What Is a Fractal? How They Work in the Real World

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What is a fractal? How are fractals useful in representing and predicting relatively unpredictable events?

A fractal is a geometric pattern that repeats at different scales. Fractals, unlike pure geometric shapes like triangles or circles, are seen quite frequently in nature.

We’ll cover what a fractal is and how it can help us make predictions in a world full of uncertainty.

What Is a Fractal?

Developed by mathematician Benoît Mandelbrot in the 1970s, “fractals” are Mandelbrot’s coinage for geometric patterns that repeat at different scales. Fractals, unlike pure geometric shapes like triangles or circles, are seen quite frequently in nature. For example, a leaf’s veins look like little branches, and a tree’s branches look like little trees: the basic shape of the tree is echoed at the smaller scales.

So can the phenomena of Extremistan be modeled—and thereby predicted—at all?  Not precisely; but they can be approximated—by the use of fractals.

What is a fractal, and how is it useful in predicting unpredictable events? The reason fractals are helpful in representing Black Swans is that their internal ratios stay constant across scales. Unlike bell curves, in which ratios decline at accelerating rates the further one gets from the average, fractals exhibit no (or mild) acceleration. They obey power laws, which describe a functional relationship between two quantities in which the relationship remains constant no matter what the initial size of the quantities.

Take the European wealth example noted just above. As the wealth number doubled, the incidence decreased by 4x (wealth greater than 1 million: 1 in 62.5; wealth greater than 2 million: 1 in 250; wealth greater than 4 million: 1 in 1,000; etc.). The “power” in this relationship is 2, because 2^2 (2 squared) is 4.

Now, imagine the power was 1: Each time the wealth doubled, the incidence would only decrease by 2x (2^1=2). The probability of extraordinary wealth would increase. This helps us understand why fractals are useful and it helps us answer the question, What is a fractal?

For Extremistan phenomena, the power laws aren’t known with any certainty, but they can be approximated. Say, for example, you want to assess the risk of a stock portfolio, and you know that, based on past data, the worst-case scenario is a -5% move once every 2 years. With a power of 2—remember that the power is estimated—you can assume that a -10% move will happen once every 8 years (2^2=4) and a -20 percent move will happen once every 32 years. (For scale, in the 1987 crash, the U.S. stock market lost almost 23% of its value, an impossibility according to Gaussian economic models.) With fractal/power-law distributions, a 1000-year flood can become a 100-year one, and a 100-year one can become a 10-year one.

What Is a Fractal? How They Work in the Real World

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Amanda Penn

Amanda Penn is a writer and reading specialist. She’s published dozens of articles and book reviews spanning a wide range of topics, including health, relationships, psychology, science, and much more. Amanda was a Fulbright Scholar and has taught in schools in the US and South Africa. Amanda received her Master's Degree in Education from the University of Pennsylvania.

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