I just like this one!
Quality of Movement
With the focus I’ve had on finishing out some of the fractal rendering code on the site, I’ve been going back and looking at some of my earlier stuff like the animated fractal demo. (This is not just a gratuitous link to some of my old demos, trust me) I’ve been leaving it open on my second monitor while I work, and it reminded me how much I like the motion.
The functions defining these images consist of 3 parameters: a central point, a rotation angle and a scale factor. My first attempts at making these things fly across the screen involved varying the position, angle and scale factors sinusoidally with random periods. It got the images moving and if you waited long enough, it was guarantied to trace out a significant volume of parameter space. However, it just ended up looking like it was constantly spinning and slowly changing shape.
My next step was to interpolate between randomly selected points in parameter space, which worked to make the motion more interesting, but it still would shrink to a point whenever it got too close to the origin and the scale factors were too small. Adding a constraint to the motion to put a minimum bound on the scales ended up fixing it and giving it the current zoomy, swooshy feel it has to day. That and adding a random factor to the time between random points along with a random pause at the point.
I use the same algorithm, with 3 dimensional fractals in the 3d bubbles demos here and here. Man, I just loaded the three.js version for the first time in a few months, and it is really fast with the latest version of chrome.
Now that I have some more experience at solving differential equations in Javascript, it might be fun to model the changes with a repulsive force at the origin and fade in attractive points at each of the destinations.
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Spiral
Work in progress…
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A Mandelbrot Image I’ve been working on.
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Something I’m playing around with.
This is something that I just stumbled onto while working on something else. I’ve taken a quick break from the atomic orbital viewer. I’m trying to make the IFS fractals more visually interesting. One approach is to create Mandelbrot versions of the IFS fractals which should break up the uniformity. Instead of showing one image that is an exact copy of parts of itself, each part of it would be locally similar to a different fractal. The images I’ve been creating up until this point have all been analogs of Julia sets. I’m hoping the Mandelbrot versions will turn out more interesting. It also opens up the potential of higher dimension renderings as Mandelbrot sets exist in parameter space, not the actual fractal space.
Voronoi and a trip far afield
In order to do that, I need to estimate the “basin of attraction” for each mapping in the fractal. A voronoi diagram is one way to do that. I started out just generating standard simple diagrams as a test. Sometimes images are just the best way to see what is going on. Using a very simple algorithm, I just iterated through a list of random points and colored pixels based on the closest one. That meant I kept track of the minimum distance and the matching index. Which lead to renders of that distance, followed by the sine of the distance.
Now that I’m looking at alternate renderings, I figured the easiest way to find the borders of domains was to track the second closest point as well and whenever the distances between first and second point were equal you were on a boundary. That looked ok, but lead to some artifacts.
Now that I was collecting this info, I figured I’d try to do something else with it, so I applied it to the alpha channel and switched over to the taxicab metric just for fun.
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Just a quick update
I’ve uploaded my very very super rough fractal canvas image editor. This is what I’ve been putting the galleries together with. It’s super rough, I’ll be improving it in random fits and spurts, and putting some demo posts up as well. Also, someone please please suggest a decent title for this one. I’ve got nothing.
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I’m just playing around with particles for this one
This has come together more quickly than any of the other demos. I wanted to see what I could do with a more abstract particle system driven through the same equations I’ve been using for fractals. The old formulas didn’t have quite enough pizazz, so I moved over to the sylvester library for matrices and bumped it up to 3d, which gave me the twisting, darting and zooming I was looking for.
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The most basic IFS Fractal of all – the cantor set
The cantor set is the simplest IFS fractal out there
What is that, Morse code? That my friends is the cantor set. It is one of the examples of basic topology and one of the objects that highlight many of the non-intuitive aspects of infinities.
The most common construction is to take a line and remove the middle third. Next you remove the middle thirds of the two remaining segments. You keep removing the middles of every segment that is left. If you could continue for ever, you would be left with an infinite “dust” of disconnected points.
We can build one with two simple transforms. Just contract by 2/3rds at two different points and the attractor is a cantor set. The next images are going to be simple tweaks of the cantor set, and the fractal space will grow from there.
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square + triangle = ???
Square angle and a triangular angle
This image has the 180 angle adjusted to roughly 60 degrees. It looks very little like the original image now.
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Now something completely different
even less square
This image has the scale for both transforms adjusted as well as the angle of the 180 rotation adjusted. You can start to see how quickly these images change into something unrecognizable as being related to the original image.