Insanely deep fractal zoom | Hacker News Reader

Insanely deep fractal zoom

kottke.org

Sounds like somebody needs to get themselves a copy of XaoS and have a play around:

    http://wmi.math.u-szeged.hu/xaos/doku.php

That said, most of my knowledge of fractals comes from poring over the (insanely complete) documentation of Fractint when I was a wee lad. I've forgotten most of it by now, but there sure was a lot to wrap my young head around.

Very neatly done. What impresses me most is the way that the creator has managed to go so far in while scarcely ever finding any interesting bits (and, when he did, zooming straight past them as quickly as possible).

The comments on Vimeo indicate that the creator is aware of this and is (or rather was, a year ago when it was posted) working on something more interesting. Here's a more recent one by the same person, which I preferred: http://vimeo.com/1973524 and another, which traverses more interesting bits of the set than either but isn't of such high video quality: http://vimeo.com/9522366

Seems like staying too close to the real line was a lot of the problem--spirals and asymmetric structures show up roughly in proportion to how imaginary the region in question is.

That and a lot of time zooming toward what I think are Misiurewicz points--if memory serves me, places on the boundary of the set where multiple "branches" meet, and tend to have very strong self-similarity along the branches and detail shrinking infinitely as the branch point is approached.

Pretty awesome. The description is great, but there's one false note: "If you were 'actually' traveling into the fractal your speed would be faster than the speed of light." 2 objections:

a) What is the speed of light in the mandelbrot set? What zoom level do you designate to be 1m?

b) The 'speed' the camera is traveling at is hugely variable. It's basically slowing down by an order of magnitude every constant window of time. So if the final image is 1m wide, you're travelling 10^n's of universes every second at the start. The above sentence doesn't come close to doing this idea justice.

Update: the sentence is from the original at http://vimeo.com/1908224

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I went over the video a second time to see how often the camera 'steers'. The entire second half drops pretty much straight down a radially symmetrical 'well'. There's 2 obvious changes in bearing between 2 and 5 minutes. In the first 2 minutes it's harder to keep count because much of the time there's no radial symmetry. I suspect the bearing is changing almost constantly.

Summary: the video was made by choosing a point about 2 minutes worth of zoom in, and then pretty much dropping straight down, except for a couple of tacks. Whoever did this was probably trying to maximize the diversity of views; at any point in the video a different tack may have ended up back at something like the starting point much sooner. It's mind-blowing to contemplate.

Travelling at The speed of light - approximated as 300,000 kilometres per second or 186,000 miles per second, It would take 8 minutes to reach the sun. After 8 minutes of the animation - the size of the original set would be bigger than that distance by a size you just can't comprehend.

I've always wanted to create a similar fractal zoom into the real number space to illustrate the position of pi. Some sort of Flash or JavaScript thing that would keep generating more digits on the fly and could run forever. I never seem to get get that kind of free time anymore, though...

It's strange that his computer would choke on high zoom levels; computing the mandelbrot set is proportional to the number of pixels on the screen -- the zoom level is not relevant. (You just run a set of computations for the coordinates of each pixel you want to know about.) I guess if you get past the point where a double can represent the coordinates, you'll hit a slowdown, but I doubt that is at "5x zoom".

The necessary accuracy keeps taking more and more bits, if you think about it. If you've only got, say, 200 bits of float to play with, then when your down at a resolution where things on the screen are closer together than that resolution can handle, you have to crank the bits up, or you'd just render the same color over and over. (Actually you have to crank it up before then, of course, I'm just reducing it to obvious absurdity for demonstration purposes.)

So, it's like the way adding integers is O(1), right? Sure... as long as your integers are always 32 or 64-bit and you don't mind a bit of light truncation. In general it's higher than that.

That's absolutely right, but it's not the only reason. The interesting bits of a Mandelbrot set plot are the ones that are on the boundary of the set itself. Remember that the Mandelbrot set is the set of points for which a particular iteration never diverges to infinity. So a point outside the set but very very close will diverge very very slowly: it will take a lot of iterations before you can tell it's outside the set. (The colours you see in a typical Mandelbrot set plot indicate how many iterations it takes.) So, if you zoom really close in then every point in your plot either lies outside the set but takes many iterations to determine that, or lies inside the set (and therefore takes as many iterations as you're prepared to do).

So you're doing more iterations, with higher-precision numbers, as you zoom closer in.

right, this kind of anim requires arbitrary precision arithmetic. i wonder how long it took to do. my 1080p work is about an hour per frame, eg http://www.archive.org/details/HighFidelityDemo would take >2 months on a regular computer, and HiFiDreams over 100 years.

That's not quite true – as you zoom in, you're no longer simply taking Re(c) = x pixel on the screen and Im(c) = y pixel on the screen. The more you zoom in, you're moving in on the Re/Im axes, which means you're no longer looking at integer values for the computation. You start looking at floating point values for x (Re(c)) and y (Im(c)) when computing the sequence. Therefore, the computer begins to do more and more floating point computations on, ostensibly, arbitrarily accurate fp.

EDIT: Added "pixel on the screen".

That was pretty amazing. The end is a real "ah ha" moment.

This site ("Dimensions") also has some great math-related visualizations (particularly the ones showing 4-dimensional objects):

http://www.dimensions-math.org/

Fractals are in episode 6 from ~8 minutes onward:

http://www.dimensions-math.org/Dim_reg_AM.htm

I see a lot of "eight-ness" in there. Some other powers of two as well. I didn't notice any "prime-ness" though.

whuh?

:) Sorry about the vague "eight-ness" there -- I just mean I see lots of patterns with eight prominent features, or "spokes." I saw quite a few with sixteen spokes, and probably higher powers of two as well, though I wasn't in the mood to count the large numbers.

And regarding "prime-ness", I mean I didn't notice any patterns with 7, 11, or 13 features -- that sort of thing.

I suppose the prominence of powers of two has something to do with "period doubling" in chaos theory, I don't know.

Neat, though best seen at a couple times faster than it currently plays at. It drills down and down and down into long-repeating stretches, so it's somewhat monotonous at times, but the overall payoff of seeing it change is fascinating. And you can certainly trance-out in the last chunk.

someone has created a faster version at http://vimeo.com/9403208 although this may have gone to far the other way at only 35 seconds long! there is also a four minute version with colour cycling found here

http://www.metacafe.com/watch/1663241/a_very_cool_deep_fract...

Yeah, 35 seconds is a bit fast, though interesting in its own way. It really shows the repetition which was hidden in the original by sheer length.

And that color-cycling one is... I dunno, hallucination / head-trip inducing o_O I can't tell if I want to keep watching or not. It's kinda too much, but it's pretty...

* stare

yeah the colour cycling one starts to look like ripples in water, kinda freaky!

Hmm I would like to know on which number is he actually zooming on :)

Loved it, thanks for sharing!

Incredible! Coloring is very 1970s. I'd like to get a T-shirt with some of the later patterns.
The ending is not to be missed.

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