
Insanely deep fractal zoom - geekpressrepost
http://kottke.org/10/02/insanely-deep-fractal-zoom
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thristian
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.

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gjm11
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>

~~~
camccann
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.

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akkartik
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>

\---

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.

~~~
teamfresh
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.

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hobb0001
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...

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jrockway
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".

~~~
jerf
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.

~~~
gjm11
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.

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

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danblick
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>

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

~~~
Groxx
whuh?

~~~
fexl
:) 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.

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Groxx
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.

~~~
teamfresh
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...](http://www.metacafe.com/watch/1663241/a_very_cool_deep_fractal_zoom_to_e214/)

~~~
Groxx
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 _

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

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

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Markus
Loved it, thanks for sharing!

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BearOfNH
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.

