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Fractal Geometry

users.math.yale.edu

183 pointsandsoitis30 comments

If you are interested in learning about Fractals in form of a MOOC, I can highly recommend "Fractals and Scaling" from Santa Fe Institute [0]. It is very accessible and taught by David Feldman.

[0]: https://www.complexityexplorer.org/courses/169-fractals-and-...

MOOC: Massive Open Online Course

For a while in the 1990's IFS (fractal) image compression was a serious competitor to JPEG.

Microsoft Encarta used it on the encyclopedia CD-ROM.

I don't know what happened to it. It was pretty slow to compress.

Sometimes things aren't "bad" per se, they just aren't as good.

Wavelets seem to be another perennial also-ran in the image compression space. They work. They have the odd useful property here and there. They just aren't as good for most purposes.

There can also be differences in terms of investment; if hypothetically there were 3 techniques with roughly equally-useful "final forms", but only one of them gets that investment, then it'll win because there's no reason to pour the additional effort into the other two.

a slow to compress format that does a good job sounds great for physical media which gets distributed, but sounds like it would be bad where the things being compressed are most probably many millions of individual images uploaded by users as you want a fast enough compression that you can start to return the compressed image to the user within milliseconds.

That just sounds to me like what happened to it, although I can't say for sure - did it perhaps also have other things about it that would make it a problem to be running lots of concurrent fractal compressing processes on a server?

I don't know - the machine I played with it on was a 16MHz 386, and the IFS software was not optimized. But it was way slower than DCT-based JPEGs to compress.

The odd thing about it though, was that it could upscale smoothly. A 320x200 source could be decompressed at 640x400, and it would sort of intelligently "fill in the gaps" on textures etc.

Some comparison between fractal compression and jpeg here: https://dl.acm.org/doi/fullHtml/10.5555/364682.364685

Slow to compress in the 1990s is probably not that slow to compress in 2024.

And you can always do fast initial compression on images needed immediately, and run background compression to replace them with the smaller versions.

hmm, yeah I never thought about that (the second part)

but slow to compress is always relative to fast to compress, and fast is related to user expectations of speed which is calibrated to your competitor's speed, thus if something is slow to compress it will probably stay slow to compress even as its speed increases.

> I don't know what happened to it.

From what I've heard, licensing costs were significant.

Presumably the patents are all expired now?

I think they were patented in the early 90s which would mean they've expired by now, yeah. CPUs and whatnot these days would make compression faster but trying to convince people to switch to fractal compression now is undoubtedly a losing proposition.

I've always been interested in how fractal geometry can be used to survey and analyze networks of all kinds. (https://en.wikipedia.org/wiki/Fractal_dimension_on_networks)

Interestingly enough, I'm now working on building intentional network interventions for tech in NYC–and it's called fractaltech.xyz

The link of paragraph '10. Panorama of Fractals and Their Uses' points to nonexistent content.

A lot of the broken links and images on that site work if you view it on the internet archive, eg the "Manufactured Fractals" images are mostly missing now, and the "Ineffective Ways to Measure" page is gone now, but still there in the archive

https://web.archive.org/web/20151110050637/http://users.math...

that "Panorama" page tho, remains broken. The source of the page on the archive says "ruffle-polyfilled" suggesting there was a flash animation there. I think a lot of it was probably the content behind the links of the "Manufactured Fractals" section - if you look you'll see those urls are all in the Panorama folder.

Yesterday someone posted this awesome 3D fractal visualizer for the browser (it runs really well on my phone):

https://news.ycombinator.com/item?id=40493981

I’ve been reading about spatial partitioning of fractals lately but haven’t found a good resource on the subject. They probably use bsp-trees?

I'm still surprised that only one mathematical form has been found with the beauty and depth of the Mandelbrot set.

One would think that like in the physical world, there should be many "species" with different types of appearances. Each a beautiful system on its own, but in comparable complexity.

Maybe the complexity of the Mandelbrot set is to the mathematical universe what is the complexity of life in the physical universe. Something that is very rare. For some unknown reason.

I think this isn't true (at least subjectively, for me). There are plenty of similar objects you can construct.

This wikipedia page has a bunch of examples of Julia sets

https://en.wikipedia.org/wiki/Julia_set#Examples

I'm also partial to Newton fractals, some of which look very cool

https://en.m.wikipedia.org/wiki/Newton_fractal

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

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!

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.

Very cool. Thanks for linking this.

"andsoitis" added 36 points to get on top of the charts, Some deeper links seem broken - I suspect this is old website

Fractals are old folly, patterns of nature - wonder if there are active use cases to represent data in some form of fractal models

Wavelets are arguably a fractal representation.

A bit of a tangential meta comment, but it's refreshing to see a cool website in 2024 like this, all content no fluff. I'm surprised to see it still using frameset. But also sort of sad that there's currently no HTML5 equivalent to have resizeable panels like this without messing with javascript, mouse events, and so many edgecases. Frameset still works in all browsers, so I suppose you could can just use that, although it's been deprecated for almost 20 years now.

I understand that the purpose of HTML is not to support such functionalities, but CSS could have filled the gap. The closest we ended up getting was "resize" but it only affects one element.

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