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Playing around with box-counting dimension using GPU:

http://mathr.co.uk/mandelbrot/j-dim/

No real documentation on how it works yet, working on that next week...

Does this suggest that the boundary of the Mandelbrot set is the set of points whose Julia sets have dimension = 2?

I don't think so: .

But you might be able to say something like "you can get arbitrarily close to dimension 2 near any boundary point of M" (perhaps by finding a minibrot and using its Feigenbaum point or so).

Good point.  Thanks, Claude.

documentation, finally:

https://mathr.co.uk/blog/2016-05-16_fractal_dimension_of_julia_sets.html
https://mathr.co.uk/mandelbrot/julia-dim.pdf (3.5MB)

nice document, would love to have an actually every day useable tool for quick checking the box counting dimension of any image :) imageJ should be capable of doing so, but never managed to make any use of it, beside never got it working to give me a fractal dimension for an image

maybe I can work on one this week, having some free time - what are your requirements?

I'd probably make a command line tool, restrict it to binary images (with a flag to set background colour), supporting only PNG, and with an option to detect dimension of foreground, or boundary of foreground, and it would output a dimension estimate based on linear regression (configurable start and stop radius in 2^pixels), and a verbose option to print a table of r,N pairs so you can generate log-log plots in other software.  For portability (losing speed, perhaps) I'll not use OpenGL, instead doing it all on the CPU.  Possibly a bonus mode to draw the boxes and save a PNG sequence (it's been a year or more since I wrote the code, will have to see how I did it!).  Should be able to compile Windows binaries too if needed.

So I hacked together a preliminary version, please test and let me know if anything should change:

https://mathr.co.uk/mandelbrot/box-counter-0.1.zip (source and Windows binaries)
https://mathr.co.uk/mandelbrot/box-counter-0.1.zip.sig (GPG signature for the paranoid)

Use -? or --help to get the documentation on what options are available.

"Does this suggest that the boundary of the Mandelbrot set is the set of points whose Julia sets have dimension = 2?"

The Mandelbrot set IS the boundary. The black solid part is the area inside the Mandelbrot set.
And the Mandelbrot set does have dimension 2... exactly 2.

However (and I don't quite understand this part), that doesn't mean that the boundary has an area... there is some sense in which it is not a large enough set to have an area, but yet still has dimension 2.

if the boundary had an area the set would not be infinite, which it is.  just sayin.

its helpful to remember that the mandelbrot set is not some organic property of the universe. it is just numbers.  it does not have to fit in with our logical thinking minds.  it is just a silly number game.  one which has no bounds.

Not quite right. The boundary of the Mandelbrot set has a Hausdorff dimension of 2. Not the set itself. See paper "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets" [Mitsuhiro Shishikura, 1991] (https://arxiv.org/abs/math/9201282 (https://arxiv.org/abs/math/9201282)).

Both the Mandelbrot set and its boundary have dimension 2.  The former is unsurprising (the Mandelbrot set has solid regions within it), the latter shows that the set is very intricate.

Agreed. What I probably wanted to say, was that it depends on the formal definition of fractals. If you accept that "a set is a fractal if its Hausdorff dimension is strictly greater than its topological dimension", then the Mandelbrot Set is not a fractal, only it's boundary is. But I think that as of now, there is no single, agreed-upon, definition of what a fractal really is. So, yeah. Very complicated stuff  :)

I found the following discussions on the topic of "what is a fractal" interesting:

• "Can we prove the Mandelbrot set is a fractal? Which maps/processes produce fractals?" http://math.stackexchange.com/questions/939947/can-we-prove-the-mandelbrot-set-is-a-fractal-which-maps-processes-produce-fract (http://math.stackexchange.com/questions/939947/can-we-prove-the-mandelbrot-set-is-a-fractal-which-maps-processes-produce-fract)
• "Search for an agreeable definition of fractals in nature" http://www.fractalforums.com/fractals-in-nature/search-for-an-agreeable-definition-of-fractals-in-nature/ (http://www.fractalforums.com/fractals-in-nature/search-for-an-agreeable-definition-of-fractals-in-nature/)