Shazam for fractals?

Monday 19 April 2021

Is there any way to find the coordinates of a Mandelbrot image from the image? Even a guess as to the rough neighborhood?

I recently saw this as someone’s avatar:

A portion of the Mandelbrot fractal, with interesting features

This is clearly the Mandelbrot fractal, but where is it? What coordinates and magnification? Without accompanying information, is it possible to find it? I’d like to explore that region, but how can I find it?

This problem reminds me of Shazam, the seemingly magical app that listens to what’s playing in your environment, and tells you what song it is.

Is there any way?

BTW, the way I solved this problem in my own long-neglected Mandelbrot explorer Aptus is to write data records into the PNG files it produces.

For example, you can download the image from the Aptus page, and use imagemagick to see what data it contains:

$ identify -verbose JamesGiantPeach_med.png
  Filename: JamesGiantPeach_med.png
  Format: PNG (Portable Network Graphics)
    Aptus State:
    "Aptus State": 1,
    "angle": 0.0,
    "center": [-1.8605327723759248, -1.270334865601334e-05],
    "continuous": true,
    "diam": [1.788139343261719e-07, 1.788139343261719e-07],
    "iter_limit": 999,
    "mode": "mandelbrot",
    "palette": [
        ["spectrum", {"l": [50, 150], "ncolors": 12}],
        ["stretch", {"hsl": true, "steps": 25}]
    "palette_phase": 190,
    "palette_scale": 1.0,
    "size": [500, 370],
    "supersample": 3
    Software: Aptus 2.0

To prove it works, here is the same place with a different viewer, using a URL crafted from the data in the PNG.

Aptus also knows how to read these files, so you can open a PNG it produced, and you will be exploring where it was captured. It’s like jumping into a photo to visit the place it was taken. I used the same technique in Flourish.

Too bad more images don’t carry metadata to help you re-find their location in mathematical space.



This isn't a scalable or certain approach, but looking at the geometry of the sticky-outy bits led me to -0.599-0.664i, and that looks like a good match, though I'm using different visualizers so the colors don't match up at all.

If I were to try to generalize and systematize my intuitions here (which is necessary to seeing if there's any there, there), I'd say that we need to look at the slope and curvature of the lines connecting the bulbs. Find parametrizations for them, check errors.

I'm also trying to figure out if there's potentially some kind of application of Farey numbers to this, as in

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