Fractals can be better understood by the well-known paradox about the length of Britain’s coastline. Basically, this paradox lies in the fact that the length of the British coastline is determined by the units and specificity in measuring it. For instance, measuring in mile-lengths excludes the curvatures in the landscape that are, say, yards long and thus will result in a smaller value than if these curvatures are included. Measuring in yard-lengths ignores smaller variations along the edges of the shore, and will result in a smaller value than a measurement that includes them, and so on. The smaller the unit of measure, the closer the fit to the coastline and the more accurate the measurement. But at what point does this conundrum of measurement end? At what point does a unit yield a completely accurate measurement? Theoretically, never, and that is why Mandelbrot considered coastlines to be fractal.
The applications of fractals in technology are surprising, varied, and powerful.
In 1978, Loren Carpenter was working with experimental aircraft engineers to simulate a flying airplane. He wished to put mountains behind the airplane to make the animation more realistic, but mountains involved thousands of hand-drawn frames. Inspired by Mandelbrot’s book Fractals: Form, Chance and Dimension, Carpenter was the first to use fractal geometry in animation. He simply began with jagged triangles, and then divided them multiple times into smaller and smaller parts, thus creating fractals. The result was amazingly realistic digital imagery of a quality that had never been seen before: a breakthrough in computer animation.
In the 1990’s, Nathan Cohen, a radio astronomer, discovered that fractal shapes make very effective antennas. After attending a conference where Mandelbrot was a speaker, Cohen had the idea to shape antennae like fractals. What he discovered was that fractal-shaped antennas were both smaller in size and picked up more frequencies than a regular antenna. He attributed this to the self-similarity of fractals, and mathematically proved that fractals are the only shapes that pick up greater frequency ranges. Virtually every cell phone in use today has fractal antennas to detect different signals like Bluetooth and Wi-Fi. This was a major advancement in communications.
There is another essential aspect to fractal science. Fractals had been recognized in nature by scientists for years, however they were regarded as non-mathematical until fractal geometry allowed scientists and mathematicians to understand and perform calculations on “rough edges” in addition to smooth curves. This was an extraordinary development. Finally natural structures and systems could be understood through the eyes of math and science.
The medical industry has benefited greatly by applying fractal math. For instance, cardiologist Ary Goldberger discovered that variations in heartbeat intervals, when plotted, formed a particular, repeating fractal edge. With this knowledge, doctors may be able to detect unhealthy hearts before serious problem arise. Biophysicist Peter Burns discovered that the overall movement of blood conforms to a fractal pattern of tiny blood vessels, and irregularities in the pattern can signify small tumors. Like the heartbeat, this could allow doctors to identify tumors very early in their growth.
Fractals provide a multi-dimensional key to the natural world; they can be applied on many different levels. Brian Enquist traveled to Costa Rica to conduct a remarkable forest experiment. How can we know how much carbon dioxide a particular area of forest can absorb? Enquist hypothesized that that one tree in a forest could reflect the structure of the whole forest. In other words, the forest system and the structure of each tree are the same fractal pattern. Enquist selected one tree in the forest and measured its branches. Then, his team measured the diameters of the trees and found that the distribution of branch sizes on one tree was comparable to the tree trunks in the forest. This means that the amount of carbon dioxide absorbed by one tree can indeed be used to estimate the capacity of the forest.
Unlocking this incredible fractal relationship between the part and the whole leads to so many exciting implications for science, technology, mathematics because the fractal is essentially a natural pattern with real-world application.
I loved Hunting the Hidden Dimension. It contained all of the mathematics that I had hoped Between the Folds would include. I also found that it was much more informative than Between the Folds. Where Between the Folds piqued my interest in the possibilities of the math and techniques therein, Hunting the Hidden Dimension provided enough substance to be genuinely exciting!
I was truly captivated by fractals and their amazing potential, particularly in technology. If fractals can be used to make computer and communication technology more compact while allowing us to further our understanding of nature, they may play a major role in the pursuit of clean and efficient energy systems. With environmental crisis at hand, easy, simple, and sensitive solutions are being demanded. Perhaps fractals can help us find these solutions to some of the biggest global issues of the day. This is definitely a topic that I would look into further!
