Mandelbrot/Julia: simple formula, infinite complexity -> how?
category: general [glöplog]
Hi,
I was 17 years old when I programmed my first mandebrot-set generator (on a C64) and now, 21 years later (after having done the same on my scientific calculator, PC CPU and recently my GPU, just like you guys :-) I'm still wondering how it's possible that such intricate structures can come from such a simple formula:
z' = z^2+c
Here's an idea that I'd like to put before you, and I'd love to hear your ideas. Hopefully Iq is reading too!
I've got this idea that by iterating the numbers, they somehow reveal something about what I've (in amateur terms) come to call 'the structure of the real and rational numbers'.
Hoping I'm using the right terms, there are of course the Real numbers, which are all non-complex numbers (right?). A subset of these are the rational numbers: all numbers that can be represented by a fraction (a ratio of one number to another).
Some numbers cannot be represented by a fraction, such as Pi.. So these numbers are real but not rational. Ok, so much for the recap..
Some rational numbers are very simple: 1/1 = 1, 1/2 = 0.50000..., 10/9 = 1.11111... Others are much more complex, such as 355/115: 3,1415929203539823008849557522124 (this is actually an approximation of Pi, see http://en.wikipedia.org/wiki/Pi)
Zooming in to the mandelbrot set is the same as finding out more digits behind the decimal point: at a high level a few digits is enough, at deeper zooms you can start discerning more and more digits behind the decimal point.
At the same time, another fraction that has a numerator or denominator that is close to the one above can have a very simple sequence of decimal digits again. This would match with the phenomenon that in the Mandelbrot set areas of great complexity can be right next to very simple ones.
Zooming in an area with no detail would correspond with looking at more and more digits of 10/9: there is no new info there, it's all 1's.
The self-similarity of the mandelbrot at smaller levels can correspond to repeating sequences decimal digits, near the decimal point (larger mini-brots) or further away from it (mini-mini-brots :-)).
Of course there are large holes in this idea: we're dealing with complex numbers, not real ones, and I don't have the mathematical tools in my toolbox to take it further toward an actual hypothesis, let alone a proof, but hopefully I've put the idea across.
To summarize: the idea is that when you look at the Mandelbrot set, the detail that you see is not _generated_ by the simple formula, but it only reveals a structure that is already there, in the numbers themselves, and this structure has to do with whether a number is real or rational.
Does this cut any wood (a Dutch expression) in your eyes?
Would love to hear your thoughs!
I was 17 years old when I programmed my first mandebrot-set generator (on a C64) and now, 21 years later (after having done the same on my scientific calculator, PC CPU and recently my GPU, just like you guys :-) I'm still wondering how it's possible that such intricate structures can come from such a simple formula:
z' = z^2+c
Here's an idea that I'd like to put before you, and I'd love to hear your ideas. Hopefully Iq is reading too!
I've got this idea that by iterating the numbers, they somehow reveal something about what I've (in amateur terms) come to call 'the structure of the real and rational numbers'.
Hoping I'm using the right terms, there are of course the Real numbers, which are all non-complex numbers (right?). A subset of these are the rational numbers: all numbers that can be represented by a fraction (a ratio of one number to another).
Some numbers cannot be represented by a fraction, such as Pi.. So these numbers are real but not rational. Ok, so much for the recap..
Some rational numbers are very simple: 1/1 = 1, 1/2 = 0.50000..., 10/9 = 1.11111... Others are much more complex, such as 355/115: 3,1415929203539823008849557522124 (this is actually an approximation of Pi, see http://en.wikipedia.org/wiki/Pi)
Zooming in to the mandelbrot set is the same as finding out more digits behind the decimal point: at a high level a few digits is enough, at deeper zooms you can start discerning more and more digits behind the decimal point.
At the same time, another fraction that has a numerator or denominator that is close to the one above can have a very simple sequence of decimal digits again. This would match with the phenomenon that in the Mandelbrot set areas of great complexity can be right next to very simple ones.
Zooming in an area with no detail would correspond with looking at more and more digits of 10/9: there is no new info there, it's all 1's.
The self-similarity of the mandelbrot at smaller levels can correspond to repeating sequences decimal digits, near the decimal point (larger mini-brots) or further away from it (mini-mini-brots :-)).
Of course there are large holes in this idea: we're dealing with complex numbers, not real ones, and I don't have the mathematical tools in my toolbox to take it further toward an actual hypothesis, let alone a proof, but hopefully I've put the idea across.
To summarize: the idea is that when you look at the Mandelbrot set, the detail that you see is not _generated_ by the simple formula, but it only reveals a structure that is already there, in the numbers themselves, and this structure has to do with whether a number is real or rational.
Does this cut any wood (a Dutch expression) in your eyes?
Would love to hear your thoughs!
my only thought is that i prefer 22/7 for an approximination of pi :D
Even if it does reveal some structure, it can't be this basic.
All of this seemed a bit misty, but a few quick questions arose: how do you define a simple sequence of decimal digits, how does it change under the change of basis for the number representation? And I guess for every point in the complex plane you could find another "interesting" number in a circle around it of any radius, no matter how small. Thus, there would be no "simple" areas.
All of this seemed a bit misty, but a few quick questions arose: how do you define a simple sequence of decimal digits, how does it change under the change of basis for the number representation? And I guess for every point in the complex plane you could find another "interesting" number in a circle around it of any radius, no matter how small. Thus, there would be no "simple" areas.
Yes, that's what I'm looking for, demystification!
This thread made me once more realize I'll get old one day - I'm 17 years old now and wrote my first mandelbrot renderer last year.
I don't completely (or at all) get the concept of the set and am still to get that nice smooth colouring some people do. I've heard about calculating the limit in infinity and colouring based on that.
But to the point; why not render the set in decimal numbers and see if you can spot any neat stuff?
I don't completely (or at all) get the concept of the set and am still to get that nice smooth colouring some people do. I've heard about calculating the limit in infinity and colouring based on that.
But to the point; why not render the set in decimal numbers and see if you can spot any neat stuff?
I'd like to add I'm really looking forward to when iq will come and give an exhaustive answer in a single sentence.
What answer, that the set of irrational numbers is dense in the real space?
<blah>
The Mandelbrot set comprises those points for which the above iteration remains bounded. As the colour of each pixel is determined by the number of iterations it takes to become unbounded, the colour patterns are not part of the set. The interesting pixels are near the set's boundary.
People often mistake the colours as part of the set, while they're really not.
I certainly don't understand the existence of the 'pattern' but the iteration can be viewed as a 2D vector rotation & scaling followed by a translation. I'm not sure if that will do you any good though...
</blah>
The Mandelbrot set comprises those points for which the above iteration remains bounded. As the colour of each pixel is determined by the number of iterations it takes to become unbounded, the colour patterns are not part of the set. The interesting pixels are near the set's boundary.
People often mistake the colours as part of the set, while they're really not.
I certainly don't understand the existence of the 'pattern' but the iteration can be viewed as a 2D vector rotation & scaling followed by a translation. I'm not sure if that will do you any good though...
</blah>
BTW.. it is interesting to plot 'z' at each iteration step by using the real and imaginary part as the x and y coordinates. For many pixels of the mandelbrot set, a.k.a the lake, you can see spirals. This feature was detected in several fractal plotting programs to provide an 'early out' for the lake, as these pixels always iterate until the maximum number. I think Fractint used this technique.
@imbusy: No, I was just agreeing with you that the whole description is a bit misty.
Good point about that circle around every point in the complex plane.
My guess was that things are different when you take the iteration process into account: by feeding the result of the computation back into itself (and adding a constant) a literal feedback-loop is created, where for some values the signal is extinguished, for others it goes to infinity, and for yet others it oscillates. Does this result have something to do with the value it all started with? Yes: this is what defines the M-set. Is the result explained by whether the starting number is rational or irrational.. or close to a rational number... That was my guess, and perhaps it is wrong.. But what _then_ explains the shape of the set?
That there are more irrational numbers than rational ones, I agree with that too :-) (if that's what you meant :-) As you might have observed, I'm a bit 'dense' on math myself)
@msqrt: haha, yeah!
Good point about that circle around every point in the complex plane.
My guess was that things are different when you take the iteration process into account: by feeding the result of the computation back into itself (and adding a constant) a literal feedback-loop is created, where for some values the signal is extinguished, for others it goes to infinity, and for yet others it oscillates. Does this result have something to do with the value it all started with? Yes: this is what defines the M-set. Is the result explained by whether the starting number is rational or irrational.. or close to a rational number... That was my guess, and perhaps it is wrong.. But what _then_ explains the shape of the set?
That there are more irrational numbers than rational ones, I agree with that too :-) (if that's what you meant :-) As you might have observed, I'm a bit 'dense' on math myself)
@msqrt: haha, yeah!
I look at the iteration as a repeated deformation of a circle or rectangle, this explains the self-repeating structures. But it does not explain why the structures become more and more complex the further you zoom in...
This can be thought of as a combination of an accumulation of various structures which are warped multiple times according to different principles depending on where along the iteration those substructures occured.
I hope this makes at least some sense... I really don't know how to put this into words.
This can be thought of as a combination of an accumulation of various structures which are warped multiple times according to different principles depending on where along the iteration those substructures occured.
I hope this makes at least some sense... I really don't know how to put this into words.
Not sure what you mean by rational number when it comes to the complex plane, but if you mean complex numbers z = a + ib where a and b are rationals, then consider this: take away all the irrational numbers. The iteration formula z_{n+1} = z_n^2 + c will keep you inside the rationals, so you can carry out all your iterations and determine a "rational subset" of the Mandelbrot set. For any discrete visualisation of the Mandelbrot set, the "rational subset" and the real thing will look the same.
I think teraflop is closer to the truth, it's more about bifurcation points of a more or less geometric process (iterated) and how proximity is defined in the complex plane, i.e. the topology of the complex numbers.
I think teraflop is closer to the truth, it's more about bifurcation points of a more or less geometric process (iterated) and how proximity is defined in the complex plane, i.e. the topology of the complex numbers.
Rational vs irrational numbers play a central role in the discussion of the connectivity of the Mandelbrot set (by means of the so called external rays), but not in the way smoothstep is pointing.
About the shape of the Mandelbrot set and all its spirals etc, one can think it as this: the iteration formula z^2+c is pretty much an angle doubling and length squaring formula. If you move far enough from the origin the c becomes irrelevant and indeed you are left with z^2 only. Most of the theory of Hubbard-Douady potential, external rays, distance estimators (usefull for raymarching!) and connectivity start from analyzing this dynamic system of doubling angles and squaring distances, and more specifically by constructing a tranformarion that undoes what the mandelbrot formula does. So, first we let the mandelbrot formula do it's job and iterate points with z^2+c. The points which move away forever and reach infinity do not beling to the Mandelbrot set by definition. Takes those points, and now undo the trip they just made and start halving angles and square rooting distances. So, if we iterated z^2+c a given amount of times, "n", now undo those things n times wihout using c: z^(1/2^n) (we half angles and square root distances n times). That brings points from infinity back close to the origin. They will not land in the same palce as we started from cause we did not use c in the return travel. Let's call the new landing point z'. So with this game we move pointf from z to z'. If the original point z was far enought already as to make c irrelevant, then z' will be really close to z. In fact, as z->inf, the mapping z'=phi(z) becomes indentity (z'=z). As you move closer to the M set, phi(z) gets funnier. See an image of phi here (first image)
from http://www.iquilezles.org/trastero/fieldlines/
The phi function/mapping happens to be a smooth function (without holes, pinching points or noncontinuous distortions) and smoothly deforms the plane, in a quite violent manner as you get clooser to the boundary of M. Theory is long, but based on phi you can derive all the theory known to date about M.
The thing is that you can think of the exterior of M as the plane whole minus a disk, ie, that has been distorted by phi. So basically, all the valleys and spirals of M are coming from the behaviour of phi, or in other words, the behaviour of different points under angle doubling and distance squaring.
So, it's all about polar angles more than complex numbers. So, the irrational vs rational character of numbers is important, but not in the real or imaginary parts of the complex number, but in it's argument (up to the 2pi factor of course)
For example, take a complex number z with an argument of 2*pi*2/7. Let's forget the 2pi and just keep in mind that any such "normalized" argument bigger that 1 will wrap in again (cause angles bigger that 2pi start from 0 again). So, argument of z is 2/7. Under one iteration (far enougth from the orgin) the new z wil have argument 4/7 (doubled). Next iteration will have 8/7 = 1/7. Next one will be 2/7. So this point is actually repeating it's own arbument every three iterations, it's a periodic point of period three, it describes a triangle spiral. That means, without justifying it now, that that point belongs to a special point in M, probably it's one of the two roots of one of the big bulbs (circles) that rest on top of the main cardiod of M.
Another example, 3/11 -> 6/11 -> 1->11 -> 2->11 -> 4/11 -> 8/11 -> 5/11 -> 10/11 -> 9/11 -> 7/11 -> 3/11, so this is a period 11 point and it will probably be related to one of those tiny bulb/spirals in the ass of the set. In fact, the spiral will have 11 branches.
In fact, there is a complete theory of combinatorics that tell you preciselly where are the bulbs, spirals, antenas and branches of M based on the argument of the complex numbers, when this arguments are rational. Because if they are rational, the argument-doubling procedure behaves periodic or preperiodically, so these are point in M and so on. On top of that, ratioanls with denominators whch prime number are interesting, and well, there is a complete game of rules involving Fibonacci numbers and stuff. Pretty fun.
However, irrational numbers are a problem. Say sqrt(2)/2, after doubling and doubing forever you still dont have repeating patterns, so mathematicians have problems to show they belong to M and therefore that M is connected. However, you can always find an arbitrarily close rational number to any irrational, so things behave "as expected".
Well, I don't think I clarified anything, Im sorry for that, these are topics I don't work in since 5 years, and anyway it's a bit complex for me to understand it well enough so I could explain more clearly. But basically, indeed rational help mathematicians understand the M set, and indeed the structure of the set has lots to do with them, but not in the way you pointed out.
Tags: "Hubbard-Douady potential", "external rays", "periodic orbit", "Misiurewicz point", "parabolic fixed point"
self promoting spam: http://iquilezles.org/www/articles/msets/msets.htm http://iquilezles.org/www/articles/arquimedes/arquimedes.htm
About the shape of the Mandelbrot set and all its spirals etc, one can think it as this: the iteration formula z^2+c is pretty much an angle doubling and length squaring formula. If you move far enough from the origin the c becomes irrelevant and indeed you are left with z^2 only. Most of the theory of Hubbard-Douady potential, external rays, distance estimators (usefull for raymarching!) and connectivity start from analyzing this dynamic system of doubling angles and squaring distances, and more specifically by constructing a tranformarion that undoes what the mandelbrot formula does. So, first we let the mandelbrot formula do it's job and iterate points with z^2+c. The points which move away forever and reach infinity do not beling to the Mandelbrot set by definition. Takes those points, and now undo the trip they just made and start halving angles and square rooting distances. So, if we iterated z^2+c a given amount of times, "n", now undo those things n times wihout using c: z^(1/2^n) (we half angles and square root distances n times). That brings points from infinity back close to the origin. They will not land in the same palce as we started from cause we did not use c in the return travel. Let's call the new landing point z'. So with this game we move pointf from z to z'. If the original point z was far enought already as to make c irrelevant, then z' will be really close to z. In fact, as z->inf, the mapping z'=phi(z) becomes indentity (z'=z). As you move closer to the M set, phi(z) gets funnier. See an image of phi here (first image)
from http://www.iquilezles.org/trastero/fieldlines/
The phi function/mapping happens to be a smooth function (without holes, pinching points or noncontinuous distortions) and smoothly deforms the plane, in a quite violent manner as you get clooser to the boundary of M. Theory is long, but based on phi you can derive all the theory known to date about M.
The thing is that you can think of the exterior of M as the plane whole minus a disk, ie, that has been distorted by phi. So basically, all the valleys and spirals of M are coming from the behaviour of phi, or in other words, the behaviour of different points under angle doubling and distance squaring.
So, it's all about polar angles more than complex numbers. So, the irrational vs rational character of numbers is important, but not in the real or imaginary parts of the complex number, but in it's argument (up to the 2pi factor of course)
For example, take a complex number z with an argument of 2*pi*2/7. Let's forget the 2pi and just keep in mind that any such "normalized" argument bigger that 1 will wrap in again (cause angles bigger that 2pi start from 0 again). So, argument of z is 2/7. Under one iteration (far enougth from the orgin) the new z wil have argument 4/7 (doubled). Next iteration will have 8/7 = 1/7. Next one will be 2/7. So this point is actually repeating it's own arbument every three iterations, it's a periodic point of period three, it describes a triangle spiral. That means, without justifying it now, that that point belongs to a special point in M, probably it's one of the two roots of one of the big bulbs (circles) that rest on top of the main cardiod of M.
Another example, 3/11 -> 6/11 -> 1->11 -> 2->11 -> 4/11 -> 8/11 -> 5/11 -> 10/11 -> 9/11 -> 7/11 -> 3/11, so this is a period 11 point and it will probably be related to one of those tiny bulb/spirals in the ass of the set. In fact, the spiral will have 11 branches.
In fact, there is a complete theory of combinatorics that tell you preciselly where are the bulbs, spirals, antenas and branches of M based on the argument of the complex numbers, when this arguments are rational. Because if they are rational, the argument-doubling procedure behaves periodic or preperiodically, so these are point in M and so on. On top of that, ratioanls with denominators whch prime number are interesting, and well, there is a complete game of rules involving Fibonacci numbers and stuff. Pretty fun.
However, irrational numbers are a problem. Say sqrt(2)/2, after doubling and doubing forever you still dont have repeating patterns, so mathematicians have problems to show they belong to M and therefore that M is connected. However, you can always find an arbitrarily close rational number to any irrational, so things behave "as expected".
Well, I don't think I clarified anything, Im sorry for that, these are topics I don't work in since 5 years, and anyway it's a bit complex for me to understand it well enough so I could explain more clearly. But basically, indeed rational help mathematicians understand the M set, and indeed the structure of the set has lots to do with them, but not in the way you pointed out.
Tags: "Hubbard-Douady potential", "external rays", "periodic orbit", "Misiurewicz point", "parabolic fixed point"
self promoting spam: http://iquilezles.org/www/articles/msets/msets.htm http://iquilezles.org/www/articles/arquimedes/arquimedes.htm
Det er så typisk... Altid når vi har en koselig diskusjon så skal Sverre komme og lage kvalm!
.. og kniv og gaffel er et språk som han ikke forstår!
Nice to read your explaination iq, thx :)
A long time ago I bought this nice book.
Sometimes the math gets complicated (complex topology / sets... some passages are really not understandable if you are not into academic level pure maths... never got those) but it has very nice images and it explains in detail all the things related to orbits, attractors, etc, etc...
hmm I should pick it back now :)
(It's fun how Google books only shows the no-heavy-math stuff in the preview :D)
btw, a technique for obtaining "smooth iteration"/colors for the M-set: http://linas.org/art-gallery/escape/escape.html
A long time ago I bought this nice book.
Sometimes the math gets complicated (complex topology / sets... some passages are really not understandable if you are not into academic level pure maths... never got those) but it has very nice images and it explains in detail all the things related to orbits, attractors, etc, etc...
hmm I should pick it back now :)
(It's fun how Google books only shows the no-heavy-math stuff in the preview :D)
btw, a technique for obtaining "smooth iteration"/colors for the M-set: http://linas.org/art-gallery/escape/escape.html
smoothstep, ya maybe want to have a look at the plants/trees!
differs, from "chaos" to "order". :)
differs, from "chaos" to "order". :)
iq: good description there, I learned quite a lot from that :)
I came across something very similar recently, with the new 'holographic hot horizons' (sounds like an mfx demo) theories (e.g. http://www.scientificblogging.com/hammock_physicist/it_bit_case_gravity - there's a link to the paper causing the interest in there too). That seems to work in a similar way: project what's going on in a system onto a holographic screen surrounding it, then work back to learn about the system in new ways.
I came across something very similar recently, with the new 'holographic hot horizons' (sounds like an mfx demo) theories (e.g. http://www.scientificblogging.com/hammock_physicist/it_bit_case_gravity - there's a link to the paper causing the interest in there too). That seems to work in a similar way: project what's going on in a system onto a holographic screen surrounding it, then work back to learn about the system in new ways.
Quote:
those tiny bulb/spirals in the ass of the set
... :D
iq's post was longer than I anticipated (D:) but still very interesting - I think I actually got some of that.
bdk, thanks for the link :)
bdk, thanks for the link :)
@iq, thanks for the great explanation! I'm going to get pen+paper (and/or compiler) and work out a couple of these things for myself:
- squaring a complex number is the same as doubling an angle and distance.. I gotta see that for myself (my Analysis course at the TU was a long time ago, but I'll manage :-)
- I remember that a complex number is a lot like a vector, so it can be described by cartesian and polar coordinates. When evaluating the points 'c' of the M-set (that are in view), the function z'=z^2+c is evaluated.. so the starting point 'c' is 'being fed into the result' in every iteration.. I should write a program to plot that..
This is fun! :-)
- squaring a complex number is the same as doubling an angle and distance.. I gotta see that for myself (my Analysis course at the TU was a long time ago, but I'll manage :-)
- I remember that a complex number is a lot like a vector, so it can be described by cartesian and polar coordinates. When evaluating the points 'c' of the M-set (that are in view), the function z'=z^2+c is evaluated.. so the starting point 'c' is 'being fed into the result' in every iteration.. I should write a program to plot that..
This is fun! :-)
btw, the answer to the original question of the thread "how to get all that complexity from such a simple formula" I guess is "chaos theory" = "strong dependency on initial conditions", which is common in almost nonlinear system. So, basically, you can say that the complexity comes from the "squaring" term in the equation z^2+c. You could read these two very old but classic books as an introduction to the topic of chaos and fratal geometry : "Does God Play Dice" by Ian Stewart and "The Beauty of Fractals" by Heinz-Otto Peitgen. Ian Stewart describes the nonlinearity in the iterations as "stretching and folding" (if I recall correctly).
iq: You recall correctly. :)
The Ian Stewart's book is, by the way, among the best books written about this topic. Its a good introduction to the work of Lorenz, Cantor, Poincare, Henon and Feigenbaum (among others) and, if I remember correctly, it has a chapter about strange atractors that is a good reading about the dynamics of nonlinear systems. And Stewart explain all this without the need of use advanced mathematics in his book.
The Ian Stewart's book is, by the way, among the best books written about this topic. Its a good introduction to the work of Lorenz, Cantor, Poincare, Henon and Feigenbaum (among others) and, if I remember correctly, it has a chapter about strange atractors that is a good reading about the dynamics of nonlinear systems. And Stewart explain all this without the need of use advanced mathematics in his book.
Hmm, I've had "The Beauty of Fractals" sitting on my shelf for countless years now, but so far I've mainly been looking at the pictures... Maybe it's time to dive into the math. :)
It's fun thinking about how much time it must have taken to produce those pictures back then (days, probably - the book is from 1986!) and now they easily run at hundreds per second on a commodity GPU...
It's fun thinking about how much time it must have taken to produce those pictures back then (days, probably - the book is from 1986!) and now they easily run at hundreds per second on a commodity GPU...
2010 - Year of the return of the fractals
