In preparation for Process & Influence, my first solo exhibition (documented here), I tried to think of a good way to convey a bit of the creative and technical process that underlies a lot of my algorithmic artwork. I decided to scan and print out a selection of my notes and sketches and incorporate them into the gallery space, so that after you experienced an artwork, if interested, you could “pull back the curtain” and learn more.

The notes cover conceptual musings, the results of creative experiments, and specific equations or programming structures. There’s even a page documenting the process of working out the show title. They were scanned at 1200 dpi into bitmap (pure black and white) TIFF images, laid out into four separate files, and produced as bond prints at Triangle Reprographics in Orlando. This type of bond print is essentially a large format photocopy, and is most often used for reproducing architectural plans.

Below, you can see all of the process documentation produced for this exhibition, including a few that didn’t make the cut to be hung.

I would like to say thanks to Lorraine Lax, with whom I had the privilege of working and who had a similar idea a few years back.

I like to visualize strange attractors in real time, and OpenCL is a promising technology to take this to the next level, as it can utilize all processing cores available on a system, be they GPU, CPU, or a combination. As long as the algorithm can be made massively parallel. This is not too hard with the equations I’m working with… they are simple iterated functions – pop in an x and y coordinate and get a new x and y coordinate. Repeat this many times, and the strange attractor appears. Although most people are used to having one thread perform this iteration, there’s really no reason that a million threads can be performing it. As long as the starting positions of the particles are randomized, you’ll still get the strange attractor.

Almost. There’s an interesting numerical side-effect… when the attractor goes into a less chaotic orbit, and then comes back out to a more chaotic orbit, everything is “sparkly”; i.e. the points in the attractor are not well distributed. Because they all started from a fixed set of points (not theoretically but in practice, as they are floats), they zip around to a fixed set of points in the attractor. Or something like that.

Anyway, I attempted to overcome that with some added randomness. In the process, I have come across two really cool “accidents”, of which this is one (I love accidents in code). Basically, I’m adding a bit of random jitter to each particle. But the random value range I first tried was big enough to create this amazing motion. WOW.

Memo’s demo on Vimeo
Memo’s post on his website with more info

The original OpenGL program was crafted to simulate a population controlled by finite state machines, using stickmen and stickwomen with simple animations to visually represent the unfolding dynamics of the population. Later, the program was changed to experiment with the idea of digital chronophotography, or a way to capture in one frame the essence of each particular simulated run. This was achieved by leaving the drawn image every frame rather than clearing it between frames. This technique, combined with a very low opacity in the drawn elements, allowed the image to slowly accumulate over the course of thousands of frames. At any point in time, a key could be pressed to save a high resolution version of the image at its current state of evolution.

Resources

A companion Art Gallery Sketch presented at SIGGRAPH 2006 describing the techniques and inspiration for these images:

Click here to download the Art Sketch pdf
(642 KB, Adobe Reader required)

Click here to download the Art Sketch presentation
(2.78 MB, Microsoft Powerpoint required)

If you would like to try out the software that was used to create these images, contact me.

Custom software was developed to calculate the Lyapunov exponent of a chaotic dynamical system over a range of its four parameters. The raw data from this 4-dimensional parameter space was fed to another program to generate thousands of individual images, where X and Y represent two of the four coefficients. Every single pixel in these images gets its value from the calculation of the Lyapunov exponent of the strange attractor generated by the four coefficients it represents. The second program outputs four 16-bit grayscale images, which represent the different “components” of the spectrum of Lyapunov exponents. These images were combined in Photoshop using a pseudo-color technique to bring out subtle coloration in the final images.

Resources

The following resources were very helpful and inspirational in my exploration of lyapunov space and in the development of my software and artwork.

Lyapunov Exponent Spectrum Software and Lecture Notes by J.C. Sprott

Extracting beauty from chaos by Andy Burbanks

Lyapunov Exponents by Jan Thor

Lyapunov fractal on Wikipedia

Lyapunov Space and Lyapunov Exponent from The Chaos Hypertextbook

Z-Lyapunov by Georges Rosset

C code for numerical calculation of the Lyapunov exponent by Paul Bourke

Complexity and ‘art’: Lyapunov graphs by Diethard Janßen

Markus-Lyapunov Fractals by Charles Vassallo

Lyapunov Exponents Lab Report by Earl F. Glynn II

Lyapunov Space – A Software Developer’s Odyssey by Bill Storage