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The quest to decode the Mandelbrot set

quantamagazine.org

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Re: The quest to decode the Mandelbrot set

#71

Post by vardump »

Baeocystin wrote: post #51
IronBacon wrote: post #37

Earlier quoted context omitted.

I remember as a teen I did mine in 68000 assembly on a Commodore Amiga to get it reasonable fast considering it was running on a 16 MHz CPU — i would say a few seconds to draw the canonical image — but IIRC it reached pretty fast the limit of math precision. At that time I didn't know what was a complex number, but was fascinated by the whole concept of fractals and how complex structures could be created with a rela…

A few seconds? Wow! I did a science fair project on the Mandelbrot set using my Amiga 2000, and it took me a good 45+ minutes to generate a single 320x200 color image. IIRC, I wrote the generator in some variant of Pascal, and was so happy with the performance increase over Basic on my C=128... I ran in to the precision limit pretty quickly, same as you. I didn't understand computers well enough to know that's what t…

The difference is probably you used floats and he/she used fixed point (integers). Of course that way you do run out of precision very quickly.

Software emulated floats on Amiga 2000 were really, really slow.

Re: The quest to decode the Mandelbrot set

#72

Post by QuadmasterXLII »

tda wrote: post #68
QuadmasterXLII wrote: post #16

Earlier quoted context omitted.

If you want to keep zooming in the browser and have snappy gpu accelerated performance while basically never hitting “reached limit of numerical precision,” check out https://mandeljs.hgreer.com . The math to make this possible gets pretty funky: I go over the tricks I used at https://www.hgreer.com/JavascriptMandelbrot/

Thanks for creating this, I had some fun exploring the set. What does the scale parameter stand for?

The color of each pixel is

    fragColor = vec4(vec3(cos(c), cos(1.1214 * c) , cos(.8 * c)) / 2. + .5, 1.);
Where c is the number of iterations divided by the scale.

Re: The quest to decode the Mandelbrot set

#73

Post by netmare »

bilsbie wrote: post #62

I’ve always wondered if you could make fractals into some sort of game.

There's MMCE†, the spiritual successor of Marble Marcher. I've only played the original some years ago and it was pretty awesome. You simply have to move a ball across a 3D fractal surface that's constantly evolving in real-time!

Sadly, I can't run MMCE to test, since I'm using a PC and gfx card from 2009.

†: https://michaelmoroz.itch.io/mmce

Re: The quest to decode the Mandelbrot set

#74

Post by Y_Y »

AlanYx wrote: post #59
hnfong wrote: post #12

The Mandelbrot set is quite well known. Yet something I learned recently blew my mind. It's about the uncanny resemblance between the images generated by the Mandelbrot set, and among all things, the popular image of Buddha. For example: https://en.wikipedia.org/wiki/Buddhabrot Even when looking at the 2D Mandelbrot set renderings, I can't help but wonder whether the similarity of the "bulbs" to the rather unique Bud…

There are a few others well-known fractals apart from the Buddhabrot that resemble real-world objects. My favourite is the burning ship fractal: https://en.wikipedia.org/wiki/Burning_Ship_fractal I found these sorts of things really helpful in getting my kids interested in fractals. They love the idea that there are "things" they can find that are only viewable through math.

The burning ship is still my favourite object in mathematics despite about twenty years of study since I first came across it. In an echo of the famous story about Tamm and the Taylor series error, I once had to explain the burning ship to border control in an impromptu test of my credentials (though I'm not sure if I'd have been shot if I got it wrong).

Re: The quest to decode the Mandelbrot set

#75

Post by fweimer »

markisus wrote: post #23

A very interesting cast of underdog characters appears in the article. You’ve got one guy with a relentless spirit to continue with mathematics in is spare time after being blacklisted from mainstream academia because of antisemitism. Another is a childhood prodigy, who set the record for the youngest American IMO team member, but got burned out as an adult and went into finance but found his way back through the men…

Note that the article refers to Soviet antisemitism: Jews were denied academic jobs, and they couldn't move abroad to work in their field, either. Not really related to mathematics as such (and definitely not about the mathematical community rejecting an antisemite).

Re: The quest to decode the Mandelbrot set

#77

Post by kjqgqkejbfefn »

derbOac wrote: post #32

Does anyone know of any good resources on Kolmogorov complexity and fractals such as the Mandelbrot set? Or even on information theory and fractals? For some reason reading this article is making me wonder about the difference between the information required to generate something like a mandelbrot, knowing the underlying rule, and the information required to represent it as it is, without following the rule. Or e.g.…

> so there's an implied total complexity as well as an "generating" one.

Dessalles's algorithmic simplicity theory of (cognitive) relevance is formulated in these terms.

>Situations are relevant to human beings when they appear simpler to describe than to generate

The discrepancy between generation complexity

>the complexity (minimal description) of all parameters that have to be set for the situation s to exist in the "world"

i.e, the "pixels"

and description complexity

> the length of the shortest available description of s (that makes s unique)

i.e. the mandelbrot formula

is named Unexpectedness in this framework.

https://telecom-paris.hal.science/hal-03814119/document

https://simplicitytheory.telecom-paris.fr/

Dessalles published a paper in 2022, Unexpectedness and Bayes’ Rule

https://cifma.github.io/Papers-2021/CIFMA_2021_paper_13.pdf

>A great number of methods and of accounts of rationality consider at their foundations some form of Bayesian inference. Yet, Bayes’ rule, because it relies upon probability theory, requires specific axioms to hold (e.g. a measurable space of events). This short document hypothesizes that Bayes’ rule can be seen as a specific instance of a more general inferential template, that can be expressed also in terms of algorithmic complexities, namely through the measure of unexpectedness proposed by Simplicity Theory.

Maybe there is a way to plug this into the https://en.wikipedia.org/wiki/Buddhabrot fractal someone mentioned above.

Re: The quest to decode the Mandelbrot set

#78

Post by neilkk »

hermitcrab wrote: post #70
neilkk wrote: post #69

Earlier quoted context omitted.

Suppose you choose such a rectangle, let's say it incorporates the 'fringe' of the main cardioid and the fringe of the biggest circle. Within that rectangle, color green all the pieces which connect to the main cardioid and blue all the pieces which connect to the circle. Local connectedness means that there won't be any points in that rectangle which have both blue and green points arbitrarily close. So there are pl…

Thanks for taking the time to explain that. So, in terms I find easier to understand, MLC would mean that if I: -take any rectanglular section of the complex plain that includes part or all of the Mandelbrot set -draw the Mandelbrot set in black -pick an arbitary black point and colour it red -recursively colour every black point touching a red point (flood fill) Then every black point would be recoloured red. And th…

No, that's not right.

Do those first four steps. You wouldn't (necessarily) cover every black point in your rectangle. Choose a remaining black point and flood fill from that, say green. Keep on doing this with different colours until you've covered every black point in your rectangle. You have a bunch of regions of different colours.

Now, if the different coloured regions are all nicely separate, then your set is locally connected. Because each point is either cleanly in one component or cleanly in the other.

If on the other hand your drawing looks like https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_... with mixed up boundaries where some points are infinitesimally close to more than one colour, then it's not locally connected.

The difficulty with intuition is that in our intuition, coloured regions always have reasonable boundaries (think countries in a map: the border can be wiggly but there's never infinitely many tiny bits of one country mixed up in the boundary of two others). In fractal geometry, things like the Newton fractal picture above are quite usual.

Re: The quest to decode the Mandelbrot set

#79

Post by [deleted] »

QuadmasterXLII wrote: post #16
infogulch wrote: post #8

Making a program to render a view of the Mandelbrot set is a fun exercise, I recommend it. I found a viewer that works in the browser: > Mandelbrot Viewer is a universal (desktop and mobile) vanilla JS implementation of a plain Mandelbrot set renderer – supporting mouse, touch and keyboard interaction. https://mandelbrot.silversky.dev/

If you want to keep zooming in the browser and have snappy gpu accelerated performance while basically never hitting “reached limit of numerical precision,” check out https://mandeljs.hgreer.com . The math to make this possible gets pretty funky: I go over the tricks I used at https://www.hgreer.com/JavascriptMandelbrot/

[deleted]

Re: The quest to decode the Mandelbrot set

#80

Post by dekhn »

crazygringo wrote: post #34
infogulch wrote: post #8

Making a program to render a view of the Mandelbrot set is a fun exercise, I recommend it. I found a viewer that works in the browser: > Mandelbrot Viewer is a universal (desktop and mobile) vanilla JS implementation of a plain Mandelbrot set renderer – supporting mouse, touch and keyboard interaction. https://mandelbrot.silversky.dev/

I made quite a sophisticated viewer as a teen, but boy was it slow to run back then, back on a 486. Now people write viewers that run lighting fast in a browser thanks to WebGPU! Like: https://www.reddit.com/r/fractals/comments/o7l4bm/please_try...

The one I wrote in BASIC on my Apple IIe never finished, it hadn't even got the interesting bits of the set after a day or so and I needed to play games.

FRACTINT was great because I could use my 286 and it was fairly fast because used integer whenever possible.

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