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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

#51

Post by Baeocystin »

IronBacon wrote: post #37
crazygringo wrote: post #34

Earlier quoted context omitted.

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...

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 the problem was, and I remember spending hours pouring over my code, trying to figure out where the bug was. Good times. :D

Re: The quest to decode the Mandelbrot set

#52

Post by QuadmasterXLII »

RBerenguel wrote: post #22
QuadmasterXLII wrote: post #2

If you enjoyed learning about the Hashlife algorithm for Conway's Game of Life, the bilinear approximation algorithm for computing whether a point is in the mandelbrot set has the same je ne sais quoi. No one has done a really accessible writeup of it yet, but this blog post and the linked forum thread are a good start: https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_pra...

I did some research on this on the side for my dissertation, but never published it. The fact centers approximate the boundary generalises to almost any point in the plane as a consequence of normality of some sequences, and generalises to most families of complex iteration under very mild conditions. I’ve had a preprint that I never felt like finishing for something like 15 years lying around.

If you ever decide to put it out as-is, I'd love to read it!

Re: The quest to decode the Mandelbrot set

#53

Post by clintonc »

OscarCunningham wrote: post #42
clintonc wrote: post #38

Earlier quoted context omitted.

MLC stands for "Mandelbrot Locally Connected". It's not obvious, but this is equivalent to the bulbs of the Mandelbrot set (the domains of parameters where almost all points get attracted toward periodic orbits) are dense in the Mandelbrot set. Everyone believes it to be true.

Yes, but how exactly are the bulbs arranged? Wikipedia says 'Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy'. Which sequences of bulbs have little copies at the end of them? And how do the little copies attach?

The combinatorics of how the Mandelbrot set is put together is well-studied, and rather independent of MLC. The arrangement of the bulbs on the boundary of the "main cardiod" (which is where there is an attracting fixed point) is described here: https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_a.... Generally, the patterns are given by something called Lavaur's Algorithm; see https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_com... for some explanation. Attachment points are always at the "root" of the Mandelbrot set, which is the cusp of the main cardioid.

A consequence of MLC is that the combinatorial picture given by Lavaur's algorithm and related analyses is "complete" -- all dynamical information is available from the combinatorial models.

Re: The quest to decode the Mandelbrot set

#55

Post by G3rn0ti »

hermitcrab wrote: post #48

I don't think I understand what 'locally connected' means. You can easily choose a rectanglular area that contains 2 areas of the set that are not joined.

It means you can choose two points inside the Mandelbrot set and always find a curve that connects the two without you ever needing to lift the pencil.

Re: The quest to decode the Mandelbrot set

#56

Post by nhatcher »

hermitcrab wrote: post #50
nonsensikal wrote: post #49

Earlier quoted context omitted.

You don't get to choose a rectangle, you choose a point.

The doesn't seem to fit with the comb analogy. I still don't understand.

Non locally connected spaces are a bit pathological. Means that given a point there is always a neighborhood of the point (might be very small) that is connected. An example of a connected but not locally connected is: https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve From (0, O) any neighborhood, no matter how small contains points that belong to the curve but cannot reach (0, 0) and stay in the neighborhood.

Re: The quest to decode the Mandelbrot set

#57

Post by hermitcrab »

G3rn0ti wrote: post #55
hermitcrab wrote: post #48

I don't think I understand what 'locally connected' means. You can easily choose a rectanglular area that contains 2 areas of the set that are not joined.

It means you can choose two points inside the Mandelbrot set and always find a curve that connects the two without you ever needing to lift the pencil.

Isn't that "connected" rather than "locally connected"?

Re: The quest to decode the Mandelbrot set

#58

Post by crazygringo »

hermitcrab wrote: post #57
G3rn0ti wrote: post #55

Earlier quoted context omitted.

It means you can choose two points inside the Mandelbrot set and always find a curve that connects the two without you ever needing to lift the pencil.

Isn't that "connected" rather than "locally connected"?

I'm aware that "locally connected" has a very specific meaning in math:

https://en.wikipedia.org/wiki/Locally_connected_space

Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.

Re: The quest to decode the Mandelbrot set

#59

Post by AlanYx »

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.

Re: The quest to decode the Mandelbrot set

#60

Post by hermitcrab »

crazygringo wrote: post #58
hermitcrab wrote: post #57

Earlier quoted context omitted.

Isn't that "connected" rather than "locally connected"?

I'm aware that "locally connected" has a very specific meaning in math: https://en.wikipedia.org/wiki/Locally_connected_space Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.

>Unfortunately I don't have the slightest idea what it actually means

Not just me then? ;0)

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