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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

#61

Post by matt-noonan »

returningfory2 wrote: post #54
hermitcrab wrote: post #50

Earlier quoted context omitted.

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

I think this part of the article is incorrect.

It's almost correct, but misses the point in an annoying way that kind of ruins the example. What does work is something like the subset of the plane given by { (x, y) | x real, y rational } U { (0, y) | y real }. This is connected, because you can walk from any point (x,y) to any other point (x',y') by traveling horizontally to the Y axis at (0,y), vertically to (0,y'), then horizontally to (x',y'). But it isn't locally connected away from the Y axis because for a tiny enough open set S around a point (x,y), there are other points in S that you can't get to from (x,y) without leaving S.

Re: The quest to decode the Mandelbrot set

#63

Post by chihuahua »

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...

I typed in a BASIC program for a C-64 from a computer magazine. That was a popular mechanism for software distribution back then.

I thought "this will never work", but I was amazed that it did in fact work. I remember letting it run overnight to find a complete 160x200 4-color image the next morning. Just clearing the screen using a loop to fill 8192 bytes with 0s took over a minute.

Everything about the computers at the time was so terrible, I don't feel any nostalgia for those days or that technology.

Re: The quest to decode the Mandelbrot set

#64

Post by LoganDark »

QuadmasterXLII wrote: post #46
sapiogram wrote: post #45

Earlier quoted context omitted.

No scrolling with the scroll wheel though :(

The whole thing is a tower of spaghetti bit hacks in the name of depth and speed. Embarassingly, it actually can't actually render zoom levels that aren't powers of two and I'm not sure how to change that.

Can you make it so that the position you clicked stays at the same position after each zoom, rather than centering it on your cursor? That way you can click multiple times to zoom in on the same point.

Re: The quest to decode the Mandelbrot set

#65

Post by dylan604 »

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…

Sounds like a couple of people decided to take an extended gap year, and then came back more focused and in a better place for the number crunching

Re: The quest to decode the Mandelbrot set

#66

Post by OscarCunningham »

clintonc wrote: post #53
OscarCunningham wrote: post #42

Earlier quoted context omitted.

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/…

Thank you, that's very helpful!

Re: The quest to decode the Mandelbrot set

#67

Post by fractaltemp »

hnfong wrote: post #31
ggambetta wrote: post #19

Earlier quoted context omitted.

> I can't help but wonder whether the similarity of the "bulbs" to the rather unique Buddha "hairstyle" (of allegedly funny lumps of hair) was just a coincidence. Honest question: what else do you think it could be, if not a coincidence?

Honest answer: some early Buddhist followers taking some form of acid/mushrooms and saw geometric visions of the Mandelbrot set and thought it was a divine manifestation of Buddha? I mean, I don't think this is likely, but that's the best I got. I hear the brain likes to go into geometry mode when hallucinogens are ingested, and I suppose the brain is theoretically powerful enough to compute the Mandelbrot sets...

I've done a wide variety of hallucinogens. While I've seen things that could be described as fractal (in that they are nested and self similar), I've never seen a literal time escape fractal. In my experience, hallucinations look more like turbulent flow, like smoke. Deepdream isn't a terrible first approximation, but may give you the impression that the hallucinations are more dramatic or total than is typical.

I can't rule this hypothesis out entirely, but I'm very skeptical. Pareidolia seems more likely to me.

Re: The quest to decode the Mandelbrot set

#68

Post by tda »

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/

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

Re: The quest to decode the Mandelbrot set

#69

Post by neilkk »

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.

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 places where 'locally non-connected parts' of the set can be close together, but there must be a border between them, rather than them being hopelessly entangled.

Strictly speaking, you would do this coloring with all connected components of the intersection of your rectangle and M. (And the rectangle could be any region.)

The example messes this up, although it is a correct example, the square on the diagram showing the local piece containing non-connected parts is wrong. The comb has more and more teeth, infinitely many in a bounded space, on the left side. Only a rectangle which includes the left edge properly shows why the set isn't locally connected. The rectangle pictured includes finitely many teeth which have a separation between them. A rectangle overlapping the left edge of the comb would include separate components which get arbitrarily close to that left edge and so can't be separated by a border.

Re: The quest to decode the Mandelbrot set

#70

Post by hermitcrab »

neilkk wrote: post #69
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.

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 this would work with a pixel based image of the mandelbrot if the image had a high enough resolution. Is that right?

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