#minibrots

[mrvelocipede]

Having finished one elaborate ceiling medallion, it becomes very tempting to make more. If only because it means I can use colors I like better. Still very rococo-influenced, though.

[mrvelocipede]

Octarine, 2008

[mrvelocipede]

Useful Tricks: Julia Switch

The small copies of the Mandelbrot set are appealing because they have all sorts of intricate patterns around them. Each one is slightly different, and they get more complex as you zoom to higher magnifications. Diving for minibrots is one of those addictive things that can easily consume way too many hours of any given day.

Once you've found a good minibrot, then what? You can put some kind of coloring on it and call it done. You can use the intrinsic symmetry of the patterns around it to make a really lovely elaborate mandala design, and it will always have the familiar mandelbrot shape at the exact center of it. But what if you don't want that? What if you want the radial symmetry, but with a nice clean empty space in the center?

You switch to a Julia.

If you choose a switch point (the ‘Julia seed’) that's exactly in the middle of the minibrot, you get all the same surrounding fractal patterns, but with a circle in the middle.

HOWEVER, this is not immediately apparent, because usually a switch will give you the Julia set at the default magnification, and to see the kinds of patterns you're looking for, you need to magnify it.

Sometimes I copy and paste the location of the Mandelbrot I've switched from, sometimes I just zoom in to the center. Usually with copy and paste, I find that I need to zoom out a little bit (and adjust the rotation) to get to the right place. In fact, with a switched Julia you can sometimes get the right kind of design at a much lower level of magnification, which speeds up rendering.

Also, if you adjust your Julia seed to slightly different places, you can have almost-identical images with any basic Julia shape in the middle.

This technique is especially nice with higher-power Mandelbrots. If you change the exponent from 2 to 3, a standard Mandelbrot formula will do beautiful three-way patterns...all with the same wonky blob in the middle. (It reminds me a little of a toothy growling mouth, and is kind of off-putting.) But the Julia switch makes everything smooth again.

[mrvelocipede]

This one is the intersection of several ideas that I've seen lately: one, that fractals are kind of like angels (infinite, hard to understand, made of strange geometries) and two, that there are angels for all sorts of familiar things, like the Angel of Water Coming to a Boil, or the Angel of Fallen Leaves Decaying into Dirt.

So here we have the Angel of Flimsy Comic-Book Logic.

[mrvelocipede]

exercises: the baroquening

This is a photograph of part of the ceiling in the Charlottenburg Palace.

When I first saw this picture, a number of years ago now, I immediately thought it looked a lot like the kinds of patterns you get around minibrots. Which made me wonder what Baroque-era designers might have done if they'd had access to actual fractal geometry, and then made me wonder how close I could get to this same effect.

If I start with something like this

and add a lot of gold filigree and whatnot

Pretty close. It's a lot of work, though. I've been tinkering with this version for at least a week (for more hours per day than I like to admit to), and according to my records I have several dozen older and less successful attempts dating back to 2018.

The minibrot at the center of the symmetry is too small to see, at this magnification, but it only takes a little bit of a zoom for it to show.

[mrvelocipede]

South Transept, 2015