Barnsley Fern Fractal: What is a Fractal?

Augustus De Morgan, a Victorian mathematician, composed the following lines to capture the IFS notion.

• “Big fleas have small fleas on their backs to bite ’em.
• Little fleas have smaller fleas, and so on indefinitely.
• And the large fleas, in turn, have greater fleas to go with,
• While they have increased much more, and so forth.”

The Barnsley fern model is exceedingly difficult to plot by hand due to its intricacy and the fact that it may need tens of thousands of iterations. While it is not impossible, using a computer is far more convenient and often preferable. Many alternative computer models of the Barnsley fern are popular among modern mathematicians. Maple’s exceptional symbolic computing capabilities make it an effective tool for modeling the Barnsley fern. Look at the Maple worksheet here.

Maple’s Fractals module allows you to easily create and explore popular fractals. The Fractals program can swiftly apply various escape time iterative maps to rectangular areas in the complex plane, producing visuals that mimic well-known fractal sets. This program allows you to investigate escape time fractals. By adjusting settings related to the formation of Mandelbrot, Julia, Newton, and Burning Ship fractals.

Over the years, Maple users have investigated a variety of natural phenomena. Gábor Domokos, the originator of the Gömböc form, is using Maple to investigate the shape of beach pebbles. His study alongside Cambridge University’s Gary Gibbons aims to characterize the morphology of pebbles and how they evolve. They are also attempting to understand the interplay of stones in their collective growth. For further information, see the user case study Maple Helps Discover the Math-Based Gömböc Shape.