Curvature of the Mind

Thoughts from a Recreational Physicist

Odds and ends

I’ve been working on getting a Mandelbrot version of the IFS fractals working. Unfortunately I’ve run into some major snags with the approach. I haven’t come up with a fast deterministic approach to detect if a particular point is in one of these fractals. I’ve been trying to build a single inverse map built by stitching together inverses of functions chosen using voronoi domains. That works for non-overlapping functions but breaks down for the overlapping case.

This results in clipping occurring at the overlaps. While it can create some interesting effects on it’s own, it is definitely not what I’m looking for.

My next approach to solve this issue was to try to converge on a particular point using forward iteration.  The corresponding issue for that one is that there isn’t any way to develop nested sets or any kind of convergence path. In order to get to a point x you might have to converge over to a point over in another region of the fractal and then hop over to the point in question. It seems the only way to make this really work is to reverse iterate every combination of maps trimming the set as the points leave the cutoff circle. In the mean time I like some of the renderings I’ve generated from my tests.

 

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New gallery – keep it simple

lots-o-touching-links

Image 1 of 35

This closeup shows that these circle images are made of nothing but circles all the way down. Very cantor setty.

I’ve added a new gallery of images. Rather than just working through more random 2 point fractals, this is a gallery of them with just a simple setup. One rotation and one purely scaling transformation. Even with this restriction I was surprised at the number of distinct shapes I was able to generate.

One of my favorite series is the circles of circles which is built of circles all the way down. I had never seen these before and they have a beautiful simplicity, plus I find the whole concept mind blowing. At least with the cantor set construction, it stuck with line segments dividing into a “dust”. The circles of circles, especially the linked ones just throws me for a loop, even though it isn’t really any different than any of the other shapes.

The other two series that stood out were the scalloped and spiral/pinwheel series. There was just something about how clean and effervescent the shapes came out that I really liked.

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The most basic IFS Fractal of all – the cantor set

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 plus 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

less less square

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.

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Things are slightly askew

less square

less square

As we continue to tweak the angle for the 180 degree center, the image loses it’s essential blocky squareness, and begins to develop what look more like “arms”.

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more square

roughly square 2

roughly square 2

This is similar to the last one, but the angles are 180 and 90 ish. You can see the similarities with the last update.

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nearly square

roughly square

roughly square

I’m working on a gui fractal image editor, and I’ve been uploading galleries of images I’ve done with it. In preparation to uploading a usable version of it, I’ll be uploading images every day or so.

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