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Hello everyone,

I like fractals a lot but I've only recently started to wonder what exactly is going on. I was wondering to which extend there is insight in the shape of the mandelbrot set. I have been asking myself many questions that I couldn't answer myself lately regarding this.

To understand the mandelbrot set better I have tried to find more information, but there isn't much information available about this or it's too difficult to understand for me. I know complex numbers, the mandelbrot function, how it is plotted in the complex pane, how the colors are generated and by simply exploring I have gotten to know some patterns in the mandelbrot set. But that's where it ends. It's only very basic things that I know while everywhere on this forum people are already discussing, what is to me, incomprehensible maths about 3d and 4d fractals. Probably the 2d mandelbrot set has been thoroughly investigated by many already but I'm still at that stage.

For example, you can see that the mandelbrot set for a significant part consists of "circles". While at some points they really look like circles, it depends on where you zoom. They're clearly not all perfectly round. Are there actually any circles in the mandelbrot set?

It looks like the largest of the circles in the mandelbrot set is connected to the weird shape on the right at -0,75, and connected to the next circle on the left at -1,25, meaning the diameter of the circle is 0,5. Is there any proof of this? Do all "circles" in the mandelbrot set have such a round diameter? Is there a pattern, a relationship between the sizes of all the circles?

On the real axis, from -2 to 0 no values go to infinity when iterating the mandelbrot function. Am I right to conclude that this means that on the antenne of the mandelbrot set lie infinitely many mandelbrot sets together forming the line?

Where does the mandelbrot set "end" at the real axis? It's obvious that it reaches from -2 to some positive value, but I wonder what that value is. Can it be defined?

Is there a method to calculate where a small mandelbrot set lies at a certain zoom level?

Are we 100% sure that the mandelbrot set is actually a fractal? Because its definition doesn't involve anything I would expect to turn out to be a fractal. I've started to question this after I read that the Riemann hypothesis (which is unrelated to fractals, just an example) is considered not proven even after some people used a computer to calculate billions of values and they were all according to that particular hypothesis. The hypothesis was still not tested for all values, which are infinite. So, are we really 100% sure that the mandelbrot set is a fractal, and how do we know that? If this has been investigated, it will probably give a huge insight in why the mandelbrot set looks like what it looks like too.

If any of my conclusions do not make sense to you I can explain them further.

in fact along the x-axis (where the imaginary components are zero) is one of the most interesting parts of the mandelbrot set. the image below (taken from wikipedia) shows the correlation between the mandelbrot set and the feigenbaum graph. you see there, that the boundary values of the mandelbrot iteration (for all bi=0) converge to one or more values. within the antenna the iteration goes chaotic. you see also, that the antenna is at full length part of the mandelbrot set (as long as bi=0) so i guess your assumtion about the antenna is correct
as the feigenbaum graph is extraordinary important to chaos theory this correlation is the main port between fractal geometry and chaos theory.

(http://upload.wikimedia.org/wikipedia/commons/b/b4/Verhulst-Mandelbrot-Bifurcation.jpg)

http://www.miqel.com/fractals_math_patterns/visual-math-mandelbrot-magic.html

here is some good info about shapes in fractals. the most interesting thing it sais is the M-set has been proven to be the longest continuous single 2d line in maths, as it is essentailly just one boundry line. Which means that it is a pretty special formula/class of formuli. I dont know if i believe that it's infinity is the only and largest infinite line formula in maths, but it's intersting.

The shape of the M-set seems to be one of those fundamental shapes in our universe that we are not presently qualified to comment on! Why are there 132 elements of different sizes and shapes that combine into millions of types of rocks, stars, and enzymes etc.

These are the patterns of the natural chaos/complexity of the cosmos in it's present state. i think that's all we can say about these shapes! our concept of shape has not evolved enough to logically compare things like the mandelbrote to anything in our tangible reality.  same with spiral galaxies etc.


It's inherently a different class of formula from any other ones in mathematics, it is simpler, it doesnt rely on any angles, shapes, or any object oriented stuff. it's results are more complicated.


the formula is very abstract and untangible, a perverse experiment by a genius mandebrot. it simply a number bending back into a square of itself or something. the fact it is an abstract logarithmic function that happens to bend back into itself in a small space means it has a kind of logarythm, pointy, shape, with large curves representing the lower expnential values before the curve picks up, and the infinitaly small details correspond to the distant reaches of the exponent.

it's kindof a perfect hybrid between a circle and an infinitely spikey exponent, the circle folds over itself into a bear shape and the spikes go into the distant spirals, same as when some 2 line maths formulas produce for you figure of 8 shapes, or a doodly but this one is infinite... a small formula like a snail shell shapes, just an infinitely folder over snail shell, in 2d.

thats my 2c :)

Thanks for the information

"Are we 100% sure that the mandelbrot set is actually a fractal?"
To be picky it isn't a fractal. Its border is a fractal. The set itself is the black bit which has an area. The border is a fractal by most definitions.

To me the most interesting feature of the mandelbrot set is that it is universal. So you can alter the formula by rotating, scaling or translating every iteration and you still get a resulting mandelbrot set. This roughly describes why the mandelbrot set pops up in many different formulas.

No other fractal is universal, it is completely unique to the mandelbrot set, actually, to the multibrot sets (Z²+C, Z³+C, .. Zⁿ+C).
It is as though these multibrots are the foundational shapes in some way. Just like the natural numbers are foundational to arithmetic.

for sure we can be so picky. But we can also be so picky, that the borderline does not fit one important definition - the namegiving dimension is not fractal. with 2 it is integer. i can't say, that i understand, how a not area-filling curve can have a haussdorff dimension of two. and it's still not clear, if it has an area at all - but at most it is finite.

i think we could count a lot more oddities without finding an end, but i guess we are in common, that this piece is still one of the most mysterious parts of math.
for me as a non-theoretician the mandelbrot set is not only a fractal, it is the mother of all fractals.

Why wouldn't a mandelbrot set have a precise area formula same as a circle? the formula for PI is "recursive" and there must be a constant like PI for fractals... i am amazed that there isnt a formula for it, surely someone will find one some day. Perhaps they should add it to the X-prize style maths challenge. i circle doesnt really have an area because PI is infinite, and M-set should be the same?

Haven't they just computationally measured the area of the m-set to a few hundred decimals by counting the black pixels at crazy zoom levels?

sorry, that i spoke unclear. i was talking about the boundary line of the mandelbrot set. it has the dimension two and might not have an area at all...

they have. it is: 1.506 591 77
don't know the unit. wikipedia remains silent.

The cusp of the elephant valley area is at exactly 0.25. Also the biggest bulb attached to the core of the set is in fact a perfect circle, and is probably the only part that is exactly circular. It's centered on -1 and has a radius of 0.25. This can be proved but requires digging into calculus and dynamical theory a bit.

The antenna is, as you surmised, essentially made of minibrots. Every point on it is either inside a minibrot or a "point of accumulation" of minibrots -- that is, given any point P on the antenna not inside a minibrot and any ϵ > 0 you can find a minibrot inside a circle of radius ϵ about P. In fact the same is true of any Mandelbrot set point along any dendrite, seahorse spiral, etc.

bumping the thread, with my orbits explorer:
http://swf.wonderfl.net/swf/usercode/3/3b/3b95/3b95ba3061861169c86ea897e54f783a01526847.swf
maybe not the best way to visualize orbits, but the best of what could be done in 30 minutes

Nice!

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

(http://hvidtfeldts.net/WebGLSet/thumb.jpg)

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

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

yes I thought about the zoom too, but then you need separate unzoomed view to see the orbit.

Sorry for typos just using the tablet. .

Very good thread here, i can not promise anything but i am working on at least a nice overview on what the mandelbrot set is and how the julia set belongs to it i have nothing to show yet
but in my research i found another question to think about, which is:

Why the heck is the mandelbrot symetrical on the x axis? The points are behaving completely differently but bail out at the exact same iteration step

next issue will be very insightful at least :)

The other things you already scratched the surface of i can answer at least some

@taurus space filling curve is a bit unprecise naming but when you watch the mathematical definition of how a space filling curve is defined you will see that this just means that every sub part of this curve has to posess infinite lenth as well

the mandelbrot set is somehow the mother of fractals because of its beauty but for sure not the first

i just spit out everything in my head right now

i wonder qhy the visualisation method is not used to understand instable behaviour of numerical functions especially for fluid dynamics, what the escape time method is basically visualising is the speed of growth

the mandelbrot set is created by an awesomly beautiful simple formula, some of you already heard from the affine transformations which are used for the plain ifs fractals like zhe fern for example

as i never grt tired of explaining is that hence complex multiplication is a rotation and a scaling it is the addition qhich completes the simple affine transforms which are rotate,scale and translate

tglad with his mamdelbox incorporated two more transforms wich are inversion and reflection, and put them in a equally beautiful fomula. . .

So regarding that chaos occurs very fast like in the logistic equation qhich formulates population growth or the three body problem from physics it is a representand for the beauty of complexity that occurs from simple processes

uff, i hope to finish my introduction to the mandelbrot set before the contest starts and before my next trip to mexico i will return talking about the mandelbrot set all the time because it is full of an awesome whole lit of interesting aspects

many of them still just lay around and are not yet appkied to more usefull processes and we fractal fanatics are often just smiled at but you will be astonished what universal insight will pop out from this object

phew . My 2 c ;)

I think the orbits are simply mirrored?
(http://hvidtfeldts.net/WebGLSet/thumb2.jpg)

Here is a orbit viewer for the mirrored point as well:
Available here: http://hvidtfeldts.net/WebGLSet/orbit2.html

Thank you for that syntopia i will examine your orbits viewer and perhaps use it as well

but my question just goes a little deeper:

Why are they mirrored?

Consider the pure imaginery points i and -i

in the iteration process when squared they both land on the -1 real position

so, what happens with the translation? Both points are moved one along the y axis
to -1,1 for the point started on positive i and -1,-1  for the point started on negative i

it seems obvious that their positions are just mirrored but isnt it quite interesting qhy this holds true for even such 'odd' locations whose distance to zero is square root of 2??

We see that they are symetrical but do we really understand why it holds true for every position on the plane?

It might aound a bit weird to bring it up, and for an expert mathematiciand it might be really trivial but i find it quite fascinating that the multiplication itself is quasi symetrical on the x axis, for the translation it is obvious but in my eyes not so clear for the rotation thing where we are at the pi circle definition

:) it is mirrored for sure, but quite puzzling for me :/

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.

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

I see thx

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.  ;D

Super cool, great orbit visualisation tool! Thanks Mikael

@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 ... ;)

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 ;)

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.

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?
(http://fc09.deviantart.net/fs71/f/2012/356/5/2/border_buddha_by_allenflusa-d5osfq0.jpg)

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

(http://i46.tinypic.com/ifmwia.png)

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.

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

(http://hvidtfeldts.net/WebGLSet/thumb3.jpg)

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

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!

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?

And where is the julia gone ??? :D
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!

woot, on my homepc it is running like a charme!!

would you mind if i modify your script, one hand to remove the double iteration, and on the other hand to give it a formula selection ;)
will send you an update...

agree, a key feature of your (makc) version was, to see the development of the iteration. I spotted some places (difficult tom aim), where the iteration oscillated some times before bailing out to infinity. Syntopia's version is still static, showing no direction - no development.
And for sure the pragmatician wishes the current coordinates of the crosshair...
But to continue the eulogy. The ammount of creative crazyness here is unbelievable. For one who waited minutes for a 640x480 version of the basic mandelbrot set in his early days, this realtime exploration is great, great, great! No words left for this...

By all means play around with it. I don't think it is possible to remove the double iteration though, if you want to render the orbit path on the GPU side: even though it seems slightly excessive to calculate exactly the same path 250.000 times for each frame :-)


It is slightly difficult, since I don't know the number of iterations before breakout (so it is hard to scale the gradient) -though I could do another pre-pass to calculate the number of iterations.

Besides that, it is still difficult to visually trace more complicated orbits:
(http://hvidtfeldts.net/WebGLSet/thumb4.jpg)

I guess animating the trace would work better.

just a thought. Leave the scale equal to the max iteration number by default. an additional slider can adjust a possible gradient. maybe it is getting more complicated to use, but ignoring the gradient slider wouldn't make anything worse for anyone. the slider would only give an additional opportunity.

Indeed. But perhaps coloring edges could work better?

an example with colored nodes.
http://www.khanacademy.org/cs/mandelbrot-spirals-2/1030775610 (http://www.khanacademy.org/cs/mandelbrot-spirals-2/1030775610)

Adding to the understanding of Mandelbrot Fractal, I was wondering, have you ever seen an animation of Minibrot evolution? If I understand correctly, any minibrot will transform into the main Mandelbrot after a finite number of transformations. What if we generated an animation, each time changing the angle orientation, position and zoom of the image to see the Minibrot evolve? I will do it in a few days if it's not done already. However I have only double precision in F77. Can someone else do it more professionally?

Ok, that didn't work.
All this time I was under the impression that points of the M-set boundary jump from place to place on the boundary. I guess that happens in Julia only, so there is no Minibrot evolution whatsoever. Now playing with the Mandelbrot Spirals. Should have paid more attention.

I made some continuous [z_n+1(t) = z_n^t + (1-t)*z0; t=1...2 ] transformations of the M-set. Hopefully tomorrow I will post an animated gif. These are just some teaser frames, initial, one iteration, 6.9th iteration, all calculated from a much bigger image of the initial M-set. (12 times larger with 3x3 AA)
(http://furan.sweb.cz/nyx/Frame0000.gif) (http://furan.sweb.cz/nyx/Frame0100.gif) (http://furan.sweb.cz/nyx/Frame0690.gif)

Coloring is not correct, just black-red-yellow-white that gets mixed. What I want to do later is to use original image without AA and follow the colors. They should slowly filter out, starting with the red diverging into the infinite.

500 frames, 0 to 50 iterations
(http://furan.sweb.cz/nyx/MandelForming.gif)

Yes, the periods multiply by denominator of internal angle as you go through touching components away from the island's root.  And for islands in the hairs, they have their own periods but the orbit will get close to 0 when a shorter nearby orbit gets close to 0 too.  You can use the indices of successive minima of the magnitude of the iterates to get a feel for which periods are influencing that point - I call that list of numbers "partials" and gave some examples here: http://www.fractalforums.com/programming/cheap-way-to-determine-angle-in-bulb-t16259/msg62419/#msg62419

Each island minibrot is like a copy of the whole set, with all its periods multiplied by a factor, plus additional hairs all over with even more details that aren't necessarily multiples of the island's root period.  Keywords to search for on this part are "renormalization" and "tuning", here's one page that gave me several flashes of intuition:  http://www.ibiblio.org/e-notes/MSet/Contents.htm (section on Renomalization)

Embedded Julia sets occur in the hairs, I'm planning on writing a couple of blog posts going into more detail on that topic in the next week or two but until then here's some rough notes I scanned: http://mathr.co.uk/mandelbrot/2013-06-14_embedded_patterns/

Playing around with the mirroriness of the Mandelbrot around the X-axis. Starting with the formula x^(800/400) + c in Gnofract4D. Then slowly drifting away from the perfect 2 by adding 1 to 800:

(http://imageshack.com/a/img404/6509/5sys.gif)

On the right you see the 3th biggest bulb of the Mandelbrot.