Modeling the Barnsley Fern

The fern is one of the basic examples of fractals. Fractals are infinitely complex patterns that are self-similar across different scales, and are created by repeating a simple process over and over in a loop. The Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas.

Modeling plant structures and other phenomena in science was a specialty of Barnsley. In fact, he is said to have inspired many graphic artists attempting to imitate nature with mathematical models. The fern code developed by Barnsley is an example of an iterated function system (IFS) used to create a fractal.

Fractals generate points to plot on a graph that are the result of iterated calculations. The answer from one calculation is used as the input value to the next calculation. One starts with shapes plotted on a graph, and then the shapes are iterated through a calculation process that transforms them into other shapes on the graph. Starting with four shapes, one of which is squashed into a line segment (this becomes the fern's rachis or stalk), one applies the shapes to the calculation to generate more shapes, feed them back into the calculation process, and so on. Eventually a pattern emerges that bears a startling resemblance to a fern. The longer the iteration process, the more intricate the tiny detail in the pattern becomes (source: Fractal Ferns, Ferns of the Canberra Region).

The Victorian era mathematician Augustus De Morgan wrote the following lines capturing the IFS concept.

“Great fleas have little fleas upon their backs to bite ’em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and greater still, and so on.”

The complexity of creating the Barnsley fern model, together with the fact that the number of iterations required could be tens of thousands, makes it extremely arduous to plot by hand. While it is not impossible, it is much easier, and often preferred, to use a computer instead. Many different computer models of the Barnsley fern are popular with contemporary mathematicians. Maple’s unparalleled symbolic computation abilities make it a powerful tool to model the Barnsley fern. Take a look at the Maple worksheet here.

The Fractals package in Maple makes it easy to create and explore popular fractals. The Fractals package can quickly apply various escape time iterative maps over rectangular regions in the complex plane, the results of which consist of images that approximate well-known fractal sets. In this application, you can explore escape time fractals by manipulating parameters pertaining to the generation of Mandelbrot, Julia, Newton and Burning Ship fractals.

Several other phenomena in nature have been studied by Maple users over the years. The inventor of the Gömböc shape, Gábor Domokos, is using Maple to study the shape of beach pebbles. His research with Gary Gibbons from Cambridge University attempts to describe the shape of pebbles, and the evolution of their shape. They are also trying to understand the interaction between pebbles in their collective evolution. For more details, read the user case study Maple Helps Discover the Mathematics-based Gömböc Shape.

For another example of the use of Maple to study naturally occurring shapes, read the case study The Art of Modeling Seashell Morphology.