Mandelbrot on real numbers (new post)

[Tater]

That is really interesting. I am teaching a course on fractals and chaos and will turn my students loose on this one.

[jehovajah]

Hiya guys and gals!

I wonder if any body could render this for me:

ø=0
For r=o until r>2
  ø=ø²+c
  r=ø
polargraph(r,ø)

c= -π*e^-3, or anu other constant you want to play round with.

I placed it here because of the similarity to Kali's original concept.

I am curious to see what results, aren't you?

I do not know the right commands in your programming language or your fractal generator, but i hope you will ask for clarification if you need it.

[Kali]

Hi jehovajah! (Yes, I'm back!)

I want to do what you ask, but I don't know if I'm getting it well:

ø is angle and r is radius? the radius then is equal to the angle? why I should use two variables then, if they have the same value?

Or I should do one iteration loop for a set of values of r from 0 to 2?

[jehovajah]

Hiya Pablo!

Firstly peace on your household, mi amigo!

Secondly the set of instructions is an outline guideline, as i had no way of giving a general description concisely at the time.

You had started this thread looking at the "mandelbrot" in one dimension, namely the x axis using x = x^2+c? Or some similar notion. These are all cartesian formulations using the x part of the x,y plane. Even rcos ø,rsinø is cartesian.

I want to explore a purely polar form that is r,ø.

Whether one uses phi or theta is irrelevant.

Polar coordinate systems are available in all graphing systems, but i do not know if a fractal generator based on them exists, or whether existing fractal generators can produce polar coordinate outputs with a little modification:
hence my outline is written in a semi script form.

What i have learned so far is that polar coordinate systems in the plane generally do not allow r to become negative, so the only form to allow this to happen is the parametric polar coordinate form.

The second thing i have found out is that exponentiation of positive values is well catered for so that one can write for example a rule y=2^x and plot it but one cannot get a plot for y=(-2)^x unless x is an integer, in which case you can write y=(-2)^round(x) or some other integer giving function.

So briefly, this would be a beginning of the exploration leading to a purely polar mandelbrot i would hope, and there are many things to define and find out along the way.

I tried a function format in one graphing calculator namely

polymand(ø)
r=0
while(r<2)
ø=ø^2-some constant
r=ø
polarplot(ø)
end
r

r at the end is the return for the function, and the plot function for this calculator was able to pass x or ø to the function polymand.

A spiral plot resulted.

It is only after really puzzling that i realised that the expected convolutions were in fact not allowed by this plotting function and that i needed a parametric plotter!
Try this one out to see what i am on about! :
This is as far as i have got to date, but i am on holiday!

[jehovajah]

ø is angle and r is radius? the radius then is equal to the angle? why I should use two variables then, if they have the same value?

Or I should do one iteration loop for a set of values of r from 0 to 2?

To do a polar plot in the plane you gnerally specify r for each ø. In the parametric form you specify both r and ø.

My initial interest is to see what spiral forms relate to the mandelbrot.

However in line with your formulation you should try

f(ø)=f(ø)^2+ø

f(r)=f(r)^2+r

for -6.3<r<6.3 and -2π<ø<2π.

You can narrow the range down by trial and error because you will get "overlapping".
The graph need to be a polar plot parametric or otherwise.

Thanks in advance! and please play around with it.

As a spicer

tap
r(t)=cos((t))^2
theta(t)=sin(t^2) + min(t,sin(t))
        or=sin(t^2) + (-1)^round(t)

into here with the polar setting to see the kind of possibilities i am expecting in draught form.

[jehovajah]

r(t)=min(1,abs(4*cos((t))))
theta(t)=(((((((((0^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)

This represents 9 iterations, placed in here

The min bracket and the cos() bracket are attempts to set constraints on the output equivalent to |z|<=1.

Without the abs the negative values of r are evident, but the core remains the same.

This kind of restraint of course leads to a disc shape overall, but i have no way of colouring those polar coordinates that do not escape to infinity.

You also have to be aware that the drawing programme includes lines which trace the course of the pen tip rather than points in the polar mandy.

[jehovajah]

r(t)=min(0.99,abs(tan((t))))
theta(t)=(((((((((0^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)^2+t)

This represents 9 iterations, placed in here

The min bracket and the cos() bracket are attempts to set constraints on the output equivalent to |z|<=1.

Probably more useful to use tan(), so i changed it accordingly above.

[jehovajah]

So i do not mind in the least admitting when i do not understand something, and i do not understand the julia and mandelbrot "processes".

Now this was due to confusion, that is: in my head; and possibly in others(let's look at the pretty pictures to check if it is right or not, cos we do not appreciate it any other way ), and this arises from my "preferred learning style".

Apparently these are all the rage amongst the more enlightened educators. Well i must admit: Thank 'h¶h for enlightened educators!!

Maths would appear to be highly visual, with its notation, diagrams symbol etc, and this can be very off-putting for us highly kinaesthetic  learners! We prefer to get our hands on what we are considering, to immerse ourselves proprioceptively and physically and pragmatically into the subject. We want to walk around the edge of a rhombus or scale the heights of seahorse valley, not just look at it.

So following this lead i constructed the "julia process" for f(x)=f(x)^2+c a recursive definition of f(x),

Now for me f(x) means a recipe with the main ingredient being x (eggs? )

So i am going to start with 0 eggs and turn the mixer and then add 1 egg.
Thats it.

Because it is recursive that means run through it again, but this time start with what you got in front of you.

Woa!  why? (wouldn't you just like to do that sometimes with your teachers? )
Erm? i really don't know why! Hey why don't we explore alternatives?

When you do that you soon find out that reverting back to the original starting point is repetetve and gets you to the same finishing point. Boring after a while , but still fundamentally useful.
So let's define recursive in such away that we can distinguish this case from any other case

Pay attention at the back of the class! (like being in school isn't it? "

Any way to cut a long explanation short and to avoid the laborious introduction of notation etc we usually ignore the "trivial" but nevertheless important recursion where we start over, and go on to the feedback case .

This is how i get
0^2+1
(0^2+1)^2+1
((0^2+1)^2+1)^2+1
(((0^2+1)^2+1)^2+1)^2+1
......

So i can write
f(0)=f(0)^2+1 to describe the above.

In the recipe the ingredients are 0 and 1, the method of combining the ingredients is recursive: f(0)=f(0)^2+1; and this is the method written in shorthand. I need one more instruction in the method: how many imes do i do it? i will call that the mixing instruction(iteration control).

That is all very good but what do i do with the mixture?

Well ususally i  separate it into pans and put it in the oven and cook until ready.

Mmmm!  Lovely Julia cake!

I now have unprecedented control over the Julia: i can vary how i start so x can run from -∞ to +∞ and i still only add one egg at each iteration! I can let x run along some limited range while only adding 1 egg. I can allow x to be periodic, hyperbolic, complex , whatever and run through a range in these fields rings or groups and still only add one egg!

The point is these variations i think would still be classed as a Julia whatever the cake baked!

However the second i start to vary the number of eggs i add each iteration thats when it becoms unclear as currently described. to see why i will need to construct the Mandelbrot process.



 
  where did this come from?
Random key pressing i guess! Those damn monkeys!
 
 Step one you say we need to talk
 
 He walks you say sit down it's just a talk
 
 He smiles politely back at you
 
 You stare politely right on through
 
 Some sort of window to your right
 
 As he goes left and you stay right
 
 Between the lines of fear and blame
 
 And you begin to wonder why you came
 
 
 
 Where did I go wrong, I lost a friend
 
 Somewhere along in the bitterness
 
 And I would have stayed up with you all night
 
 Had I known how to save a life
 
 
 
 Let him know that you know best
 
 Cause after all you do know best
 
 Try to slip past his defense
 
 Without granting innocence
 
 Lay down a list of what is wrong
 
 The things you've told him all along
 
 And pray to God he hears you
 
 And pray to God he hears you
 
 
 
 Where did I go wrong, I lost a friend
 
 Somewhere along in the bitterness
 
 And I would have stayed up with you all night
 
 Had I known how to save a life
 
 
 
 As he begins to raise his voice
 
 You lower yours and grant him one last choice
 
 Drive until you lose the road
 
 Or break with the ones you've followed
 
 He will do one of two things
 
 He will admit to everything
 
 Or he'll say he's just not the same
 
 And you'll begin to wonder why you came
 
 
 
 Where did I go wrong, I lost a friend
 
 Somewhere along in the bitterness
 
 And I would have stayed up with you all night
 
 Had I known how to save a life
 
 

[jehovajah]

The Mandelbrot process

f(c)=f(c)^2+x

I think right thar is the source of 1 level of confusion ! That change right thar! Uh huh huh!

Call the po-lice!

So a mandelbrot cake has a different ingredient called c(seed? )

So lets start with no seed in the bowl. i mix it then i add any amount of seed i like!
Thats it

So now i do the recursive bit and suddenly the amount of seed i added becomes fixed!

so f(c)=f(c)^2+x =>f(c)=f(a)^2+a where a is the amount i added after the first mix which then becomes  f(c)=f(c)^2+a by the 3rd mix.

So this is how i get

0^2+x
(0^2+x)^2+x
((0^2+x)^2+x)^2+x
(((0^2+x)^2+x)^2+x)^2+x
((((0^2+x)^2+x)^2+x)^2+x)^2+x
...........
f(c)=f(c)^2+x would be the shorthand version ,

This formula therefore obscures a very important procedural step which occurs at the outset within the first 3 iterations.

So to see the difference that the mandelbrot procedure makes we have to get past 2 iterations!

Now the above formula apparrently says that the mandelbrot is a combination of a constant part and a variable part, like the julia but the variable is not squared and a constant added rather the value of the function is kept constant between iterations then squared ans a variable added.

This is much more complex combining than in the julia and equivalent to spooning in ingredients into a cake mixture,stirring until well mixed and then spooning in some more!

Anyone know why bakers do that?

Any way, it should be clear that x can be anything variable, so it could be a function of any sort, a variable on any ring group or field, etc and as lonf as it is combined in this precise and complex way we can call it a mandelbrot cake (EU regulations permitting! )

But again also i now have unprecedented control over the recipe: i can change where i start what i keep constant between iterations etc etc.

I hope this has been as useful to you as to me, revealing the complexity of what we are looking at in these 2 processes, and revealing an infinite landscape of other dynamic processes behind the formulation.

At the end of it all i have to concur: the best way to evaluate a Julia or Mandelbrot  (et al) cake is to eat it !

I'll have that big slice of Julia cherry pie, with custard! Yummy

How does this relate to the polarmandy?

try r(t)=min(2,((((t^2-2.000)^2-2.000)^2-2.000)))
theta(t)=t      here

[Kali]

Hey, I was on a mini-holiday with my familiy to have some rest and almost totally disconnected from the world 

I promise to try your ideas tomorrow

Enjoyed your posts, amigo!

[jehovajah]

Hey, I was on a mini-holiday with my familiy to have some rest and almost totally disconnected from the world 

I promise to try your ideas tomorrow

Enjoyed your posts, amigo!

That is good to hear Pablo. Hope you have been refreshed

Using the mandelbrot process try

r(0)=r(0)^2+t
theta(0)=theta(0)^2+t

Here

Has anyone made any inroads into programming a fractal generator that polar plots?

[jehovajah]

My fiddling about has helped me to see the possible controls for the polarmandy

r(t)=(((((0)^2-0.9)^2-0.9)^2-0.9)^2-0.9)^2-0.9

Theta(t)=(((((0)^2+t)^2+t)^2+t)^2+t)^2+t

0<t<0.2π

The Mandelbrot process maps (r,theta) onto circles of differing radii. some radii shrink to zero some expand to infinity and some negative radii hang about near the radius 1,

So the set identified under this version would be mainly consistent of negative radii.

The theta variation expands or concentrates on a circle so hopefully does not contribute in this format to the definition of the polar mandy.

However we have options of using trig ratios and other functions so the possibilities are endless.

My experience confirms one other thing:The Mandelbrot set and the deformation of the plane into these fractal regions cannot be drawn, it and they can only be sculpted. As such the 3d mandelbrot- the mandelbulb can only be sculpted from space. Thus we have not seen the real mandelbrot until we have carved it from the plane, and similarly the real 3d mandelbrot has to be carved from space.

That we have conception of it is hope that we will find its real life counterparts in free space.

[Kali]

In the last days I was focusing on fractal art and I totally forgot about the polar mandy you proposed... sorry!

I'm on it now, but I think there's a misconception in your ideas. First of all, the Mandelbrot formula doesn't behave like with complex numbers. The interesting range is (-2,0), where the values converges into waves forming a bifurcation pattern if you plot all the iterations. You should saw this in my posts about Mandelwaves.

Outside the range, the values just tend to infinity in a rather uninteresting exponential way. I've learned that the patterns of the 1D Mandy I "discovered" and showed in this thread, are some kind of "interference pattern" between the waves on x and y axis.

I will look for a polar use of this methods, but unfornately I tried what you proposed and nothing relevant occurred, at least for me.

I think you are wrong here:

The Mandelbrot process maps (r,theta) onto circles of differing radii. som radii shrink to zero some expand to infinity and some negative radii hang about near the radius 1,

Because of the plotting you use, that connects the points with lines, you may think that the radius changes, but I tried it and it just draws a circle of the same radius, in a "accelerated" way, with the points of the perimeter getting more separated... this leads eventually to a great distance between the points within the circle's perimeter, and when the lines connects the sequence of points plotted, they cross the circle's insides forming strange patterns but they are not values within the formula results... I hope you are getting it, because it's difficult for me to explain it in english ( otherwise I should try in spanish )

Anyway it's a good idea to look for a polar form of plotting, I will try to do something with it and come back to you...

[jehovajah]

Hiya Pablo,

Yes , i am wrong somewhere! 

I have tried to track it down and i think it is in understanding how the graphers/generators plot and colour the mandy. I am not very skilled at colouring the mandelbrot set, or playing with the escape conditions.

The range (-2,0) i had not even explored properly until now, pursuing the polar mandy!

The bifurcation graph, map i do not fully grasp yet, but i kind of hoped that the plots i would "happen" upon would be versions of that, and you can see it in the drawings but in a "folded" way. In some of the conditions as you limit r to smaller values you get an equally dense lattice in the circle, like the bottom of the bifurcation maps.

Other things i have looked at are limiting the range of theta etc.

This has increased my understanding, but highlight the problem of not having a polar plotter, and not identifying the polar coordinates of points in the mandelbrot set and colouring accordingly.

`if you can Pablo, i wouid appreciate some help,and i too am busy elsewhere at the moment but i will return to correspond.

[jehovajah]

Ok, Pablo

I have a little time to gather my polarmandy thinking!

First, Why do the polarmandy?
Because we have not explored the mandelbrot in any other reference frame. Maybe we might find it has a different shape or the same shape.

The behaviour of the parts under the iteration may be more naturally explained or not. The distribution of features etc may reveal more coherent features or not,

`much has been done with variations to the mandelbulb,and maybe the polarmandy can do the same for the mandelbrot? Who knows?

For me it is an exploration worth pursuing, but only if we can take time to smell the roses on the way.

Because of my lack of understanding i wanted to explore by trial and error, from simple cases to more complex. Your 1D approach appealed from that point of view, but i have no real clue as to how to proceed,or even what to expect.

The first things i have done is clarify and examine some assumptions, both in the idea of the mandelbrot process, and the visualisation of the set.

My first assumption was that the set cannot be "drawn" only "carved" or printed, but i wanted to see what happened if i applied the form of the madelbrot equation to a drawing programme. It seemed to draw spirals or circles .

Gradually i realised that parametric graphers were necessary, and the relations between the measures(r,theta) and the parameter could be explored.

This was obscuring the fact that drawing was being pursued not sculpting. So i realised that playing around with the parameter, even setting up an iterative mode was not the same as developing the polarmandy .

Thats where i got to really, but i have doe some exploring i have not posted.