Fractal Guide

Fractals are very visual and mathematical based constructs. They are never-ending patterns that result from a geometric shape that is created over and over again.

The amazing thing about fractals is that when divided into parts, each section appears as a smaller replica of the entire fractal shape. That means a key property of fractals is self-similarity, a mathematical concept where an object is exactly or very close in similarity to one or more of its parts. Because the pattern is built on the same shape throughout, a fractal can look the same no matter how long or closely you zoom in on it. In other words, it is a scale-free pattern.

Fractals Origin and History​

The history of fractals first began in the 19th century with the mathematician Gaston Julia (1873-1978) who discovered the first concepts of “worlds within worlds” found in fractals in nature.

However, it wasn’t until the 20th century and the availability of computer technology that fractals were first accurately described as “fractal geometry” and seen as a revolution in mathematics.

The mathematician Benoit Mandelbrot (1924-2010), at one time deemed the “father of fractals” by a mathematics scientist, may have been the first person to accurately put fractals into words and to use the actual term “fractal”, which is based on the concept of fractional dimension.

It is adapted from the Latin word “fractus” which means “fractured” or “broken”. This term makes sense because each piece of the fractal pattern can be broken off and still resemble the larger fractal.

Mandelbrot built upon the previous work of Gaston Julia, this time with the benefit of newly developed computers at IBM, where he worked for 35 years of his life. This new computer technology allowed Mandelbrot to plot the patterns and create the graphics for a fractal with computer graphics code, revolutionizing the way we look at fractals.

Mandelbrot’s work began a rise in new discoveries flowing from the visual field of mathematics.

How​ Fractals Are Created

To better understand how fractals are created, here’s a basic demonstration of their structure.

Picture a basic, three-pointed triangle. Then, in the middle of each flat side of that triangle, add an equilateral triangle with sides that are ⅓ of the original triangle’s side. Then, in the middle of each flat side of that new shape, add a triangle that is ⅓ of the new shape’s sides (or 1/9 the size of the original triangle’s sides)—and so on, and so on, until you reach your desired fractal shape. On a computer, this process continues and continues.

This method - taking a shape and then populating it with smaller versions of the original shape - is the most general way to create and understand a fractal.

Fractals in Nature

When Mandelbrot first began studying fractals and before computer technology was available to create them, he was able to predict the complexity involved by using mathematics.

His first phase of research began with fractals in nature, where there are many examples. Natural fractals can appear as spirals, trees, stripes, cracks, symmetries, bumps and more!

Examples of fractals in nature include:

  • Lightning branches
  • Bumps on broccoli
  • Mountain ranges
  • Coastline
  • DNA
  • Snowflakes
  • Leaves
  • Heart rates
  • Ocean waves

Let’s explore a few of these fractals found in nature.


The equiangular or logarithmic spiral happens in many systems in our universe and at large, amazing scales.

The largest known examples we have of spiral fractals are galaxies, and the largest spirals on planet Earth are cyclones and hurricanes.

In plants, spiral fractals are found in flowers, cacti, pine cones, and vegetables. A great example of spiral fractals in nature is found in the Romanesco broccoli as shown below.

Romanseco broccoli displaying spiral and bump fractals

The repeating patterns and shapes look almost computer generated. Yet the wonders of nature’s creativity are at play here and demonstrate fractal design.

The nautilus shell is another great example of spiral fractal design. As the organism continues to expand its home through the simple, repetitive process of adding sections to its shell, a spiral is maintained.

Nautilus shell spiral fractal construction


A lightning bolt, formed within microseconds, contains a natural fractal pattern, as it travels in a jagged path rather than a straight line. Scientists are able to duplicate this fractal pattern by create small-scale lightning in laboratories using a particle accelerator.

Lightning Bolt Showing Fractal Design

Lightning Bolt


The Earth’s river networks are formed by rain falling and then flowing downhill. This causes erosion and creates a small channel, which gets carved deeper each time there is rainfall. As you can see, rivers are formed by repeating a process again and again—the fractal process!

Fractal River Branches

River branches displaying fractal designs


Besides being in the world all around us, we even have fractals inside of us! Many of the internal structures and organs of our bodies have fractal properties.


If you look at our lungs upside down, they have the same branching pattern as trees. It makes sense that they would have similar structures because both our lungs and trees are meant to perform the function of respiration—breathing!

Blood Vessels

Like rivers, blood vessels contain a fractal branching network in which the blood vessels branch out smaller and smaller, all the way down to an 8 micron-diameter to the width of a capillary. This allows each cell of our bodies to be close enough to the vessels to get necessary nutrients and oxygen.


The same brain you’re using to read and understand this article contains its own sets of fractals! In fact, its fractal structure is necessary for allowing each neuron’s sections to communicate with all the other brain cells.

 Brain Neurons Showing Fractal Patterns

Brain neurons showing fractal patterns

Fractals in Mathematics

Besides the natural world, fractals have been implemented in artistic spaces through creations like mandalas, overlapping patterns, and Jackson Pollack paintings.

They are also used in different technology applications today. Models of fractals are generated with specific software that uses the creation process of fractals. The two most famous patterns of fractals are the Mandelbrot set and Julia set, named after Benoit Mandelbrot and Gaston Julia respectively.

Each are easily generated on the computer, yet are more complicated to describe. Both sets are based on the multiplication of complex numbers and use the same formula—but apply it in different ways.

Get ready—we’re about to get more heavily into the mathematics!

The Mandlebrot Set

The Mandelbrot set is built using an algorithm based on the formula zn+1 = zn2 + c.

A different number c represents each point on the plane. For each single point, these steps are repeated:

  • An infinite sequence of numbers is created is this pattern: Begin with z0. For each new number, square the previous number, plus c.
  • If the number sequence always increases and tends to infinity, the point is coloured white. But if the sequence doesn’t increase past a certain amount (meaning it is bounded), the point is coloured black.

The process is repeated for each point on the coordinate, and all of the black points equals the Mandelbrot set.

Mandlebrot Set Fractal Image

Example of a Mandelbrot set fractal

The Julia Set

The Julia set is closely connected with the Mandelbrot set and uses the same function when created.

To make a picture of a Julia set, each point c on the complex plane must be constant during the process while z0 varies. The shape of the set is determined by the value of c.

A Julia set is connected if it’s associated with a point in the Mandelbrot set, while otherwise it’s disconnected. In contrast, the Mandelbrot set is always a single connected piece.If this all sounds like a foreign language to you, don’t worry! It’s not necessary to understand the full mathematics of fractals in order to appreciate them. The cool thing about these complex patterns is that they aren’t exclusive to those just interested in math. In fact, they may have emotional or state of mind benefits as well as you’ll see.

Julia Set Fractal

Example of a Julia set fractal

If we take the fractal patterns generated by the Julia set, add some colourful blends and surface texture effects, you can see from the image below how digital fractals can become a mesmerizing piece of art.

Digital Fractal

Colourful digital fractal

Fractals May Promote Stress Reduction

Fractal uses are not exclusive to mathematics and physical processes.

Studies that have been conducted using eye tracking equipment, fMRI imaging, and other brain measurements show that response to fractal patterns within nature can reduce stress levels by up to 60%! This is likely due to a physiological resonance in the eye.

It has also been shown through some research that certain artwork, landscapes, and architectural designs containing fractal patterns can cause a relaxation effect. The self-similar patterns are soothing. This is worthy for further investigation because it could be useful in stress reduction in both personal living environments and workplaces.

Just think about those basic and repetitive pictures of nature you see on the walls of exam rooms in doctors’ offices. The scenery often incorporates simple images of mountains, rivers, coastlines, or lightning, right? Perhaps there is a deeper reason we choose images like these for a calming effect: their fractal patterns!

For those looking to reduce stress and increase happiness, fractals can easily be used to your own advantage this way—and it’s as simple as going for a walk in the park or watching for patterns in the clouds.

Try sitting in a fractal-rich place for around 20 minutes each day and notice the difference in your stress levels before and after. No matter how you use them, fractals are a fascinating part of our lives, our world, and our universe, continuing the process of life as we know it while connecting us all through nature, technology, and wellness.

Please check out my other guides on Kaleidoscopes and Mandalas.