First by fractals i mean fractals which will produce by iterating a formula on polar coordinate, like where we'll make Mandelbrot fractal.
I was always wondered how to guess what a formula is going to look like as it calculate, or how can we know what is the fractal formula of an image.
Its unlikely for me to find a way, but how can we get nearer to guess them?
I agree with Mohamed that it is not easy at all. But the number of colors in the fractal image can determine the order of the complex polynomial which define fractal.
Yes, you can judge formula for deterministic fractals. For example, in Sierpinski triangle, you take initial image reduce to its one third and join three pieces in such a way that it becomes of the size of original size and connecting the tree pieces. So if you know a priory how the fractal is generated then corresponding affine transformations can be designed. Mandelbrot set is non-linear attraction formula and can not be guessed but can do experiments to know. They are generated by chance and can not be guessed.
I speak about fractals one can meat in infinite double (or many dimensional sequences) over finite sets (alphabets), without bounding the number of letters of the alphabet. They are just functions f : An --> A and they start with initial conditions, like first infinite column and first infinite row given. The recurrence looks like a(m,n) = f(a(m-1,n),a(m,n-1)) for the easiest example. Those sequences sometimes produce fractals, sometimes don't (like also by S. Wolfram). However, these sequences (already the form given here) interprets all Turing Machines. Well, by the Theorem of Rice, it is already undecidable if such an arbitrary sequence produces some fractal or (contrary) kind of chaos. It is also undecidable if it produces one particular fractal, we already know that one of them has produced, or also if it produces a set that can be projected on a particular (feasible) fractal by reducing the colors. Maybe this is not an answer to exactly the question you have put, but I think that it gives a hint about what can happen there.
Thank you, yes it does. but as we can't prove its impossible, to me its like a door way to search for a way.
Dear Saleh, "undecidable" means that there is no an algorithm to solve that given problem, and that cannot be. So it really means "impossible" because you have asked about a "method of computation", which seems to be some (wanted, or asked for) algorithm.
Really a good question to think over it and try to solve this puzzle. In case of IFS, somewhat we may be able to design an algorithm by seeing the fractal image as it follows a deterministic pattern. But, quite difficult to achieve success for escape time fractal.
Really a good question to think over it and try to solve this puzzle. In case of IFS, somewhat we may be able to design an algorithm by seeing the fractal image as it displays a deterministic pattern. But, quite difficult to achieve success for escape time fractal.