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.