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Comment by zmgsabst | Hacker News Reader

What do you mean by “mathematical form” here?

A way to colorize the two-dimensional plane.

Nothing compares to the complexity of the Mandelbrot formula when it comes to colorizing the two-dimensional plane.

With complexity I mean the human impression of complexity. The reaction of "Oh, there is a lot of stuff going on in there" after looking at some parts of it.

There are many fractals and variations of them. The Mandelbrot is definitely famous, but it is hard to say if it is "the best". Look for example at this strange creature: https://en.wikipedia.org/wiki/Burning_Ship_fractal

I’ve never felt more staring at the Mandelbrot than, say, pendulum settlement patterns or random noise.

Often less, in that the Mandelbrot set has much more regularity than those examples.

I suppose we have different aesthetics.

The Mandelbrot fractal has all it's detail on the boundary, which is nice and all but there are other formulas (or settings for formulas) that instead seem to fill up the whole plane with detail everywhere as if you could zoom in to any location and see more details! Mandelbrot gets smooth if you zoom somewhere that isn't on the boundary.. which I see as a bit of a shortcoming personally.

Even the generalised Mandelbrot fractal formula starts to do this plane-filling via compounding 'branch cuts' in the 'lake'/inside areas when the exponent (a complex number) is a negative non-integer like -1.5+0.0i instead of the usual 2.0+0.0i.

Here's an example I tried to add to Wikipedia but unfortunately it was rejected:

https://commons.m.wikimedia.org/wiki/File:AlanTIsVeryHelpful...

Could be an illusion that it's actually filling the plane, but if so it's an illusion I'm rather fond of!

From a single image, one cannot get a good grasp.

If you want to show the world a new fractal, I think the best way is to write a js+webgl program and put it online, so people can explore it themselfes.

Yeah that wasn't actually a new fractal though, just generalised Mandelbrot with a negative non-integer exponent.. but yes some kind of interactive exploration of it would be fun I agree!

There's something I'd be much more keen to show the world (and which might render faster than negmandel!) which I call 'mandelfield', here's an old blogger post about it (yeah sorry it's blogger@): https://ultraiterator.blogspot.com/2009/10/hidden-mandelfiel...

And I think Flickr is the best way to browse my fractals at present: https://www.flickr.com/photos/57934548@N02/ My avatar pic there is a 'mandelfield'. I think it's a really interesting phenomena!

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