and earth (Lauwerier 115)
Figure 7: Fractal lunar landscape (Barnsley 32)
Rivers and coastlines are commonly modeled by fractals.
We will consider the Missouri River’s length.
River length (known as G-length) is defined from the source of the river
to its mouth, approximated by a self-similar line of dimension D, which is
greater than 1. Using what is known
as fractal length-area relations and algebra, mathematicians have found the
relationship
, where k is a constant and S is the straight distance from the source of the
river to its mouth (Mandelbrot 111). This
formula yields a value of D of 1.2 (meaning the formula closely approximates the
real thing!). Interestingly, if one
were to measure the degree of irregularity by D, that person would find that the
degrees of irregularity of smaller rivers (Potomac River, etc.) and large rivers
(Amazon, etc) would be identical (Mandelbrot 111).
The study of coastlines is similar. Lewis
Richardson in 1961 posed the question, “How long is the coastline of
Britain?” (Liebovitch 33). It is a
simple question, but his research showed that finding the answer can be somewhat
paradoxical, and in the process showed the relevance to fractals.
Obviously, the length must be finite, and to find the distance, one
measures from one end to the other with a straight edged device.
However, if one wants to be exact, that person will measure the coast
itself, meaning we must examine the coastline at a higher resolution to get an
accurate reading. If we look at the
coastline on a 1/100000 scale, bays and peninsulas are visible.
Looking further at a 1/10000 scale, sub bays and sub peninsulas are
visible, and so on. The point is
that if we keep looking at the coastline under increased resolutions, we see
more and more land formations, each of which adds length to the coastline.
This gives the impression that the coastline is increasingly large and so
difficult to measure that it could easily be considered infinite, as we could
keep looking at it under a finer and finer resolution (Mandelbrot 25-6).
Finally, it is interesting to note that the fractal dimension, D, of
coastlines is around 1.2, just like rivers (Barnsley 35).
As our bodies are naturally created, it is possible to find fractal
relations in our own flesh and blood. As
mentioned earlier, the structure of the bronchial tubes in the human lung is
fractal in nature. We will use the
variable E, which represents the diameter exponent.
The diameter exponent comes from the equation
, where d is the diameter of the main bronchial tube, and d1 and d2
are the diameters of the branches of the main tube (Mandelbrot 156-7).
For all practical purposes, E = 3, and Mandelbrot makes an argument as to
why this number works well with human physiology.
He argues that since the tubes essentially start from a single bud, which
produces two branches, each of which produce two branches, and so on, then the
“outcome of growth is determined fully by the branches’ width/length ratio
and the diameter exponent” (Mandelbrot 158).
Based on the value of E, the outcome of growth can either be: 1) branches
run out of room and stop growing; 2) branches never fill more than a part of the
available space; or 3) they find the space they need (Mandelbrot 158).
Either way, our bronchial tubes have been successfully formed in a finite
amount of space. Finally, other
areas of the human body that have had fractal dimension measured include:
blood vessels in the heart, textures of x-rays of bones, blood flow in
the heart, and growth of bacterial colonies (Liebovitch 69).
Finally, fractals have made their appearance in popular culture as well. Inspired by images in Mandelbrot’s The Fractal Geometry of Nature, three mathematicians/computer programmers, Alain Fournier, Don Fussell, and especially Loren Carpenter, found ways to compute beautiful fractal landscapes quickly through the use of computers (Barnsley 9). Carpenter moved to Lucasfilm and soon took a leading role in the production of Star Trek II: The Wrath of Kahn, where several computer-generated sequences involve fractal landscapes. The best known is the Genesis planet transformation sequence. Carpenter now works at Pixar Studios (Barnsley 9-10). Another company, Digital Productions, also included fractal landscapes in their movie, The Last Starfighter (Barnsley 10). Fournier and Fussell went on to create programs for fractal landscapes similar to those of Carpenter, but are not well known for their images in Hollywood (Barnsley 9-10).
Mandelbrot and other mathematicians have shown through calculations, experiments, and research that fractals and fractal patterns are apparent in many aspects of our daily life. Fractals are an excellent way of describing plants in nature, such as ferns and broccoli. Also, geographic terrain, both on earth and other planets, can be represented and recreated with a high degree of accuracy using fractal programs on powerful computers. Some fractal-based structures, such as Brownian landscapes, have been used in science fiction movies. Finally, our own bodies are harbors for fractal designs, such as our hearts and even bones. Clearly, fractals are interesting from a mathematical standpoint, but also help us to learn more about our world and even ourselves.
Barnsley, M.F., et al. The Science of Fractal Images. Ed. Heinz-Otto and Dietmar Saupe.
New York: Springer-Verlag, 1988.
Hastings, Harold M. “Fractal.” The World Book Encyclopedia. 2002.
Lauwerier, Hans. Fractals: Endlessly Repeated Geometrical Figures. Trans. Sophia
Gill-Hoffstadt. New Jersey: Princeton University Press, 1991.
Liebovitch, Larry S. Fractals and Chaos Simplified for the Life Sciences. New York:
Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman
and Company, 1983.