Fractal Showcase: Images, Simulations, and Videos

My curiosity into fractals was piqued by a full-length film on Google Video called Hunting the Hidden Dimension. Then I began exploring the web for examples of fractal and generative art. My search led me to an ex-physicist named Tom Beddard whose site is a virtual cornucopia of 3D fractals, butterfly curves, generative structures and videos of Mandelbulb fractals. Beddard creates simulations using Adobe After Effects, allowing one to experience the intricacies and geometric beauty of fractals. The work he does is truly inspiring as it gets as close as possible to the intersection between mathematics and art. Both the right and left brain are stimulated by the infinite possibilities rendered in these visual wonders.

Beddard took an interest in Guilloché patterns after reading about them on the blog, Ministry of Type. He explains that “Guilloché machines (also known as geometric lathes) have been used since the 17th century by watchmakers and goldsmiths, such as Fabergè, for ornamentation.” Using a set of equations, he built a flash application to experiment with different parameters. You can visit this page and create your own Guilloché patterns.

Set to the atmospheric music of We Are The Formula, this is a wild fractal video where our 3D view of the Mandlebulb fractal creates the illusion of being inside the space of its structure. According to Beddard, the video was made from “two sequences of continuously morphing Mandelbulb fractals.” He achieves this effect by using a plugin he developed for Adobe After Effects. About Mandelbulbs, which I had never heard of before, Beddard writes, “Mandelbulbs are a new class of 3D Mandelbrot fractals. Unlike many other 3D fractals the Mandelbulb continues to reveal finer details the closer you look.”

This is a fascinating simulation which produces examples of the Gumowski-Mira pattern. Beddard tells us that the Gumowski-Mira was developed to calculate the trajectories of sub-atomic particles. He continues, “Its nonlinear nature creates a wide variety of organic looking images ranging from galaxies to feathers and marine-like creatures to the cross-section of fruit.” The application allows you to move your cursor over the patterns and watch them zoom, transform, and cause a number of parameters to change.

According to Beddard, the “harmonograph was a 19th century curiosity for visualising the mathematics of music.” He describes an example of a harmonograph where two pendulums swing at right angles of each other, and a pen attached to the pendulums traces shapes on a piece of paper based upon the relative swinging frequencies. In Beddard’s interactive harmonograph, we can set the amplitude, x and y ratios, spin phase, and various other parameters. When you check the boxes beside the parameters, the harmongraph simulation moves side to side as a pendulum would, responding to the exact changes you’ve made.

The recursive images and patterns that made MC Escher famous are explored by Beddard with a filter he created for After Effects and Photoshop CS4. You can download this filter on his website. In an interesting blog post about Escher’s Droste Effect, he describes the process of how the technique was developed and expanded upon. He writes, “(MC Escher) only got so far with pen and paper leaving an enigmatic white centre. It wasn’t until the Dutch mathematician, Hendrik Lenstra, worked out the maths behind the image transformation that the complete effect could be visualised.” Also check out Beddard’s gallery of images using Escher’s Droste Effect.

Beddard’s gallery of generative structures.

Beddard’s gallery of fractals using scripts available on his site.