Fun Science: Fractals in Nature and Fractal Measurement

This post continues Wednesday’s post about fractals and the Mandelbrot set. Fractals are a branch of mathematics that we can observe in our daily life. Something is said to be fractal when a small piece of an object resembles a larger part of itself. The featured image is of romanesco broccoli; as you can see, each small cone on the broccoli resembles the overall structure of the vegetable. For this reason, the mathematical terms “fractal” and “self-similar” are closely related.

Examples of fractals in nature abound. The heartbeat of a healthy person is fractal when plotted in time; interestingly, people with various health problems show less fractal character to their heart rate. For a great slide show with images of fractal-ness in nature, check out this Wired article. Fractals have been observed in ocean waves, mountain structures, fern, lightning, city layout, seashell, trees, and many others. Many computer graphics of natural phenomena are generated using fractal processes.

Koch Snowflake (Wikipedia)

The Koch snowflake (above), is a fractal generated from a line. As the fractal pattern is repeated, the length of the curve grows infinite. A line segment does not have infinite length, and yet the Koch Snowflake clearly does not fill space. So what is the dimension of this object? Through a method called the “box counting method“, we can determine the dimensionality of a fractal object. The box counting method is used to estimate area and coastal length from satellite pictures, as demonstrated below.

Using the box counting method to estimate the area of Great Britain (Wikipedia).

In short, we can see how the number of boxes needed to define a length or space changes as the box size changes. For a line, the number of boxes needed grows as 1n. For a space, the number of boxes grows as 2n. The method is explained in more detail here. Intuitively, we can tell the Koch Snowflake has a dimension between 1 and 2. It turns out that, using the Box Counting method, we can determine that the Koch Snowflake has a fractal dimension of log(4)/log(3), or about 1.26.

Lorenz attractor from Wikipedia

Fractal dimensions turn up in strange places. For example, chaotic attractors have fractal dimension. The Lorenz attractor, above, has a fractal dimension of 2.06. In the future I will discuss chaos and chaotic attractors. Check out my previous science posts on synchrony and art resembling science.

Fun Science: Fractals and the Mandelbrot Set

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.

from Capturedlightning.com

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.

Koch snowflake from Wikipedia

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.