I’ve completely revamped the way I generate motion for the IFS fractals. I think it is much more entertaining. Now I generate new points and linearly interpolate between them with the constraint that the sum of the scaling factors is a fixed constant. Before I just generated a fixed path which ended up with a rather repetitive sequence of images. It didn’t matter what image you were shown, you could pretty much see how it was going to twirl into the future.

With this rev, the destination is more random, so the images themselves seem to ebb and flow in a much more organic fashion. I added a 1.5 second pause at the randomly generated point and those images are rendered with more detail.

I’m disappointed that I haven’t tracked down what is causing chrome to crap out rendering, but I’m hoping it’s a bug in the engine that will get patched or eventually I’ll get it figured out.

Anyway, here is the latest version: http://curvatureofthemind.com/wp-content/uploads/2010/10/fractal-lerp.htm

WARNING javascript development geek alert.

I haven’t posted in a while as I’ve been refactoring the javascript. I’ve broken the script down into a jQuery plugin to draw the fractals, and moved the ifs animation portion into another level of code. I’m thinking about releasing the code as a canvas based replacement to those “loading” animated gifs on websites. That way users can see something a little bit more interesting than a spinning circle while something is going on with the server.

On the math side of things, I’ve finally worked out a scheme to produce more evenly distributed paths through the fractals. I’ve two minds about this. On one hand, the outlines of the images are more thoroughly filled out, on the other the image is missing some of the nebulous detail I liked in the previous rendering image. I love this. It means I get to choose how each rendering looks. Right now I can choose between a distribution which is evenly selected per transformation or the area traced out by that map. Either the senate or the house of representatives in political terms. Each choice produces a different set of images and I am free to choose the ones that I like best.

So I’ll be playing with those choices and an infinitude in-between. I’ve also got my eye on creating something that will let my viewers create and share images of their own, as well as get into the nitty gritty details of how these images are generated and all the mathematics that goes into them.

Far from being vacuous beauties, there are a number of advanced mathematical concepts that can be explored through them. Topology, probability, measures, geometry, chaos theory, group theory are just the tip of the iceberg. I’ve plans to delve into group theory and generating one dimensional fractals, though it’s going to be a challenge for me. I’ve been more comfortable with short descriptions of the code that I’ve generated, and now I’ll be focusing on the mathematics behind them; I’ll probably be spending a lot more time on those.

Link to the gratuitous new version of the engine here:

http://curvatureofthemind.com/wp-content/uploads/2010/09/fractal_balance1.htm

I’ve updated yet again. This time I’ve increased the refresh rate and changed the way that I do the motion blur. Instead of keeping a buffer of past images and redrawing them all for each frame, I’m clearing the screen with an alpha factor. That gives a fading trail. I also added a pause on click feature, so you can see a more detailed image if you pause it. Things are starting to come together, and I’m thinking about adding some UI elements to control rate, pause and resume, also hide an show the map indicators with the ability to drag them around the screen.

I also want to work on the movement algorithm to add some more random and compelling movement, like speeding up/slowing down, changing direction, or pulsing through the same point in different directions for a while before zooming off into a completely new area.

Of course I want to add color back too, and experiment with different ways to color the image based on path through the fractal, location, randomness, etc. I’d also like to add movement through the color palette and algorithm as well.

Pause

The continual evolution of the animated fractal page. I’ve worked a simple palette into the images giving a slight scintillating effect. I’m testing out markers to show the center, scale and rotation of the maps. They also include more kinds of movement. In the first version, I only included animating the scale and rotation of the maps which lead to a predictable pattern of movement, so this update includes moving the centers as well.

animate2