At the heart of them lay the Mandelbrot set, which today has achieved fame even outside the field of mathematics. |

"Fractal geometry is not just a chapter of mathematics, but one that helps us see the same world differently."

Benoît Mandelbrot, 'The Fractal Geometry of Nature' (1982)

Antennas, fractal compression, understanding the stock market, and designing more efficient computers have all be possible and dramatically improved because of our understanding of fractals. Before French-American mathematician Benoit Mandelbrot launched his branch of fractal mathematics, our understanding of how things function and why things repeat themselves was limited. Whilst-contemplating algebra one day Mandelbrot claims he began seeing vivid, geometric images in his mind. He soon realized that he had actually transformed algebraic formulas and equations into the pictures he was seeing. However, he kept this discovery to himself, for he figured it would not help him in the long run. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton.

Modern mathematics began with Georg Cantor, a Russian mathematician. In 1883, he created the first mathematical monster: He formed a straight line and broke it into thirds, erasing the middle portion. He repeated this process time and time again, in an attempt to fathom the mathematical mystery. He soon discovered that the length between the lines grew infinitely small and appeared to approach zero. However, the length never actually shrank to zero, and the pattern continued to repeat itself. Mandelbrot realized that these were types of fractals. He also saw different uses for them other than just “Mathematical Monsters.”

A short while later, Mandelbrot stumbled upon another monster, the Julia set. Julia was named after the French mathematician Gaston Julia who studied it during the First World War. Gaston Julia tried numerous times to iterate an equation in a feedback loop; however, each time he plugged a number into the formula, he would get a new number. Unable to create an image, he gave up; it was too much for one man to accomplish. Luckily for Mandelbrot, he had computers to aid him in his problem solving and was able to graph Julia using an IBM computer. It was a breathtaking image.

Modern mathematics began with Georg Cantor, a Russian mathematician. In 1883, he created the first mathematical monster: He formed a straight line and broke it into thirds, erasing the middle portion. He repeated this process time and time again, in an attempt to fathom the mathematical mystery. He soon discovered that the length between the lines grew infinitely small and appeared to approach zero. However, the length never actually shrank to zero, and the pattern continued to repeat itself. Mandelbrot realized that these were types of fractals. He also saw different uses for them other than just “Mathematical Monsters.”

**Julia Set**A short while later, Mandelbrot stumbled upon another monster, the Julia set. Julia was named after the French mathematician Gaston Julia who studied it during the First World War. Gaston Julia tried numerous times to iterate an equation in a feedback loop; however, each time he plugged a number into the formula, he would get a new number. Unable to create an image, he gave up; it was too much for one man to accomplish. Luckily for Mandelbrot, he had computers to aid him in his problem solving and was able to graph Julia using an IBM computer. It was a breathtaking image.

**Mandelbrot Set**

It was time for Mandelbrot to create his own equation. In 1980, he took all of the Julia sets and used them to create one iconic image. He named this image “The Mandelbrot Set.” Little did he know that it would become the most famous of all fractals. The Mandelbrot set and fractals were embraced by artists but not by mathematicians. His peers criticized and ridiculed him, claiming he was no good at math and that his fractals were merely nothing. His friends did not even speak to him. They called his fractals “artifacts from the computer” and called them utterly useless. Ignoring the responses from his colleagues, Mandelbrot created his soon-to-be-famous book, The Fractal Geometry of Nature. In it, he demonstrated how to measure things in nature and provided more ideas on the applications to his newly discovered branch of mathematics. Soon afterwards, fractals were everywhere. Now, as Mandelbrot points out… “Nature has played a joke on the mathematicians,” believing the 19th-century mathematicians may not have been lacking in imagination, but Nature was not either. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.

Some mathematicians, professionals and amateurs, are approaching this realm with the tenacity of adventurists and explorers. And one of the ways these "fractonauts" go about exploring fractals is by the use of applets such as the one below. Try it for yourselves, but don't go in too deep, you might get lost...

**Fractals Today**

In the early 1990s, fractal antennas were introduced. One of the very first of these was in the shape of the infamous “Koch snowflake.” It worked extremely well and had the advantage of being far smaller than a normal antenna with a greater surface area, leaving the antenna able to receive a greater number of frequencies. Mobile phone companies were also having similar issues at this time: every single portion of the phone ran at a different frequency. However the introduction of the fractal antenna enabled companies to all these frequencies to run on one single phone, thus kick starting and enabling the mobile phone revolution that continues today.

Fractals are also used heavily in movies. For example: It is possible to create natural-like scenes such as mountains and even entire planets using only iteration. Fractals premiere in the film industry came in Star Trek II: The Wrath of Khan, where a planet was made entirely from a fractal. Even the last Star Wars movie had fractals. In one of the scenes, the protagonists and antagonists fight in a chamber filled with lava. That lava was created purely using fractals. The artists added swirls to the 3-D model and shrunk them. They repeated this layering each upon another until the entire lava background was made of fractal swirls.

Fractals are not only used in technology but are being found applications throughout our modern world, recently being used to understand the human body. For example: our heartbeats do not actually act like metronomes; instead, they fluctuate to several extremes, forming familiar patterns exactly like fractals do. These fractals also show up in blood vessels, the nervous system, the respiratory system and brain patterns, even the movements of the eye. All these discoveries are helping doctors in diagnosis and treatments. As you can see, throughout several different fields, fractals have initiated revolutions of all kinds. We are fractals, and fractals are our future as well.

###### Ponder this

If the form of physical world is governed by mathematical sets, what about the forces that affect it? What about plant or animal behavior? The force and direction of the wind? The undulating waves of the ocean?

What if fractal patterns are just simplifications of nature's randomness to satisfy our need for meaning and order? The same as how we see shapes in clouds, or star constellations.

###### Discuss

How can you tell if something is a fractal, or if it is just random? Is everything that looks random a fractal? Give examples and counter examples. Figure our the mathematical rule as a way to prove those examples.

###### Further readings

Benoit Mandelbrot, a French mathematician, who showed that there is order in the chaos of nature. Discoverer of the Mandelbrot set.

Fractal geometry, a mathematical set that reiterates itself. The visualisation of these sets allow for many solutions of real world problems.

Patterns in nature, much of what nature exhibits involve mathematical rules, which also includes fractals.

Fractal Foundation, an organisation that promotes the study, understanding and appreciation of fractal geometry.