How Fractals Work

By: Craig Haggit  | 
Mandelbrot set
This partial view of the Mandelbrot set, possibly the world's most famous fractal, shows step four of a zoom sequence: The central endpoint of the "seahorse tail" is also a Misiurewicz point. Wolfgang Beyer/(CC BY-SA 3.0)

Fractals are a paradox. Amazingly simple, yet infinitely complex. New, but older than dirt. What are fractals? Where did they come from? Why should I care?

Unconventional 20th century mathematician Benoit Mandelbrot created the term fractal from the Latin word fractus (meaning irregular or fragmented) in 1975. These irregular and fragmented shapes are all around us. At their most basic, fractals are a visual expression of a repeating pattern or formula that starts out simple and gets progressively more complex.


One of the earliest applications of fractals came about well before the term was even used. Lewis Fry Richardson was an English mathematician in the early 20th century studying the length of the English coastline. He reasoned that the length of a coastline depends on the length of the measurement tool. Measure with a yardstick, you get one number, but measure with a more detailed foot-long ruler, which takes into account more of the coastline's irregularity, and you get a larger number, and so on.

Carry this to its logical conclusion and you end up with an infinitely long coastline containing a finite space, the same paradox put forward by Helge von Koch in the Koch Snowflake. This fractal involves taking a triangle and turning the central third of each segment into a triangular bump in a way that makes the fractal symmetric. Each bump is, of course, longer than the original segment, yet still contains the finite space within.

Weird, but rather than converging on a particular number, the perimeter moves towards infinity. Mandelbrot saw this and used this example to explore the concept of fractal dimension, along the way proving that measuring a coastline is an exercise in approximation [source: NOVA].

If fractals have really been around all this time, why have we only been hearing about them in the past 40 years or so?


Fractal Terminology

Mandelbrot set
In the Mandelbrot set, points remaining finite through all iterations are shown white; values diverging to infinity are shown darker. Encyclopaedia Britannica/Contributor/Getty Images

Before we get into any more detail, we need to cover some basic terminology that will help you understand the unique qualities that fractals posess.

All fractals show a degree of what's called self-similarity. This means that as you look closer and closer into the details of a fractal, you can see a replica of the whole. A fern is a classic example. Look at the entire frond. See the branches coming out from the main stem? Each of those branches looks similar to the entire frond. They are self-similar to the original, just at a smaller scale.


These self-similar patterns are the result of a simple equation, or mathematical statement. Fractals are created by repeating this equation through a feedback loop in a process called iteration, where the results of one iteration form the input value for the next. For example, if you look at the interior of a nautilus shell, you'll see that each chamber of the shell is basically a carbon copy of the preceding chamber, just smaller as you trace them from the exterior to the interior.

Fractals are also recursive, regardless of scale. Ever go into a store's dressing room and find yourself surrounded by mirrors? For better or worse, you're looking at an infinitely recursive image of yourself.

Finally, a note about geometry. Most of us grew up being taught that length, width and height are the three dimensions, and that's that. Fractal geometry throws this concept a curve by creating irregular shapes in fractal dimension; the fractal dimension of a shape is a way of measuring that shape's complexity.

Now take all of that, and we can plainly see that a pure fractal is a geometric shape that is self-similar through infinite iterations in a recursive pattern and through infinite detail. Simple, right? Don't worry, we'll go over all the pieces soon enough.


Before They Were Fractals

Katsushika Hokusai used the fractal concept of self-similarity in his painting "The Great Wave Off Kanagawa" the early 1800s. Public Domain

When most people think about fractals, they often think about the most famous one of them all, the Mandelbrot set. Named after the mathematician Benoit Mandelbrot, it's become practically synonymous with the concept of fractals. But it's far from being the only fractal in town.

We mentioned the fern earlier, which represents one of nature's simple and limited fractals. Limited fractals don't go on indefinitely; they only display a few iterations of congruent shapes. Simple and limited fractals are also not exact in their self-similarity — a fern's leaflets may not perfectly mimic the shape of the larger frond. The spiral of a seashell and the crystals of a snowflake are two other classic examples of this type of fractal found in the natural world. While not mathematically exact, they still have a fractal nature.


Early African and Navajo artists noticed the beauty in these recursive patterns and sought to emulate them in many aspects of their everyday lives, including art and town planning [sources: Eglash, Bales]. As in nature, the number of recursive iterations of each pattern was limited by the scale of the material they were working with.

Leonardo da Vinci also saw this pattern in tree branches, as tree limbs grew and split into more branches [source: Da Vinci]. In 1820, Japanese artist Katsushika Hokusai created "The Great Wave Off Kanagawa," a colorful rendering of a large ocean wave where the top breaks off into smaller and smaller (self-similar) waves [source: NOVA].

Mathematicians eventually got in on the act as well. Gaston Julia devised the idea of using a feedback loop to produce a repeating pattern in the early 20th century. Georg Cantor experimented with properties of recursive and self-similar sets in the 1880s, and in 1904 Helge von Koch published the concept of an infinite curve, using approximately the same technique but with a continuous line. And of course, we've already mentioned Lewis Richardson exploring Koch's idea while trying to measure English coastlines.

These explorations into such complex mathematics were mostly theoretical, however. Lacking at the time was a machine capable of performing the grunt work of so many mathematical calculations in a reasonable amount of time to find out where these ideas really led. As the power of computers evolved, so too did the ability of mathematicians to test these theories.

In the next section, we'll look at the mathematics behind fractal geometry.


Math Behind the Beauty

A Julia set fractal is the boundary of the filled-in set (the set of "exceptional points"). There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust). Encyclopaedia Britannica/UIG Via Getty Images

We think of mountains and other objects in the real world as having three dimensions. In Euclidean geometry we assign values to an object's length, height and width, and we calculate attributes like area, volume and circumference based on those values. But most objects are not uniform; mountains, for example, have jagged edges. Fractal geometry enables us to more accurately define and measure the complexity of a shape by quantifying how rough its surface is. The jagged edges of that mountain can be expressed mathematically: Enter the fractal dimension, which by definition is larger than or equal to an object's Euclidean (or topological) dimension (D => DT).

A relatively simple way for measuring this is called the box-counting (or Minkowski-Bouligand Dimension) method. To try it, place a fractal on a piece of grid paper. The larger the fractal and more detailed the grid paper, the more accurate the dimension calculation will be.


D = log N / log (1/h)

In this formula, D is the dimension, N is the number of grid boxes that contain some part of the fractal inside, and h is the number of grid blocks the fractals spans on the graph paper. However, while this method is simple and approachable, it's not always the most accurate.

One of the more standard methods to measure fractals is to use the Hausdorff Dimension, which is D = log N / log s, where N is the number of parts a fractal produces from each segment, and s is the size of each new part compared to the original segment. It looks simple, but depending on the fractal, this can get complicated pretty quickly.

You can produce an infinite variety of fractals just by changing a few of the initial conditions of an equation; this is where chaos theory comes in. On the surface, chaos theory sounds like something completely unpredictable, but fractal geometry is about finding the order in what initially appears to be chaotic. Start counting the multitude of ways you can change those initial equation conditions and you'll quickly understand why there are an infinite number of fractals.

You won't be cleaning the floor with the Menger Sponge though, so what good are fractals anyway?


Practical Fractals

After Mandelbrot published his seminal work in 1975 on fractals, one of the first practical uses came about in 1978 when Loren Carpenter wanted to make some computer-generated mountains. Using fractals that began with triangles, he created an amazingly realistic mountain range [source: NOVA].

In the 1990s Nathan Cohen became inspired by the Koch Snowflake to create a more compact radio antenna using nothing more than wire and a pair of pliers. Today, antennae in cell phones use such fractals as the Menger Sponge, the box fractal and space-filling fractals as a way to maximize receptive power in a minimum amount of space [source: Cohen].


While we don't have time to go into all the uses fractals have for us today, a few other examples include biology, medicine, modeling watersheds, geophysics, and meterology with cloud formation and air flows [source: NOVA].

This article is intended to get you started in the mind-blowing world of fractal geometry. If you have a mathematical bent you might want to explore this world a lot more using the sources listed on the next page. Less mathematically inclined readers might want to explore the infinite potential of the art and beauty of this incredible and complex source of inspiration.


Originally Published: Apr 26, 2011

Fractal FAQ

What are fractal patterns?
Chaotic equations that form complex patterns that increase with magnification are known as fractals.
What is the most famous fractal?
The Mandelbrot set introduced by John Briggs is known to be the most famous fractal in modern mathematics, mainly because of its haunting beauty.
Where do you find fractals?
The world is full of complicated patterns called fractals. From the tiny, little patterns of seashells to the magnificent wonder of galaxies, they can be found in nature quite easily.
How are fractals used in real life?
Fractals are used to detect and capture the complexity of various structures. They are also used to analyze bacterial patterns and other biological processes.

Lots More Information

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  • Bales, Judy. "Thinking Inside the Box: Infinity Within the Finite." Surface Design Journal. Pages 50-53. Fall 2010.
  • Cohen, Nathan. "Fractal Antennas, Part 1." Communications Quarterly. Summer 1995.
  • Eglash, Ron. "African Fractals: Modern Computing and Indigenous Design." Rutgers Univ. Press. 1999.
  • Falconer, K. J. "The Geometry of Fractal Sets." Cambridge Tracts in Mathematics, 85. Cambridge, 1985.
  • Fractal Foundation. "Online Fractal Course." (April 17, 2011)
  • Mandelbrot, Benoit. "The Fractal Geometry of Nature." Freeman. 1982.
  • Mandelbrot, Benoit. "Fractals: Form, Chance, and Dimension" Freeman. 1977.
  • Mandelbrot, Benoit. "How Long is the Coastline of England?: Statistical Self-Similarity and Fractional Dimension" Science, New Series. Vol.156, no.3775. May 5, 1967.
  • NOVA. "Hunting the Hidden Dimension." PBS, 2008. Originally aired on Oct 28, 2008. (April 17, 2011)
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