More May Mandelbrot

I had thoughts about a second May Mandelbrot post that got a bit deeper into the weeds, but a couple attempts today went nowhere (except the trashcan). But I been having some fun exploring the Mandelbrot with Ultra Fractal, and I thought some pictures might be worth a few words.

Click on any to see bigger versions.

The satellite Mandelbrot above (one of an infinite number) has nice simple spikes on it because it’s in the region of the “needle” — that long narrow part of the Mandelbrot that sticks out to the left (from about -1.25 to -2.00 on the real number line).

Zooms in that region have a simple spiky structure.

In other regions of the M-set, things are more complicated. The image above shows the needle of a satellite deep in the region to the “lower left” of the main Mandelbrot.

Here again is the needle of a satellite in a very deep zoom. This region of the Mandelbrot is from the lower side of a bulb off the main lake. At the top of those bulbs, the pathways are shaped a bit like lightning (as in the image above this one).

As you move down the sides of those bulbs, the pathways twist more and more (which way they twist depends on which side you go down). The image above shows that twisting.

As I discussed in the previous post, the boundaries of the M-set are extremely chaotic. As the various images here show, that’s especially pronounced when you zoom into the area around the satellites.

The image above (a “Mandelbrot Doily”) shows that chaos beautifully. This one is from the region in the “valley” between the main cardioid and the biggest bulb directly “west” of the cardioid.

The satellites aren’t the only interesting regions, though. The pathways themselves create some fascinating structures, and those are very dependent on the region. The general spikiness of this image above says it comes from the needle region.

One thing that fascinates me about images like this one is that they are essentially entirely structure. all those lines, those pathways, contain zero-thickness lines of M-set. These fill the space (but never touch).

Here’s another example of pure structure. This is a seriously zoomed in bit of that space-filling curve. This is not the needle of a mini-Mandelbrot. That thick black portion is just more structure. Zooming in would open it up to look like the other parts of this image.

The image above illustrates how these pathways (or threads or spines) of Mandelbrot really have no thickness. The zoom level here is 10100, which makes it one of the deepest shown here (it took 20 minutes to render).

The thing is, it looked essentially the same at a zoom level of 1030, so expanding to several times the size of the visible universe hasn’t revealed anything.

Blows my mind to think about the dance of numbers with such precision.

Here’s another deep zoom of just 1D lines. The clean spikiness shows how this came from the main Mandelbrot’s needle region.

This is from the same area as the third one down above. I zoomed in a little deeper here on the tip of one of tendrils of the satellite. This is, again, pure (very complicated) structure.

And I think it looks really cool. (Took almost a half-hour to render.) I’m planning to render a very large wallpaper version of this one overnight!

And finally:

A little pastel pretty, a satellite fairly deep, surrounded with candy-colored chaos. The general clean spikiness telling us it lives in the needle region.

So there ya go, to start your weekend, 10,000 words in pictures.

Stay chaotic, my friends!

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

2 responses to “More May Mandelbrot

  • Arlene Blackman Yolles

    Hi there, haven’t been to your blog in ages but you hit upon one of my favorite subjects. Before I retired as Math chair ( & 8th grade math teacher), I used to show my classes the Nova video about fractals, for enrichment. Afterwards, I asked them to give me a paragraph on what fractals are … I was amazed that ALL gave me an accurate and quite cogent answer!

    • Wyrd Smythe

      Hi, Arlene, long time no see! Hope all is well with you and yours!

      Sounds like it captured their attention! It always did mine (which I talked about more in the previous post). Fractals, in general, are pretty cool, but the Mandelbrot seems to stand (considerably) above them.

      Sadly, my various math-related posts get only slightly more traffic than my baseball posts (which is to say “almost none”). [sigh]

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