Understanding the mandelbrot set

[Pauldelbrot]

I think the orbits are simply mirrored?

Yes.

(a - bi) + (c - di) = (a + c) - (b + d)i, so the sum of the x-axis mirrors (or conjugates) is the conjugate of the sum.

(a - bi)(c - di) = ac - bci - adi + bdi² = (ac - bd) - (bc + ad)i, so the product of conjugates is the conjugate of the product, too.

So, if we have f(z, c) = z² + c, f at the conjugates of z and c will be the conjugate of f(z, c). And if we start with z = 0 (which is its own conjugate) and the conjugate of c and iterate, by induction the entire orbit is conjugate to what we get by starting with 0 and c.

If you want asymmetry, the orbits have to escape at different points, and you either need a different function or you need a bailout trap that isn't symmetric in the x-axis. The usual bailout traps (complements of disks of various radii about 0) are all symmetric.

[makc]

...or go for complex power, then it will no longer mirror.

[cKleinhuis]

I see thx

[taurus]

This is so cool! "Understanding the mandelbrot set" gets a whole new quality here. It's a very physical expierience with theese tools - more like apprehension, than understanding.
Thanks makc and Syntopia - tools like that would be perfect for teaching purposes. 

[bib]

I also made an similar orbit viewer (in WebGL) a few weeks ago:

Available here: http://hvidtfeldts.net/WebGLSet/orbit.html

No splines, but you can zoom in realtime, if you have a fast GPU :-)

Super cool, great orbit visualisation tool! Thanks Mikael

[cKleinhuis]

@syntopia great visualisation, but let me give you one tiny modification that i find is very important:

do not scale the orbits, perhaps show 2 coordinate systems, one for the current zoom location, and one for the orbits, which should be a normalized view of
the bailout radius, the bailour radius can be shown as a circle, and should not scale with the zoom location as well, because it is very interesting to zoom
very very close to the mandelbrot set and examine how long the orbits can be before the bailouting happens, but when zoomed in it is not clear what
happens to the orbit

@all others, the most interesting points in the orbit view are the centers of the bulbs, in each such center you find a pure periodic behaviour of the orbit
the bigger centers produce small loops like 3 values that are visited throughout the whole iteration, which explains easily why they do not diverge each center circle of the mandelbrot owns such a period, which can be nicely viewed when the orbits would not scale ...

[taurus]

the most interesting points in the orbit view are the centers of the bulbs, in each such center you find a pure periodic behaviour of the orbit

same is within the mini brots exactly there, where the origin is in big brot...

[cKleinhuis]

same is within the mini brots exactly there, where the origin is in big brot...

the periods are in every minibrots-circle, they only get longer and longer, a period can be a really really long, depending on the zoom level minibrots a period can easily reach 1 million before it reaches its starting point

[Dinkydau]

This is so cool! "Understanding the mandelbrot set" gets a whole new quality here. It's a very physical expierience with theese tools - more like apprehension, than understanding.
Thanks makc and Syntopia - tools like that would be perfect for teaching purposes.  

Yes I absolutely agree. Those two programs combined give a very great insight.

Aside from difficult mathematics, I thought I knew everything about the mandelbrot set, but I hadn't thought of studying the WAY numbers go or don't go to infinity.

This thing is so inherently fractal, it's amazing.

[eiffie]

I like this thread! Thanks for starting it Dinkydau. I also find the orbits fascinating and made some pics by taking points near the border (and near each other) and plotting their orbits. You can see that the seed values are near the upper large bulb (period 3) so it has three main branches, like a julia. Some points veer off into smaller minibulb orbits. I believe you can tell the "lineage" of a minibrot by studying its period. It will be multiplied by its parent bulb and its grandparents. That is why the periods get so long. I may be wrong about this, can someone verify?

[makc]

Btw, here is the reason I went into this orbit study business:



I had to wait over 5 hours for this random minibrot at 1e20 zoom and 350K iterations. Now, how much faster would it render if actual orbits were taken into account, and loops for orbits that are obviously converging after, say, 1000 iterations were terminated? (edit: this assuming that minibrot took most of the time and loop for outer region pixels terminated early - if it's not the case, we'd have to detect divergence, which is probably harder)

Taking it further, if we simply switch to coloring based on orbits behavior in 1st 1000 iterations, it could already provide enough data to render equally detailed (but different) image but 350 times faster nah, that doesn't work, because until the iteration actually happens there is no information in numbers that could give us a hint if it will converge or not - in other words, this kind of coloring has the same amount of details as classic method.

[Syntopia]

@syntopia great visualisation, but let me give you one tiny modification that i find is very important:

do not scale the orbits, perhaps show 2 coordinate systems, one for the current zoom location, and one for the orbits, which should be a normalized view of
the bailout radius, the bailour radius can be shown as a circle, and should not scale with the zoom location as well, because it is very interesting to zoom
very very close to the mandelbrot set and examine how long the orbits can be before the bailouting happens, but when zoomed in it is not clear what happens to the orbit

I updated the example as you suggested and it is indeed more usable now:



Available here: http://hvidtfeldts.net/WebGLSet/orbit3.html
(Slider for escape radius, two independent views)

[cKleinhuis]

wow,congratulations, this really gives the insight!

i was fiddling around to adjust the orbit zoom at work, but i figured out you are doing a double-iteration for visualising the orbits
pretty nice to see the behaviour from the deepeer areas, and you see how mini-starting locations variations just affect the end behaviour
and the periods from the minibrots are really "stable" some minibrot orbits really look like "alphabets" great, it was hard to get an idea of
the behaviour inside the minibrots, but every minibrot produces another pattern ... awesome tool!

[makc]

may I suggest another tweak: color orbit vertices from, say, red to green, so we could see the direction easier at a glance, without having to trace the lines?

[cKleinhuis]

And where is the julia gone
a formula selector would be just awesome perhaps just for simple mods like burning ship and other powers?

Great work dude! And such a stylish presentation!