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.

Years ago I wrote a Windows program to show the dynamics of strange attractors, but that experiment quickly moved over to computing the dynamics of iterated function systems, iterated rational polynomials, and Kleinian groups. And I always wanted to use these systems to generate sounds.

All that remained on the back burner until this year. I recently discovered that the Flash Player now supports on-the-fly sound generation. And an implementation in Flash would conform to my self-imposed mandate to write only software that was browser based. So I downloaded the latest Flex Builder beta from Adobe and set about learning enough Actionscript to get this project going.

It’s been a tough slog because Flex 4 has many departures from Flex 3, in particular the Spark component set, which is sort of the same as Halo, but also different. So hard times for a newb.

Anyways, I’ve got a bare example experiment going, which is in my beta area. It comes with a video that should explain what’s going on.

Last year, I dabbled with creating some ways of interactively zooming into images using the browser.  Here are some examples, one using Javascript, and another using Google Maps.  The first one was kind of a Javascript learning project for me, as was the second.  Both suffer from latency problems.

Then I discovered Seadragon.com, a Microsoft site for creating and hosting zoomable images.  Seadragon technology comes in various flavors, including one that uses pure Javascript, and another that uses Silverlight.  The functionality is similar to the zooming functionality of Google Maps, but has a nicer feel.  An example, using one of my images, can be found at http://seadragon.com/view/4bj, or seen below.

Note that there are controls for zooming in and out, and for using the full browser window.  You pan by clicking and dragging on the image.  The controls have a nice springy physics to them, and interactivity feels like real time.  And even though you may not be able to exactly contr0l the zoom levels as finely as you like, the zoom transitions are continuous. The original image was 24000 x 13500 (324 megapixels)

Hyperbolic Kaleidoscope from Peter Liepa on Vimeo.

I recently stumbled across this mating movie 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.

I just got word that a UK professor was going to show these movies to her hyperbolic geometry class today, as a treat for the last class of the term.

Reminds me of when I was in public school and we got ice cream on the last day of school.

In another day and age, they would have been asked to get out their rulers and compasses and draw the thing, as shown here. Up until recently, that’s representative of the best depiction mathematicians had.

The movie that corresponds to it is here.