We may not know it, but fractals, like the air we breathe, are all around us. Their irregular, repeating shapes are found in cloud formations and tree limbs, in stalks of broccoli and craggy mountain ranges, even in the rhythm of the human heart.
In this documentary film, NOVA takes viewers on a fascinating quest with a group of maverick mathematicians determined to decipher the rules that govern fractal geometry.
For centuries, fractal-like irregular shapes were considered beyond the boundaries of mathematical understanding. Now, mathematicians have finally begun mapping this uncharted territory.
Their remarkable findings are deepening our understanding of nature and stimulating a new wave of scientific, medical, and artistic innovation stretching from the ecology of the rain forest to fashion design.
The documentary highlights a host of filmmakers, fashion designers, physicians, and researchers who are using fractal geometry to innovate and inspire.
Produced and directed by Emmy and Peabody Award winning filmmakers Michael Schwarz and Bill Jersey, the documentary weaves cutting edge research from the front lines of science into a compelling mathematical detective story. The film introduces a number of distinguished individuals who have used fractal geometry to transform their fields, like Loren Carpenter, who created the first completely computer-generated sequence in a movie.
In the late 1970s, Carpenter stumbled across the work of a little-known mathematician, Benoit Mandelbrot, who coined the word “fractal,” from the Latin word fractus, meaning irregular or broken up. Based on Mandelbrot’s mathematical descriptions of fractals in nature, Carpenter was able to create detailed computer simulations of organic forms in a way that had never before been possible. The groundbreaking computer generated sequence Carpenter produced in 1980 for Star Trek II: The Wrath of Khan marked a milestone in movie history, and owed its creation to fractal geometry.
It took a maverick with a hard-won aversion to authority to stand up to the conventional wisdom that nature stood outside the bounds of mathematics. Through interviews and personal artifacts, Mandelbrot shares the story of his struggle to survive as a Jewish teenager in Nazi-occupied France, his journey to America, and his lifelong fascination with a corps of European mathematicians whose explorations of the so called “mathematical monsters” laid the groundwork for his own discoveries.
Filmmaker Bill Jersey believes Mandelbrot’s approach to fractals might ultimately become as significant as the cracking of the genetic code. “As fractals continue to revolutionize the way scientists develop theories and conduct research, the inevitable results will be innovations that dramatically change health care, environmental policy, design, and technology,” Jersey says.
In 1980, Mandelbrot published a mesmerizing image known as the Mandelbrot set. (To explore the set, go to A Sense of Scale.) The intricate, mysterious beauty of this image, which was generated by a single mathematical function, won him acclaim from an unexpected quarter in the world of popular culture. But fractals are more than pretty pictures. Almost all living things distribute nutrients through their bodies via branching networks, such as systems of blood vessels, that obey the rules of fractal geometry.
In Toronto, physicist Peter Burns is making a mathematical model of blood vessels to find ways to diagnose cancer earlier than is now possible. In Boston, cardiologist Ary Goldberger has discovered that, contrary to centuries of belief, a healthy human heartbeat does not have an even pattern like a metronome but rather a jagged, variable fractal pattern. This discovery that may help doctors diagnose cardiac disease before damage is done.
“This film is about looking at the world around us in a completely different way,” says producer Michael Schwarz. “If you pay attention, you can see that fractals appear throughout nature. But until Benoit Mandelbrot came along, no one really understood what was there all along.”
A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetryor evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge. Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set Fractals also include the idea of a detailed pattern that repeats itself.:166; 18
Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). But if a fractal’s one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal’s topological dimension.
As mathematical equations, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.:15:48
The mathematical roots of the idea of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term “fractal” was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning “broken” or “fractured”, and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.:405
There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as “beautiful, damn hard, increasingly useful. That’s fractals.” The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal.
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