Fractals are often immediately visually appealing, even if the underlying equation is harder to understand. For this reason, fractals have reached a wider audience than many branches of mathematics. Beyond their visual appeal, fractals give us a way to look at many natural systems that math was not previously able to examine. How long is a winding and convoluted coastline? How does a one-dimensional system like the circulatory system serve our three-dimensional bodies? How does lightning disburse its energy when it strikes? (The image below shows how electricity dissipated through a block of plexiglass, more details here.) These are all concepts related to fractals.
One very famous fractal is the Mandelbrot set (pictured at the top of this entry), named after pioneer Benoit Mandelbrot. The Mandelbrot set is generated by the iterative equation zn+1 = zn2 + c. This equation indicates that at a specific value of c, we get to the next z (that is, zn+1), by squaring our current z and adding the constant c. Let’s say that c is 1. z0 is 0, so z1 is z0 squared plus 1, and z1=1. Then z2=z12+1=2, z3=z22+1=5, and so forth. A value c is in the Mandelbrot set if zn→∞ goes to a constant value (so that zn=large is roughly equal to zn=large+1). When c=1, each z keeps getting bigger and bigger, so clearly it is not a part of the Mandelbrot set. c is a complex number, so we generate a map in two dimensions of which values of c belong to the set. The video below shows the Mandelbrot set (color giving rate of divergence, black giving a member of the set) and continues to zoom in. Even at incredible zoom scales, fine and self repeating structure can be seen.
Fractals can also be generated in a more directly visual way. Below is a fractal called the Koch Snowflake. The Koch Snowflake is generated iteratively as well. The base unit is a triangle. The middle third of each leg of the triangle is replaced by a tent. For the next step, the middle segment of all the legs of the new structure are replaced by a tent, and so on. You can see in the graphic that the Koch Snowflake gets complicated quickly. Many other visual fractals have been explored. The java applet here has a few that you can play with.
I will have another post about fractals on Friday, where I discuss more numerical properties and examples of fractals in nature. Food for thought: what is the perimeter length of the Koch Snowflake? Also check out my previous science posts on synchrony and art resembling science.