It's virtually impossible to determine the exact length of a natural coastline. The problem is of knowing where to stop—or if a stopping point even exists. To completely measure the coastline, one would have to find the length of each irregularity, from the large curves of bays and inlets, to the medium-sized jagged outgrowths within those curves, to the tiniest sand grains within those jags. As the units of measurement shrink, the coastline's length approaches some incalculable number. We can say that it's undefined.
Natural coastlines are fractals—shapes that repeat patterns on a countless number of diminishing scales until the length of the whole becomes near impossible to measure. New patterns exist at every scale, making the entire shape more complex. This is known as the coastline paradox.
Theoretical fractals, which mathematicians generate on computers, iterate perfect patterns infinitely, while examples in nature iterate rough patterns that eventually approach some endpoint—just one that's maddeningly difficult to pin down.
Finding fractals in nature is almost disappointingly easy. Zoom in on a tree branch, and that branch resembles the whole tree. Zoom in on an offshoot of the branch; it also resembles the tree. Repeat until you hit the smallest twig.
Finding fractals in the built environment is harder. For the most part, our buildings, parks and transit systems are neat and modular. With our love of uniformity, we humans find fractals strange and marvelous; they belong outside our realm of order. But nature has a different sense of order: one in which fractals dominate and linearity across scales is scarce.
We have more to gain from emulating nature's fractals than admiring them from afar. By incorporating many scales, fractals are often more resilient and adaptable than things built with uniform regularity. They are organized, but with a degree of random variation that allows for flexibility.
Here are some ways we could use "fractal thinking" to our advantage:
1) STREET SYSTEMS
Transportation networks with fractal layouts may allow cities to expand efficiently, thereby relieving high population densities, researchers at the Santa Fe Institute found recently.
The authors sampled 105 cities and calculated their street networks' fractal dimensions, a measurement of "fractal-ness." To do this, they overlaid grids on road maps of each city, decreasing the size of the grid's boxes until each box had a least part of one road in it. Then they counted the number of boxes with a road—a process called the box counting dimension method.
Maps with streets that branched into smaller and smaller patterns needed more boxes, and therefore had a higher fractal dimension. The researchers found that more densely developed cities, like New York and Tokyo, had higher fractal dimensions.
The key here is taking advantage of multiple scales, the authors believe. Mixed road layouts maximize use of space, the same way filling up a jar with a combination of big rocks, medium-sized pebbles, and sand grains fills up more space than a collection of pebbles that are all the same size.
Roads of many sizes allow for more efficient traffic flow. A network of large highways branching into smaller secondary roads, then tertiary roads, and then neighborhood streets can serve a more dispersed population than a network of roads that are all the same size. Incorporating fractal streets into city layouts could optimize how overpopulated cities grow, facilitating dispersal from a dense urban core into less dense areas.
2) IMAGE COMPRESSION
Computers can take advantage of recurring fractal patterns to compress images. A fractal compression program identifies large image patterns and then searches for those patterns on smaller and smaller scales, using these redundancies to reduce space. To recover the original, the program does this iterative process in reverse.
Fractal compression can produce images that are less pixelated than its main competitor, JPEG compression, because it can fill in details on smaller scales without depending on image resolution. Still, JPEG compression is much more commonly used today, partly because the benefits of fractal compression are not necessarily worth its longer compression time.
3) ART APPRECIATION
Research shows we find fractals aesthetically pleasing. This might explain why we love gazing at natural landscapes, which are chock-full of fractals. Scholars think we are visually drawn to fractals because we enjoy images that are complex and interesting, but underscored by some degree of coherency. Preferring coherent but complex environments might have even given an evolutionary benefit to early humans, who thrived in flat, homogenous grasslands scattered with trees.
But we don't just love gazing at trees and mountains. Scientists think we also like manmade art with fractal patterns. Looking at the preferences of more than 200 participants without artistic training, researchers found that people enjoyed works of art with fractal complexity, which they measured using the fractal measuring techniques described above.
People have explored the visual appeal of fractals in all sorts of applications, ranging from art therapy to furniture design to art analysis. Computer programs mimicking human cognition can actually incorporate fractal algorithms to accurately categorize paintings by artistic movement—for instance, grouping together High Renaissance Artists like Raphael, Leonardo da Vinci, and Michelangelo. In some cases, these programs can even make connections beyond human detection. In 2012, computer scientists found a higher degree of similarity between Vincent van Gogh's and Jackson Pollock's paintings than between those of van Gogh and impressionist painters traditionally thought to be similar to van Gogh, such as Monet and Renoir.
In fact, analysts have claimed they've found fractals in both van Gogh and Pollock's paintings. In van Gogh's paintings, large swirls that give way to smaller eddies display fractal turbulence with a surprising degree of mathematical logic (you can watch a great animation about the math of turbulence in van Gogh's paintings here). Using the box counting method described in the first example, researchers have found that the writhing splotches in Pollock's drip paintings contain patterns that repeat themselves at different levels of magnification. Furthermore, these researchers claimed, fractal analysis could even be used to discern authentic Pollock paintings from forgeries, although this is disputed.
4) MEDICAL DIAGNOSIS
All of the examples we've looked at so far are visual fractals. But fractals can occur over time as well as space—one example is how hearts beat across time. Robust hearts have fractal heartbeats, according to Ary Goldberger, a professor at Harvard Medical School. Using graphs of heart rate time series (like the 30-minute time series show below), he quantified the "fractal-ness" of heartbeats using a method called detrended fluctuation analysis, which identifies similarities in curves across different scales.
He found that heartbeats that are either too uniform or too irregular correspond to problems like congestive heart failure or irregular heart flutters. On the other hand, healthy heartbeats demonstrated fractal complexity, which Goldberger believes indicates a healthy ability to respond to stressors. He thinks doctors could eventually use fractal analysis to help diagnose heart problems.
Maintaining an overall fractal rhythm might allow hearts to store and organize the different rhythms they need to adopt in different scenarios. For instance, our hearts must adapt their behavior, across a wide range of time scales, to variables like stress, exercise, and disease. Goldberger suggests that a fractal pattern allows for regularity but also incorporates variability to help our hearts adapt quickly to these stressors. Without this fractal organization, hearts might lock themselves in one mode and narrow their range of responsiveness.
Goldberger believes the same logic can be applied to other medical problems. In general, he observes, declining health corresponds with a breakdown in fractal complexity. For example, neurons and blood vessels show a loss of branching with aging and disease. And studies of other diseases have also found that temporal fractal patterns might allow for adaptive variability. For instance, researchers have found that people with Parkinson's disease and epilepsy show decreased fractal complexity in their walking patterns and brain waves respectively.
5) THE STOCK MARKET
It may seem like sudden spikes and falls in the stock market are anomalous flukes, but they happen all too often to just be random, according to Benoit Mandelbrot, a mathematician who is often called "the father of fractals." Based on his belief that market fluctuations follow fractal geometry, he has created fractal-based financial models that better account for extreme events than traditional portfolio theory, which is based on a normal bell curve. Furthermore, these models can be applied to any timescale, from years to hours.
Many researchers have corroborated Mandelbrot's theory by successfully characterizing stock market behaviors using detrended fluctuation analysis, the fractal-measuring method discussed in the previous example. Furthermore, fractal behavior in stock markets seems largely universal—with studies having been done in many countries, including the United States, Brazil, and China. However, most of this research applies fractal analysis to past stock market events, or as a tool to assess how efficient stock markets are over time.
Scientists have yet to develop a model that can reliably predict crashes before they happen. While researchers have used fractal-based models to successfully predict events like earthquakes, it is much harder to apply those models to financial systems. This is largely because stock markets are more complex, constantly adapting, and not bound by the same physical and geological constraints as earthquakes. Nonetheless, an entire new field of science, called econophysics, has emerged with the hope that building off fractal-based earthquake models will help economists better understand stock markets.