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This post consists mainly of pictures relating to elliptic functions and the conformal map from a square to disk. Useful background can be found in Chamberlain Fung’s Analytical Methods for Squaring the Disc, which inspired the work here. See also the Wikipedia article Jacobi Elliptic Functions.
I’ve been interested in the subject of non-Euclidean tessellations for many years now, although the reality is that I’ve only managed to work on them infrequently. My most recent spurt was in early 2009, when I created this image (click on it to open a larger version in a new window).
Continue reading “A non-Euclidean tessellation and its fundamental domain.”
In Part 1 I described a WebGL pilot project for a client. After experimenting with a couple of WebGL frameworks I reverted to basic principles and wrote a purpose-built display app that was able to display 506K textured triangles at interactive rates. The demo let the user navigate through a pseudo-architectural scene using first-person-shooter style keyboard navigation.
Continue reading “WebGL in the real world – a short case study – Part 2”
I started following WebGL a few months ago when it was in beta in several browsers. Many creative web folks were already working with it, and some of the experiments were spectacular. Fast forward to the present, and Google Chrome now officially supports WebGL (although your computer may not be up to it), and Google has a WebGL Experiments website.
Continue reading “WebGL in the real world – a short case study – Part 1”
I’ve long been fascinated by non-Euclidean tessellations and symmetries , like this one based on an example in the book Indra’s Pearls. The most widely known examples of these are the circle limit images by M.C. Escher. I’ve always wanted to create an interactive way of producing these images, and with the advent of WebGL I’ve made some preliminary steps to doing this in the browser.
Even though I’ve created a game or two, I have to admit I don’t play them much. But now and then I get caught up in playing. iSlash by Duello Games is one of the few iPhone games that I’ve admired, played compulsively, and recommended to anyone who would listen. You can get a good idea of the gameplay by viewing the above video or, better still, downloading and playing the game. According to Duello Games, there have already been 450,000 downloads, and the game has reached #1 status in Turkey, France, and Belgium.
Continue reading “Tune up your Ninja senses: An Interview with the creators of iSlash”
Chaos theory and its little cousin – strange attractors – have been around for a long time. Pictures of chaotic systems and strange attractors abound, and they are a mainstay for computer math experimentalists, although still in the minor leagues relative to the Mandelbrot set.
Most implementations tend to ignore the fact that these systems represent dynamics, that they move and evolve. Still pictures can hide the fact that, for example there are sink states, and that was supposed to represent 20,000 iterations only shows 5,000 because the system hit a fixed point at iteration 5,001.
There’s something about music or a soundtrack that really enhances what would otherwise be just a silent movie. So I plan to add music to my visualizations as time and inspiration allow. Here’s an example that I did a week or two back. The original movie was 24 seconds long, but now it’s been slowed down to accommodate an atmospheric soundtrack.
Hyperbolic Kaleidoscope from Peter Liepa on Vimeo.
A few weeks ago I discovered noteflight.com, a website that lets you create musical scores. I’ve always wanted to use scoring software, but never got around to it until – well you know, the price was right.
Years ago I wrote a computer game called Boulder Dash. The music for that was composed in a very basic soundtrack editor I wrote for the Atari 800, and was never actually played on a real-world instrument. I’ve always wanted to convert that music to a real score and hear what it would sound like on, say, the piano.
I recently stumbled across this movie of a quadratic mating at Curt McMullen’s site. I used to play a lot with fractals that were hybrids between quadratic Julia sets. This is apparently a movie of one of these matings evolving.
To the best of my knowledge, the movie is by Xavier Buff.
Complex Dynamics 