A Path to Fractals

Nature is filled with a beauty easily perceived at many levels. Trees and forests, rocks and mountains, streams and mighty rivers, wisps of fog and great billowing clouds all contribute to an immediate, undeniable impression upon our visual sense. Although the impression may be hard to quantify, it is no less real.

Artists often use poetry, paintings, or photographs in an attempt to communicate this impression of beauty. All these forms share in that they are patterns — of words, pigment or light — which gain permanence as they are connected with ideas. In this way, mathematicians also make patterns with permanence.

To understand nature, we must also understand the language with which it speaks to us. Galileo said that this language is mathematics “its characters are triangles, circles and other geometrical figures.” The attempt to understand nature on the basis of mathematical patterns has a long tradition, which includes Pythagoras and Kepler. Though scientists since may have generally agreed with Galieo’s statement, they found the triangles and circles of Euclid’s geometry to be limiting. Mandelbrot acknowledged that:

clouds are not spheres,
mountains are not cones,
coastlines are not circles,
and bark is not smooth,
nor does lightning travel in a straight line …

and he provided an alternative with fractal geometry. Mandelbrot coined the word fractal from the Latin root fractus to capture the rough, broken and irregular characteristics of the objects he intended to describe. In fact, this roughness can be present at all scales, which distinguishes fractals from Euclidean shapes.

In the twenty-odd years since its introduction, fractal geometry has found a wide purview. Fractals seem to be everywhere, as the word fractal is now used to describe the universe and some of the very smallest things in it. Fractals, which belong to Galileo’s realm of “other geometrical figures,” add a very rich vocabulary for describing nature concisely. Now, the ruggedness of a coastline and the roughness of a rock can be quantified.


This calendar celebrates the deep symbiotic relationship between fractal geometry and computer graphics, which has grown as each nascent field has developed. The use of computer graphics for the study of fractals represents one of the first applications in the now popular realm of scientific visualization. As mathematical objects of infinite complexity, the detail of fractals can only be appreciated with the help of computers and this dependence represents the epitomy of scientific visualization’s goal to see the unseen. Apart from fractals with geometric constructions, the computer is also essential in the study of systems which exhibit fractal behavior over time. Here, computer-generated fractal images provide a glimpse of nature not so easily seen, one filled with deterministic chaos. Many physical systems are governed by deterministic laws, yet behave unpredictably. The unpredictability comes from sensitive dependence on initial conditions — two states which begin arbitrarily close together may become arbitrarily far apart after a finite period of time. Though the rules which define them are exceedingly simple, the complexity of shape is derived from repetition of those rules at a level that only the computer can manage. Computer graphics allows the communication of this complexity, with an eloquence not possible through words alone. Surprisingly, even these abstract images can appear organic and natural.

Many of those who know of fractals were introduced to them through this application of computer graphics. With the increase of available computing power, the rewards and dangers of exploring the Mandelbrot set have become more evident. Arthur C. Clarke, in Ghost of the Grand Banks, speculates that in the future, Mandelmania will become a serious affliction for many.

A sense of fractals can be gained without the computer, as nature abounds with forms possessing fractal characteristics. Although the largest and smallest examples of these cannot be seen by the naked eye, many remain easily accessible. More formally, fractals can be seen as abstractions of the natural qualities found in mountains, trees, and clouds. This process of abstraction leads to algorithms which allow computer graphics to produce astounding forgeries of nature that capture our impressions.

One hopes that the result of the computer graphics process, approached from either of the above perspectives, is a pleasing picture. Each image may be created with a different purpose in mind, but ultimately the pictures are a means for effective communication of the material. And if the picture is pretty enough, it may even help to convince some of the beauty in mathematics. After seeing computer-generated pictures of fractals, some people will ask if there are any real uses of fractal geometry. In this case, it is important to emphasize the connection of the images to the reality just beyond the computer screen.

There are many examples of the struggle to evaluate and apply the new idea of fractals and acceptance has not been unanimous. Mandelbrot gave a tentative mathematical definition for his discovery, but many have been concerned by its lack of precision. Then again, who could argue with the beautiful images, produced from simple equations, like the famous:

Zn+1 = Zn2 + C

which defines the Mandelbrot set.

Natural objects possess detail at several levels but manufactured objects generally do not. In becoming focused on or bound to a particular scale, an important connection with nature is lost. Self-similarity is the key to much of the detail in natural forms. Similarity, in a mathematical sense, means that objects may differ only in size or position but not shape. Self-similarity means then that parts have the same shape as the whole. Fractals may also possess the self-similarity in a statistical or approximate sense, as many natural objects do. In its simplest form, fractal dimension can be said to measure the degree of this similarity.

Experience has shown that it is difficult to be precise when defining fractals. Mandelbrot’s original definition based solely on dimension was considered too narrow, but Falconer has tried to capture the essence of what it is to be fractal. He has done this by adopting the approach of the biologist who would define life — most living things possess certain properties, most of the time. In the case of fractals, these properties are:

The connection to the definition of life seems appropriate, since fractals have been linked at many levels with nature and life. Even the human body exhibits fractals in its most basic functions, both spatially and temporally: in the circulatory system, the respiratory system, and at the cellular level across nerve membranes. It is remarkable that fractal geometry, born in an abstract mathematical universe, has come to have such an impact in our own physical one.

Although one may appreciate equally the beauty of the Mandelbrot set and a mountainous landscape, it is not always easy to see a unifying theme if it even exists. One may understand the recipes for generating images of the Mandelbrot set, but it is quite another matter to see fractals when looking away from the computer screen. It is revolutionary to think that fractals really are much more than the pictures which represent them and it may be more reasonable to view fractals only as isolated curiosities. Is fractal geometry a set of techniques for computer graphics or a new way to look at the world? Many of those involved with fractal geometry advocate the latter, but in order to arrive there from the former position, a significant transition must take place. Several paths may be followed, for example:

Once the process has begun, it becomes possible to develop an eye for fractals.

Although fractals are used mainly in physics, they have appeared of late in such diverse areas as aesthetics and psychology, indicating again that fractals are a very powerful idea. As researchers have struggled with the implications of fractals and chaos, so too have philosophers and artists. Just as the flapping of a single butterfly’s wings may affect global weather patterns, the actions of any individual may have far-reaching implications. These changes in world view are already being echoed in the arts. Historically, the introduction of a new geometry has been inspirational for artists. Although fractal geometry is significantly different from Euclid’s, the challenge may be to understand how these different puzzle pieces fit together.

Fractal art is in a unique position since computers wait to become a widely accepted artistic medium. The procedural, or algorithmic nature of fractal art places it at a crossroads between mathematics and art — a fertile ground waiting to be fully explored. But more than just in visual arts, fractals are being actively explored in books, television, motion pictures, dance, theatre and music. In fact, a great deal of music exhibits the fractal distribution of 1/f noise over time, a sort of middle ground between unstructured white noise and highly structured brown noise. It contains enough structure to be regular, and enough variation to be surprising. An early example of fractally-informed music with computer-generated fractal imagery was Mandelbrot and Wuorinen’s experimental performance entitled New York Notes.

Fractals, as a language of nature, give a new understanding of the finite world around us and provide a glimpse of infinity. Represented in this calendar is a collection of exciting and diverse fractal images. The Mandelbrot set is depicted in some interesting new ways. Both deterministic and random fractals are represented — some closely related to their mathematical roots. I hope you will be encouraged to explore fractals in new ways.