Fractal math can be described as both simple and yet one of the most sophisticated mathematics there is.
In basic form, a fractal can be described as a recursive formula, one where the output goes back into the equation as input in the complex plane.
After learning this fact, the graphs and depictions of fractals make more sense. Sierpinski's Triangle is graph that has these recursive qualities. As the graph develops
you see there is a pattern created through the omission of triangle space. Each new pattern is called an iteration, where each new iteration continues the pattern one
step further. Looking at the following will give you the opportunity to look at Sierpinski's Triangle up to the sixth iteration.
A Look At Fractals: Sierpinski's Triangle
Looking at the Sierpinski Triangle applet you get to see iterations of the graph, but what we are really seeing is a recursive pattern.
When we look at the graphs of these formulas or functions we end up seeing a graph where when you zoom in on the graph,
you begin to see a more detailed depiction of the original graph, meaning it shared the same qualities as the original drawing or function. In the applet above these
same properties you could idenitfy as eliminated the middle triangle each iteration. If you look at the following website you can look at both the
Mandelbrot set and a Julia set (both of which are fractals) and zoom in to see the recursive properties of each (How to use, 1997).
The Fractal Microscope
Mandelbrot described fractal graphs to measuring the coast of Britain. When one measures the length with continuously smaller devices, you get a longer length.
For instance, driving a plane over the coast of Great Britain and measuring the mileage you flew will you give a number of miles. However, if the coast were to measured
with a yardstick the recorder would have to measure the different bends in the coastline. This means that you would record a longer length of the coast when measuring with
a yardstick then when driving over in a plane where you can fly straight. The idea then goes the smaller device you use for measuring, the longer length you would see.
Look at the following applet that demonstrates this type of recursive formula as it shows the length of a Peano curve (a space-filling curve) in the shape of Koch's snowflake:
(The first six iterations of Koch's Snowflake are displayed below the applet link in order to compare.)
Peano's Curve and Koch's Snowflake
This phenomenon is seen in many other aspects of our world. We also see other properties of fractal math in the world arounds us.
For instance, Mandelbrot once said, "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth,
nor does lightning travel in a straight line. (Debnath, 2006)"
If you investigate each of these naturally occurring fractals you can identify the different recursive properties that each has.
In order to better explain fractals, I found a video clip of the founder of fractals himself,
Benoit Mandelbrot speaking on them and giving a brief explanation of the math we see around us.
TED Talks-Benoit Mandelbrot (TED, 2010)
Overall, I think the take away from the mathematics that defines fractals is that fractals are functions mapped in the complex plane with recursive properties.
