Important Discoveries in Mathematics

Since the word fractal was popularized by Benoit Mandelbrot in the 1970’s, he has become the father of fractals. Mandelbrot had been fascinated by discoveries of mathematicians from the early 19th century who were attempting to define their understanding of what a curve is. Experiments like George Cantor's discovery that a single line could be divided infinitely, and Koch's triangle, a shape that has an infinite perimeter but finite area, resulted in the term ‘monsters’. These monsters were beyond the comprehension of mathematics of that time, however, Mandelbrot used the modern computing powers developed by IBM to run these ‘monster’ equations millions of times over.

With the benefits of the computing powers of these computers, Mandelbrot was able to make some breakthroughs.

With the help of the discoveries of mathematicians that came before Mandelbrot, like Helge von Koch and what is known as the Koch Snowflake, Mandelbrot was able to further his understanding of fractals. To the eye, the Koch Snowflake appears to be perfectly finite, but mathematically it is infinite, which means it cannot be measured. At the time of its discovery, the snowflake was called a pathological curve, which was beyond the understanding of Euclidian Geometry of the time (Koch Snowflake 2015). The Koch snowflake became crucial to a lingering measurement problem, how to determine the length of a coastline. In the 1940’s, British scientist Lewis Richardson observed that there can be great variations between different measurements of coastline.

Mandelbrot saw that the finer and finer indentations of the Koch Snowflake was exactly what was needed to model a coastline.

Mandelbrot wrote a very famous article in a science magazine called ‘How Long is the Coastline of Britain?’ Mandelbrot explained that a coastline, in geometric terms, is a fractal and though he knew he couldn’t measure its length, he suspected he could measure its roughness. To do that required re-thinking one of the most basic concepts of math, dimensions. While one dimension is a straight line, two dimensions is a square with a surface area, and three dimensions is a cube. But Mandelbrot thought something could exist with a dimension in between two and three, and this was a fractal.

Mandelbrot decided to tackle another mathematical monster, this time a problem introduced by Gaston Julia. At the time, the early 1900’s, Julia was looking at what happens when you take a simple equation and you integrate it through a feedback loop. which means you take a number, plug it into the formula and you get a new resulting number. Take that new number back to the beginning and plug it into the same formula, get another new resulting number and you keep iterating that over and over. The output of one operation becomes the input of another, resulting in a series of numbers, the Julia Set. Working by hand, you could never really know what the complete set looked like. You'd have to iterate this millions of times, so the development of this kind of mathematics had to wait for computers.

Working at IBM, Mandelbrot was able to do something that Julia could never do, use a computer to run the equations millions of times. This process led him to a breakthrough equation combining the patterns found in previous ‘monsters’ resulting in his own set of numbers. This would become known as the Mandelbrot Set, an infinite geometric visualization of a fractal (Fractal Geometry n.d.). In 1980 Mandelbrot created an equation of his own f(z) = z2 + c, where c is a complex number. For each c, you begin the function by plugging in 0 for the initial value of z. Then you take the outcome and plug it back into the function. As you continue to iterate the function, there are two possible outcomes: the iterates get arbitrarily large, moving further away from 0, or the iterates are bounded, never getting too far from 0. The values of c for which these iterates remain close to 0, make up the Mandelbrot Set (Lamb, E. 2017).

One of the most amazing things about the Mandelbrot Set is that theoretically, if left by itself it would continue to create infinitely new patterns from the original structure, proving that something could be magnified forever. Through this discovery emerges Fractal Geometry which gives us a very precise way of looking at the world in which we live in, specifically, the natural world. The Mandelbrot Set quickly became known as the emblem of all fractal geometry.

The shape of the Mandelbrot Set is very reminiscent of many forms in nature and embodies an example of an important aspect of how the world works. Its self-similarity is infinite and ever changing complexity is mesmerizing. With this image, Mandelbrot was able to challenge the long standing ideas about the limits of mathematics. What is so interesting about the Mandelbrot Set is that while it is infinitely complex, it is based on incredibly simple principles. By using only addition and multiplication you can understand the principles on which it is based.

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