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Wonderful article on fractals and fractal zooming/rendering! I had never considered the inherent limitations and complications of maintaining accuracy when doin

Wonderful article on fractals and fractal zooming/rendering! I had never considered the inherent limitations and complications of maintaining accuracy when doing deep zooms. Some questions that came up for me while reading the article: 1. What are the fundamental limits on how deeply a fractal can be accurately zoomed? What's the best way to understand and map this limit mathematically? 2. Is it possible to renormalize a fractal (perhaps only "well behaved"/"clean" fractals like Mandelbrot) at an arbitrary level of zoom by deriving a new formula for the fractal at that level of zoom? (Intuition says No, well, maybe but with additional complexities/limitations; perhaps just pushing the problem around). (My experience with fractal math is limited.) I'll admit this is where I met my own limits of knowledge in the article as it discussed this as normalizing the mantissa, and the limit is that now you need to compute each pixel on CPU. 3. If we assume that there are fundamental limits on zoom, mathematically speaking, then should we consider an alternative that looks perfect with no artifacts (though it would not be technically accurate) at arbitrarily deep levels of zoom? Is it in principle possible to have the mega-zoomed-in fractal appear flawless, or is it provable that at some level of zoom there is simply no way to render any coherent fractal or appearance of one? I always thought of fractals as a view into infinity from the 2D plane (indeed the term "fractal" is meant to convey a fractional dimension above 2). But, I never considered our limits as sentient beings with physical computers that would never be able to fully explore a fractal, thus it is only an infinity in idea, and not in reality, to us.

ttoinou 2y ago
1. No limit. But you need to find an interesting point, the information is encoded in the numerous digits of this (x,y) point for Mandelbrot. Otherwise you’ll end up in a flat space at some point when zooming 2. Renormalization to do what ? In the case of Mandelbrot you can use a neighbor point to create the Julia of it and have similar patterns in a more predictable way 3. You can compute the perfect version but it takes more time, this article discusses optimizations and shortcuts
- ccvannorman 2y ago
  1. There must be a limit; there are only around 10^80 atoms in our universe, so even a universe-sized supercomputer could not calculate an arbitrarily deep zoom that required 10^81 bits of precision. Right? 2. Renormalization just "moves the problem around" since you lose precision when you recalculate the image algorithm at a specific zoom level. This would create discrepancies as you zoom in and out. 3. You cannot; because of the fundamental limits on computing power. I think you cannot compute a mathematically accurate and perfect Mandelbrot set at an arbitrarily high level of zoom, say 10^81, because we don't have enough compute or memory available to have the required precision
  - ttoinou 2y ago
    1. Mandelbrot is infinite. The number pi is infinite too and contains more information than the universe 2. I dont know what you mean or look for with normalization so I can’t answer more 3. It depends on what you mean by computing Mandelbrot. We are always making approximations for visualisation by humans, that’s what we’re talking about here. If you mean we will never discover more digits in pi than there is atoms in the universe then yes I agree but that’s a different problem
    - adrianN 2y ago
      Pi doesn't contain a lot of information since it can be computed with a reasonably small program. For numbers with high information content you want other examples like Chaitin's constant.
      - thaumasiotes 2y ago
        > Pi doesn't contain a lot of information since it can be computed with a reasonably small program. It can be described with a small program. But it contains more information than that. You can only compute finite approximations, but the quantity of information in pi is infinite. The computation is fooling you because the digits of pi are not all equally significant. This is irrelevant to the information theory.
pepinator 2y ago
In the case of Mandelbrot, there is a self similar renormalization process, so you can obtain such a formula. For the "fixed points" of the renormalization process, the formula is super simple; for other points, you might need more computations, but it's nevertheless an efficient method. There is a paper of Bartholdi where he explains this in terms of automata.
LegionMammal978 2y ago
As for practical limits, if you do the arithmetic naively, then you'll generally need O(n) memory to capture a region of size 10^-n (or 2^-n, or any other base). It seems to be the exception rather than the rule when it's possible to use less than O(n) memory. For instance, there's no known practical way to compute the 10^100th bit of sqrt(2), despite how simple the number is. (Or at least, a thorough search yielded nothing better than Newton's method and its variations, which must compute all the bits. It's even worse than π with its BBP formula.) Of course, there may be tricks with self-similarity that can speed up the computation, but I'd be very surprised if you could get past the O(n) memory requirement just to represent the coordinates.
calibas 2y ago
> What are the fundamental limits on how deeply a fractal can be accurately zoomed? This question is causing all sorts of confusion. There is no fundamental limit on how much detail a fractal contains, but if you want to render it, there's always going to be a practical limit on how far it can accurately be zoomed. Our current computers kinda struggle with hexadecuple precision floats (512-bit).

ttoinou 2y ago

1. No limit. But you need to find an interesting point, the information is encoded in the numerous digits of this (x,y) point for Mandelbrot. Otherwise you’ll end up in a flat space at some point when zooming 2. Renormalization to do what ? In the case of Mandelbrot you can use a neighbor point to create the Julia of it and have similar patterns in a more predictable way 3. You can compute the perfect version but it takes more time, this article discusses optimizations and shortcuts

ccvannorman 2y ago
1. There must be a limit; there are only around 10^80 atoms in our universe, so even a universe-sized supercomputer could not calculate an arbitrarily deep zoom that required 10^81 bits of precision. Right? 2. Renormalization just "moves the problem around" since you lose precision when you recalculate the image algorithm at a specific zoom level. This would create discrepancies as you zoom in and out. 3. You cannot; because of the fundamental limits on computing power. I think you cannot compute a mathematically accurate and perfect Mandelbrot set at an arbitrarily high level of zoom, say 10^81, because we don't have enough compute or memory available to have the required precision
- ttoinou 2y ago
  1. Mandelbrot is infinite. The number pi is infinite too and contains more information than the universe 2. I dont know what you mean or look for with normalization so I can’t answer more 3. It depends on what you mean by computing Mandelbrot. We are always making approximations for visualisation by humans, that’s what we’re talking about here. If you mean we will never discover more digits in pi than there is atoms in the universe then yes I agree but that’s a different problem
  - adrianN 2y ago
    Pi doesn't contain a lot of information since it can be computed with a reasonably small program. For numbers with high information content you want other examples like Chaitin's constant.
    - thaumasiotes 2y ago
      > Pi doesn't contain a lot of information since it can be computed with a reasonably small program. It can be described with a small program. But it contains more information than that. You can only compute finite approximations, but the quantity of information in pi is infinite. The computation is fooling you because the digits of pi are not all equally significant. This is irrelevant to the information theory.

pepinator 2y ago

In the case of Mandelbrot, there is a self similar renormalization process, so you can obtain such a formula. For the "fixed points" of the renormalization process, the formula is super simple; for other points, you might need more computations, but it's nevertheless an efficient method. There is a paper of Bartholdi where he explains this in terms of automata.

LegionMammal978 2y ago

As for practical limits, if you do the arithmetic naively, then you'll generally need O(n) memory to capture a region of size 10^-n (or 2^-n, or any other base). It seems to be the exception rather than the rule when it's possible to use less than O(n) memory. For instance, there's no known practical way to compute the 10^100th bit of sqrt(2), despite how simple the number is. (Or at least, a thorough search yielded nothing better than Newton's method and its variations, which must compute all the bits. It's even worse than π with its BBP formula.) Of course, there may be tricks with self-similarity that can speed up the computation, but I'd be very surprised if you could get past the O(n) memory requirement just to represent the coordinates.

calibas 2y ago

> What are the fundamental limits on how deeply a fractal can be accurately zoomed? This question is causing all sorts of confusion. There is no fundamental limit on how much detail a fractal contains, but if you want to render it, there's always going to be a practical limit on how far it can accurately be zoomed. Our current computers kinda struggle with hexadecuple precision floats (512-bit).