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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

#41

Post by BeetleB »

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 did it in QuickBasic (also on a 486).

Now that's slow.

Re: The quest to decode the Mandelbrot set

#42

Post by OscarCunningham »

clintonc wrote: post #38
OscarCunningham wrote: post #29

What is the conjectured topology of the Mandelbrot set if MLC is true? My understanding is that there's a certain number of bulbs, each centred around a point which becomes periodic with period p after k steps. But how do they all stick together?

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?

Re: The quest to decode the Mandelbrot set

#43

Post by OscarCunningham »

qazxcvbnm wrote: post #39
bongodongobob wrote: post #36

Earlier quoted context omitted.

The entire set is connected iirc.

But what would be its homology, for instance?

It's known that it's connected and simply connected. So if it's locally connected then I think it has to be contractable.

Re: The quest to decode the Mandelbrot set

#45

Post by sapiogram »

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/

No scrolling with the scroll wheel though :(

Re: The quest to decode the Mandelbrot set

#46

Post by QuadmasterXLII »

sapiogram wrote: post #45
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/

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.

Re: The quest to decode the Mandelbrot set

#47

Post by bunabhucan »

auroralimon wrote: post #44

i’ve often wondered if mandelbrot is what you get when you do a simple quadratic iterator in complex numbers, what are the comparable sets for quaternions and octonions??

https://en.m.wikibooks.org/wiki/Pictures_of_Julia_and_Mandel...

There's a whole world of this as well as iterating different functions.

Re: The quest to decode the Mandelbrot set

#50

Post by hermitcrab »

nonsensikal wrote: post #49
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.

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.

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