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Author Topic: complex not so complex  (Read 2257 times)
Description: a geometrical theory for designing fractals
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puntopunto
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keep it simple


« Reply #15 on: June 02, 2013, 12:05:09 AM »

note for Kali:
Thanks for your gentle reaction. I will answer later and show you the Mandelbrot construction. But after I have make very clear what I mean by folding and stretching. I must also disappoint you. It is no step further to the holy grail. In my opinion it is almost certain there is no equivalent of the Mandelbrot set in 3D. But, well, that's also something a holy grail should do, shouldn't it?

PART 3

First I want to unify the idea of a line fold and circle inversion. The circle inversion is often called reflection in a circle. That's because reflection in a line and inversion in a circle shares important properties. For instance:
A circle that intersect the inversion circle perpendicularly, is invariant (does not change) under the circle inversion and a line perpendicularly to the reflection line is also invariant. 
Both transformations shares the property that if the image of point a is b then the image of b is a.
Etc.

Moreover we can extend the real plane with ONE point, called infinity (see for a model of that plane wikipedia, Riemann sphere) . Then for a circle inversion the image of the centre of the inversion circle  is that point infinity. So the inversion is defined for every point. The point infinity lies on all lines. Going along a line in both directions you are going to that same point infinity. So you can think of that line as a (infinity large) circle. On the extended plane circles and lines plays the same role.
 
A reflection in a line consist just of two fold, one to the left and one to the right. I have to be very careful here. Up until now I was thinking about folds as executing them one after another. Reflection is more as doing two folds at the same time. It is better to think point by point if we describe  the relationship between folds and reflection.
As an example, reflecton  in the Y-axes (x=0) is the same as:
      for every point P(x,y)
      If x<0 then fold to the right and
      if x>0 then fold to the left
We can do the same thing with the reflection (inversion) in a circle:

I define fold_in as
      
      if P(x,y) lies outside the circle apply the circle inversion

And

fold_out as

      if P lies inside the circle apply the circle inversion

And now we can say:

Circle reflection (inversion) is the same as
      for every point P(x,y)
      if P lies in the circle fold_out
      if P lies outside the circle fold_in

With these reasonable definitions for folds we have a fairly  complete set: fold left and right, fold up and down, fold skew left and skew right, fold in and out.

All the fractals you can create with folding_general_1 are created with folds and nothing else.

Now we have access to fold_in and fold_out we can made a really very simple formula, capable to generate fractal images, namely:

fold_left followed by fold_out

Here some results:






The first two has folding line x=0. The last x=-.5
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Roquen
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« Reply #16 on: June 02, 2013, 01:05:13 AM »

puntopunto: Isn't the real problem "how" complex numbers are presented?  The algebra itself is very straight forward and accessible to far wider audience than domain and conformal mappings.  And more importantly it makes forward and inverse trig functions drop like flies and the properties of affine transforms likewise show up by algebraic manipulation.  The bottom line is that if you really want to understand geometry you can't just be given a small set of black-boxes.

As an example in your last post you describe building a reflection out of a fold and conditionals.  Now you could go forward and build all of the affine transforms out of reflection...but there's no algebra structure to manipulate and I doesn't really seem any easier to understand.
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Ryan D
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« Reply #17 on: June 02, 2013, 07:28:38 AM »

That fractal I know already for a long time. I made it with Fractint about 15 years ago.

As an aside, I would be happy to see some of your old Fractint formulas / parameters posted somewhere.  (I'm a dinosaur, I still have more fun with Fractint than anything else.)  The only work of yours I have been able to find in the past is from the Fractint mailing list from 2002 and 2003.  Some of the images shown on your first fractalforums post are very interesting to me.

http://www.fractalforums.com/mandelbulb-3d/first-post-something-fundamental-about-fractal-generation-t11008/

Ryan
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s31415
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« Reply #18 on: June 02, 2013, 11:07:12 AM »

Hi,

Picturing the complex plane plus a point at infinity is indeed a good way to construct the most general formula of this type. You then have generalized foldings identifying the disks on each side of a circle on the sphere. For instance the fold along a straight line going through the origin is associated to a great circle going through the point at infinity. The circle inversion fold associated with the unit circle is a fold along the equator of the sphere, if you pictured it with the point at infinity at the a pole. More generally, you can pick just any of these folds, and compose it with a Moebius transformation. In this way you can obtain a fold along any circle.

More generally, this type of fractals are obtained by iterating a "quasi-conformal" map of the sphere. By "quasi-conformal map", I mean a map which is conformal except possibly on a subset of measure zero. For instance the fold is conformal everywhere, except on the circle you are mirroring about. You can obtain more general conformal maps by composing the folds with more general conformal maps of the sphere, like for instant rational or holomorphic functions.

This is a huge class of maps. There will probably still be many people who will for some reason pick a random family of maps in this class, get similar patterns and claim a discovery. A more interesting task would be to identify in this class of maps the ones which are in some aspects special.

For instance the fact that the Ducks formula is obtained by composing a fold with the logarithm function:
http://algorithmic-worlds.net/blog/blog.php?Post=20110227
implies some rather unusual properties for the associated fractal patterns:
http://algorithmic-worlds.net/blog/blog.php?Post=20110724

Also, you cannot get the Mandelbrot set from folding transformations as they are defined above. The Mandelbrot set is obtained by iterating a conformal map, while the folds are necessarily non-conformal at the fold. And indeed, there is no Mandelbrot set in 3d because there is no 2-1 global conformal transformation of the 3-sphere or R^3 to themselves. People like to ignore this fact to preserve the hope...

Sam


« Last Edit: June 02, 2013, 11:17:44 AM by s31415 » Logged

kram1032
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« Reply #19 on: June 02, 2013, 11:20:31 AM »

With a point at infinity, you turn your space into a projective one. In such a space, n-spheres and n-planes are equivalent to each other and all translations in non-projective space can be represented by rotations of the projective space.
Now, each rotation geometrically is equal to two reflections. So if you allow any reflection along a projective plane or sphere in projective space, you get any combination of translations, rotations and reflections in non-projective space.

This, as has been shown before, has the potential to cover the entire space with nice fractal patterns. All you need to add to that is some kind of scaling.
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Kali
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« Reply #20 on: June 02, 2013, 04:17:11 PM »

Code:
This is a huge class of maps. There will probably still be many people who will for some reason pick a random family of maps in this class, get similar patterns and claim a discovery.

I hope that wasn't for me  embarrass

Anyway I gave you the proper credits and I've said that I took the basic idea of your ducks formula, and your explanations about how this patterns are created. Maybe I claimed a discovery because it's so mathematical simple, computational cheap, and also extendable to 3D.

Btw, if you can run webgl, take a look at this:

https://www.shadertoy.com/view/MssGD8

I didn't see any other 3D formula capable of this results using so little code, so, don't you think that I can claim somehow a discovery?


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s31415
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« Reply #21 on: June 02, 2013, 07:12:58 PM »

@Kali: No, but this was a silly comment, sorry about that.

Sam
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puntopunto
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keep it simple


« Reply #22 on: June 04, 2013, 11:38:52 AM »

It looks like that there is some misunderstanding about my  intentions. My intentions lies in the title of this topic: complex not so complex. I mean that as in "complexity" as well as complex in the mathematical term "complex". I am trying to reformulate basic fractal generation in such a way that you have a small collection of transformations, geometrically easy to grasp, but capable of creating many fractals.
My drive is "didactically". So even name giving is important to me. That's why I talked about the Riemann sphere. I tried to explain why fold_in and fold_out are logic names. It appears to me that the folds as defined earlier together with stretches, which I will define later in an obvious way, is such a collection.
You can, then, loosely spoken, talk like "take a point, make a fold, where does your point land, now push or pull a little in some direction, the point will have gone again to another place. Now do this operation over and over again until it lands in a forbidden part. If it hasn't land in that part after let's say a hundred times, you stop as well. Do this for all points. You will have a number for all points and you can color the point according to that number". Many people will  understand that.
Also a piece of code like
fold_left (p,...}
fold_down(p, ...)
stretch(p, ...)
is  then easy to understand.

For the record, I didn't say nowhere in my posts that I could construct the Mandelbrot set with folds. I said I could construct it with folds and stretches. I think sam among others, posting on this topic, are aware of this. Here is the construction:

Loop for point P:
With folding line the X-axes:
   if the point lies under it fold_up, If not fold_down.
With folding line the line through  P, with an angle with the X-axes that is half the angle of the line through O and P and the X-axes :
   if the point is to the left fold_right
   if the point is to the right fold_left
stretch_circular ( even without defining it, this will be clear) with factor length OP.

The construction is not unique. Every two lines through O and with  angle 1/2 phi, phi being the angle (OP,X-axes) will do.
(this serves also as an answer to Kali)

I don't know if I should use this in an explanation. Rotate and scale looks simpler. But there are some advantages: You can stay in the same terminology and in the calculations you need no more then to know how to decide on with side of a line a point lies, how to calculate the image of a folded point, definition of tan.
The main aim is to have a good collection of simple formulas to construct fractals. With that you can go to point multiplication, and together with point addition showing commutativity, associativity  etc. justifying the name multiplication. And you have the fractals to show their usefulness.

It looks to me as a good way to introduce complex numbers. And you have the basic concepts, with their geometrical meaning without even mention the words complex or imaginair. After all the number one is as real (meaning existing in the real world) as i. They both, like every mathematical object, are constructions of our minds.  That the number one feels very real is because we have so many useful representations for it.

I, myself, are not so happy with the usual introduction of complex numbers. Why?
Mostly they start with something like:

We introduce a totally new number, with the name i and i=square root of -1

or

We introduce a new number, not one of the reals and for that new number, represented by the symbol i. we have i^2=-1

Take the first one.
They have told you over and over again that the square root of a negative number don't exist, that it is not defined. So here they define something, named i. to be the same as something that don't exist.
That make no sense to me. Moreover even after i has been defined properly, and with that the square root of -1 has been given meaning, even then, the expression is somewhat obscure. Indeed, except zero, all complex numbers has two square roots, so the square root of -1 is not only  i but also -i.

In the second one, the -1 is oke, but what is the 2? Short for ixi? Multiplication? I don't know a multiplication for i

After that, if you are good in manipulating expressions with reals. working with complex expressions is easy. All is the same, the i acts in the same way as the a,b and c's. It's easy to learn the tricks. Only if you encounter i^2 you replace it by -1. But what is the meaning, what is the use. You don't have a representation that gives meaning to what you are doing.

Back to the folds. I started to extend the formulas I already had. For me the transitions of Julia sets, by changing positions of folding lines to two copies or more was new, also the M sets combining folds and changing the fold line position with multiplication ( apart from the burning ship fractal). That they are new to me does not say very much. Between 2003 and 2011 I never looked at fractals or gave it a thought. But I looked a bit around, finding some interesting  developments involving folds, like the log(laps(z)) formula on the site of Samuel and the examples in the topic "A very simple formula". But that was not exactly the same thing I did. So I put it in a topic. To find out if it is interesting, if it was done before and if people has ideas about another way of introducing fractals or ideas of improvement of my code.
By analogy  (left-right, up-down) I tried in-out. And I was particularly pleased with that last formula in my last post. So simple, so easy to understand.

Isn't the real problem "how" complex numbers are presented? 
Certainly it is also about representations

As an example in your last post you describe building a reflection out of a fold and conditionals.  Now you could go forward and build all of the affine transforms out of reflection...but there's no algebra structure to manipulate and I doesn't really seem any easier to understand.

Yes I think you can define multiplication and addition by folds. And no, as far as I can see, that does not look very useful. But from folds to rotations and squaring is a short step and then you have a meaningful opportunity to show (0,1)^2=(-1,0) and square root (-1,0)=(0,1) or (0,-1). Somewhere in the process there is a moment to step over, introduce i as (0,1) for short and a as (a,0) for short. Then it is obvious that i^2=-1 etc.

I thought, fold and stretch are well known as being fundamental parts of fractal imagery?

Very true. I liked the example of kneading dough in a book of Peitgen ea.


.....did you publish anything before?

Yes, I know there is also a 1 fold version, as you can see in my entangled tree image similar to yours. Also I've been exploring a lot of combinations of foldings, symmetries, rotations, hybrids, including the 3D version of this kind of fractals, and I've been posting some of my findings in other topics here in FF.
 

No, not on fractals.

To be very clear: I don't claim anything. I am not interested in it. If I do, accidentilly, something special, people will give me credits. And I will look to those other topics and your findings.

And indeed, there is no Mandelbrot set in 3d because there is no 2-1 global conformal transformation of the 3-sphere or R^3 to themselves. People like to ignore this fact to preserve the hope...

But do those people understand the fact. Is there an argument, without using advanced math? And 2-1 is 1-1?

As an aside, I would be happy to see some of your old Fractint formulas / parameters posted somewhere. 

Sorry, I don't have them any more. But it couldn't be that difficult to translate the formulas in this post. I think you have to restrict the number of parameters.

Below some zooms into the same area of the M-set using the formula folding_general_2 with two skew folds. I use the absolutely fantastic coloring function Mandelbrot Multiwave Coloring from Pauldebrot. Although I don't understand how to use the parameters, it's awesome. Thanks for that.













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kram1032
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« Reply #23 on: June 04, 2013, 02:10:57 PM »

I feel like the mistake about introducing i isn't so much how the introduction is phrased but rather how before that, the existence of such a thing is denied.
Just don't tell the kids that there is no square root of -1 if they are wondering about it, but rather tell them, that this will be covered later on but at the same time encourage them to try it for themselves. Just give a hint, that they should name the corresponding number as an unknown for now and see what they can discover from there by themselves.

One of the biggest mistakes in almost all education seems to be, that it's all geared against self-exploration. You'd just need some semi-mysterious nudges to pique a kid's curiosity and if you give the opportunities, they might explore and discover things on their own.
But raising ahead is highly discouraged in most official teaching/learning environments.

Anyway, rant over.
Giving a more geometrically inspired introduction to it all would also help. Though, if you try to not use i, you will be able to do the same things but in way less elegant ways. "i" is essentially a construct that simplifies a ton of things, even if the initial construction doesn't do it justice and even is based on a contradiction of previously learned.
« Last Edit: June 04, 2013, 10:00:13 PM by kram1032 » Logged
Roquen
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« Reply #24 on: June 04, 2013, 05:53:56 PM »

@kram1032:  Ah but sqrt(-1) is always non-sense under the rules of the game called the algebra of reals.  If you want to give it meaning, you have to make up a new game with new rules.

@puntopunto: First off, I'm not attempting to discourage you in any manner w/r to your write-up.  Nor am I attempting to discourage anyone from taking what they can from it.  However from the contents of your last post you're discussing complex numbers from a history standpoint, which ignores modern algebra.  For example: 'i' is not a number.  It can't be.  If it were the algebra doesn't work (SEE: complex number paradox).  Likewise none of the offspring would work.  Dual numbers: i2=0 -> i=0 therefore it collapses to Reals.  Split-complex: i2=1 -> i=+/-1.  The second component always has a concrete value, so this doesn't work either.  Quaternions: i2=j2=k2=ijk=-1, collapses to complex.  None of these are numbers...they symbols used to denote the basis set of the vector space.  It certainly is a problem that most introductions are based on this broken historic notion.  The second problem with complex number is that vectors replaced them for working geometric problem of the type under discussion, so there are few reference that discuss them.  And lastly, you look at complex differently for geometric problems than you do for analysis of real valued functions.  For example, for geometry you virtually always are only interested in the principle power.
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s31415
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« Reply #25 on: June 04, 2013, 08:41:01 PM »


But do those people understand the fact. Is there an argument, without using advanced math? And 2-1 is 1-1?


The argument goes as follows. We presumably want two properties for a hypothetic "3d M-set":
1) It should be self-similar.
2) The small scale details can be deformed, but should be recognizable.
In all the fractals based on the iteration of a map, self-similarity occurs because we are iterating a many to 1 map (i.e. a map such that each point in the image has several preimages). This crucial property imply that the orbits merge at each iteration. When two distinct regions are mapped onto one, then the fractal pattern they will ultimately contain will be related, because all the subsequent iterations will be the same. This is how fractal structures and self-similarity occur. This means that property 1) imply that we need a many to 1 map, and in particular a 2 to 1 map if we want to stay as close as possible to the original Mandelbrot set.
Property 2) requires the map to be conformal. Conformal maps deform patterns, but preserve angles, which implies that they approximately preserve shapes locally. If we use a non-conformal map, the small scale pattern will get irremediably stretched, as is familiar from the Mandelbulb or the quaternionic M-set, for instance.

Now it can be shown that in dimension 3 or higher, the only conformal maps of the sphere to itself are 1 to 1. This is shown by analysing infinitesimally what it means for a map to be conformal, and one deduce that in dimension 3 or higher, the only conformal maps are the translations, the rotations, the dilatations and the special conformal transformations. The proof can for instance be found in Chapter 4 of
http://books.google.ch/books?id=keUrdME5rhIC&printsec=frontcover&hl=fr#v=onepage&q&f=false
but there might be better references.

One way of going around this obstacle is to slightly relax the condition of conformality, and allow maps which are "quasi-conformal" in the sense that they are conformal everywhere except on a subspace, like the folds. This is how the Kaleidoscopic IFS fractals are generated. These are the fractals that are closest to satisfying the conditions above, and imho they are also the best looking 3d fractals.

Sam
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Syntopia
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« Reply #26 on: June 04, 2013, 09:47:37 PM »

1) imply that we need a many to 1 map, and in particular a 2 to 1 map if we want to stay as close as possible to the original Mandelbrot set.

I'm not sure I follow why the map should be 2-1 when generalizing to three dimensions. For instance, a 2D fractal using the component-wise abs(...) operator for folding on the x and y axis, is a 4-1 mapping. In 3D dimensions it generalizes to a 8-1 mapping (folding in the xy,yz,xz planes). Of course this doesn't help much when there is only 1-1 conformal mappings.
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s31415
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« Reply #27 on: June 04, 2013, 09:53:22 PM »

I'm not sure I follow why the map should be 2-1 when generalizing to three dimensions. For instance, a 2D fractal using the component-wise abs(...) operator for folding on the x and y axis, is a 4-1 mapping. In 3D dimensions it generalizes to a 8-1 mapping (folding in the xy,yz,xz planes). Of course this doesn't help much when there is only 1-1 conformal mappings.
True, I have no good reason to ask for a 2-1 map, except for the fact that the 2d M-set uses a 2-1 map.
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kram1032
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« Reply #28 on: June 04, 2013, 10:03:24 PM »

Roquen: I suppose so. But then I guess, what should be said to kids is, that the corresponding number does not exist in what we call the Real Numbers (with the rest of the explanation basically unchanged)
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M Benesi
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« Reply #29 on: June 05, 2013, 02:28:48 AM »

The argument goes as follows. We presumably want two properties for a hypothetic "3d M-set":
1) It should be self-similar.
2) The small scale details can be deformed, but should be recognizable.

hehehe.  Found that a long time ago. 

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