As you all already know, the Mandelbrot set has little "copies" of itself strewn throughout the boundary region (some of them distorted somewhat), and these are all connected.

The point $i$ (or $x = 0, y = 1$) seems to correspond to the tip of a filament. I have zoomed in to a window bounded by roughly $-0.0001 + 1.0001i$ at the top left corner and $0.00001 + 0.9999i$ at the bottom right corner and I can only see filaments here (I switched to a black and white color map to be extra sure).

Is there a little Mandelbrot somewhere along these filaments, but they're just too small for my computer to be able to zoom in on them? Or is it necessary to travel some distance along these filaments to find a little Mandelbrot?

I am not interested in a path that goes along points that escape to infinity.

Also, if it's relevant, I used an old Windows NT computer at work. I should have saved an image of the area I'm talking about, since I had some difficulty finding it. WinFract won't run on my Windows 10 computer at home.

Edit: It finally occurred to me to take advantage of Julia set toggling. Here's the Julia set for $i$: Julia set of 0 + 1 Then I toggle to the Mandelbrot set and use the zoom bar to zoom in as much as possible: Mandelbrot zoom 0 + 1 It doesn't show on here, but the zoom box was in the light gray "band" which suggests the filament I'm looking for is way too thin for my computer to be able to zoom in on it.

However, I did spot a little Mandelbrot that the old computer can zoom in on:

another Mandelbrot zoom

I should've written down the coordinates, but I do remember it is "northwest" of $i$.


1 Answer 1


$i$ is indeed the tip of a filament. However, mini-Mandelbrot copies are dense in the boundary of the Mandelbrot set, which means that for any radius $r > 0$ you can find a little Mandelbrot copy in a circle of radius $r$ centered on $i$ (actually infinitely many of them, but one will be the biggest). There are numerical methods you can use to find them.

Here's a picture centered on $i$ with radius $0.00001$, coloured with distance estimation:

Mandelbrot set near i

As you can see, the filaments are quite wriggly, and are infinitesimally thin in many places, so finding the path to a particular mini-Mandelbrot is quite a challenge. The filaments spiral around $i$ so you wouldn't even be able to say which direction to start travelling in. Finding the shortest path is like finding the smallest positive number, not really possible.


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