• April 19, 2021, 09:50:14 PM

### Author Topic:  a way to accelerate Mandelbrot (etc) deep zoom world record attempts  (Read 826 times)

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#### claude

• 3f
• Posts: 1833
##### a way to accelerate Mandelbrot (etc) deep zoom world record attempts
« on: February 16, 2018, 06:37:12 AM »
You've seen the boring zoom videos: go towards -2+0i for half the video, then go offcenter and zoom the rest of the way to the miniset.

The boringness can be quantified like this: the set is asymptotically self-similar about Misiurewicz points like the antenna tip.  The (complex) multiplier of a Misiurewicz point is the derivative of its periodic cycle (the preperiodic part is ignored).  The asymptotic self-similarity means that zooming (and rotating) by the multiplier leaves the image the same, provided you are zoomed far enough, and the colouring is not too dependent on iteration counts (for example, distance estimation works, as does monochrome edge detection of atom domains, I guess binary decomposition would work fine too though I haven't tried it yet).  Then when you go off-center you take the square root of the multiplier (or whatever root for other powers) at the doubling point, and square root again at the quadrupling point, etc.

If you align the zoom keyframes just right compared to the multiplier, you end up with repeating loops of keyframes (in general this means spinning, but you can unspin when assembling the zoom video).  So you don't need to really render the vast majority of keyframes, you can just render a few and repeat them, splicing the repeating parts together with non-repeating transition sections.

For example, in the Mandelbrot-Burning Ship-Mandelbrot-Mandelbrot hybrid, zooming into the point 0-1i with 2x zoom factor key frames, the first few frames (about 20-30) are unique, but then there is a repeating loop of 10 key frames.  You could repeat this loop 160 more times, then go off-center with a transition of 40 frames, arriving at another repeating loop, this time of 5 frames.  Repeat this loop 160 times and double up with a transition of 20 frames.  The next loop could be 2.5 frames, but to simplify things (avoid changing keyframe interval) you can extend it to 5 frames as before.  Repeat it 80 times, and we're already at zoom depth 2^2900 or so with only 100 or so key frames really needing to be rendered.

Howver, this method is not very practical for accelerating rendering of videos anyone would actually want to watch

See also https://mathr.co.uk/blog/2017-06-06_asymptotic_selfsimilarity.html which just covers zooming into Misiurewicz points, without the transitions when going off-center or doubling.

#### claude

• 3f
• Posts: 1833
##### Re: a way to accelerate Mandelbrot (etc) deep zoom world record attempts
« Reply #1 on: February 16, 2018, 12:30:56 PM »
render time at 1920x1080 without the acceleration technique (3325 key frames):  real 4972m27.057s (3.5 days) / user 16928m6.956s (11.75 days) / sys 13m45.016s
will now try rendering a 4k version with the acceleration technique as proof-of-concept (~205 key frames)

#### gerrit

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• Posts: 2402
##### Re: a way to accelerate Mandelbrot (etc) deep zoom world record attempts
« Reply #2 on: February 17, 2018, 06:03:11 AM »
Maybe a more pleasing video would result if these repetitive sequences could be summarized by just a short fragment.
I tried to do something like this for the bifurcation zooms here: https://fractalforums.org/fractal-movie-gallery/19/m-set-zoom/401/msg2769#msg2769

Very deep zooms into Misiurewicz points (like 1e200) as part of a zoom method (esp. terminal points with either preperiod or period =1) often lead to nice looking results (but boring video of course). Manually zooming to Misiurewicz by 1e200 is also boring of course. I wrote some code to find those points to desired accuracy and then input them into KF. Probably takes longer overall  to find right (pre)period etc. but less boring

More generally speaking, the usual zoom movie strategy maybe needs some new ideas as we now routinely zoom to 1e1000 resulting in an hour long video which nobody wants to watch. Some "algorithmic" way to skip the boring parts seems to be in order.

When new notes/rythms appear in your music patterns is that related to new properties of the thing we're looking at? Couldn't quite figure that out. (Obviously the soundtrack is not generated completely automatically, it's nice.)

#### claude

• 3f
• Posts: 1833
##### Re: a way to accelerate Mandelbrot (etc) deep zoom world record attempts
« Reply #3 on: February 17, 2018, 11:58:32 PM »
Nice warp speed effect in your video.  So far (almost) all my videos have been constant zoom speed, which is a bit boring.  I think I did one using "modified atom domains" that had interesting zooming-combined-with-translation based on Poincar? hyperbolic metric, but that wasn't deep at all.

Finding the preperiod if the period is known can be done by a variant on the box-period method (or probably Taylor ball interval arithmetic too).  I think we discussed two variants of that already, one iterating the box further than the center, the other vice-versa.  "structural" Misiurewicz points tend to have periods of the outer influencing minibrot(s).  Finding the points is via Newton's method I guess.

Soundtrack credits (Dataman and aAirial) for mine are on the web page (below the embedded player).  I did synchronize the video zoom speed to the beat of the music, because the first try was almost-but-not-quite in sync by accident, and the out of sync was very annoying, so I slowed the video down (one number went from 5.7 to 5.0) and added another short piece of music at the end.  Other than the beat-sync, there is no other connection between video and soundtrack.

#### claude

• 3f
• Posts: 1833
##### Re: a way to accelerate Mandelbrot (etc) deep zoom world record attempts
« Reply #4 on: February 19, 2018, 04:46:10 PM »
4k version:

https://youtu.be/SsvL03ir5xo

2.5 days to render and assemble zoom video, over a day faster than the 1080p version.  It pays well not to render repetitions!

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