I have been thinking about the (I think) still unsolved problem that I got told about in grad school and recently saw mentioned in a 1987 issue of Byte magazine.
Namely people make these Poincare section plots for Hamiltonian systems like
That section that looks like a bunch of random dots is where chaotic motion is observed. There's a lot of reason to think that area should have more structure in it because the proof that there are an infinite number of unstable periodic orbits in there starts with knowing there are an infinite number of stable periodic orbits and that there is an unstable orbit on the the separatrix between them. Those plots are probably not accurate at all because the finite numeric precision interacts with the sensitivity to initial conditions. The Byte article suggests that it ought to be possible to use variable precision math, bounding boxes and such to make a better plot but so far as I know it hasn't been done.
(At this point I care less about the science and more about showing people an image they haven't seen before.)
"In 1920, Pierre Fatou expressed the conjecture that -- except for special cases -- all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. ... can be interpreted to mean that for a dense set of parameters 'a', an attracting periodic orbit exists [in the logistic map]."
Friggin' blew my mind at age 20. Still kinda does. :)
> There's a lot of reason to think that area should have more structure in it because the proof that there are an infinite number of unstable periodic orbits in there starts with knowing there are an infinite number of stable periodic orbits and that there is an unstable orbit on the the separatrix between them.
An analogous result is true for the number line: between any two rational numbers, there is an irrational number.
But that isn't cause to suspect that, if you look at a random piece of the number line, you should be able to see a lot of rationals. They're there. But you can't see them. The odds of any given number you look at being irrational are 1.
In a plot like that one those groups of circles that you see are resonances where the winding ratio on the torus (how many degrees you go around per turn) is rational. The area between two resonances (or non-resonant behavior) has a little bit of chaos in those regular regions if you look closely: there is a certain chaotic zone. Those areas of pervasive chaos happen when the chaos area around the rationals covers everything, see
which involves doing a sum over all the rationals.
In the 2 degree of freedom case those tori are a solid wall so even if you have a chaotic zone around a resonance the motion is constrained by the tori. For N>2 though there are more dimensions and the path can go "around" the tori. You could picture our solar system of 8 planets having 24 degrees of freedom (although the problem is terribly non-generic because the three kinds of motion in a 1/r^2 field all have the same period) It sure seems that we are in a regular regime like one of the circles in that plot but we cannot rule out that over the course of billions of years that Neptune won't get ejected. See
which is poorly understood because nobody has found an attack on it. You have the same problem with numerical work that you do in the plot because sensitive dependence on initial conditions magnifies rounding error. Worse than that normal integrators like Runge-Kutta don't preserve all the geometric invariants of
but other than preserving that invariance those perform much worse than normal integrators. This is one of the reasons why the field has been stuck since before I got into it. This
came out of people learning how to find chaotic trajectories to use for transfer orbits and is an exciting development though. Practically though you don't want to take a low energy-long time trajectory to Mars because you'll get your health wrecked by radiation.
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.
> 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).
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
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
1. You asked about the fundamental limits, not the practical limits. Obviously practically you're limited by how much memory you have and how much time you're willing to let the computer run to draw the fractal.
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
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.
> 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.
No, it does not contain more information than the smallest representation. This is fundamental, and follows from many arguments, e.g., Shannon information, compression, Chaitan’s work, Kolmogorov complexity, entropy, and more.
The phrase “infinite number of 0’s” does not contain infinite information. It contains at most what it took to describe it.
Descriptions are not all equally informative. "Infinite number of 0s" will let you instantly know the value of any part of the string that you might want to know.
The smallest representation of Chaitin's constant is "Ω". This matches the smallest representation of pi.
„Representation“ has a formal definition in information theory that matches a small program that computes the number but does not match „pi“ or „omega“.
No, it doesn't. That's just the error of achieving extreme compression by not counting the information you included in the decompressor. You can think about an algorithm in the abstract, but this is not possible for a program.
You seem wholly confused about the concept of information. Have you had a course on information theory? If not, you should not argue against those who’ve learned it much better. Cover’s book “Elements of information theory” is a common text that would clear up all your confusion.
The “information” in a sequence of symbols is a measure of the “surprise” on obtaining the next symbol, and this is given a very precise mathematical definition, satisfying a few important properties. The resulting formula for many cases looks like the formula derived for entropy in statistical mechanics, so is often called symbol entropy (and leads down a lot of deep connections between information and reality, the whole “It from Bit” stuff…).
For a sequence to have infinite information, it must provide nonzero “surprise” for infinitely many symbols. Pi does not do this, since it has a finite specification. After the specification is given, there is zero more surprise. For a sequence to have infinite information, it cannot have a finite specification. End of story.
The specification has the information, since during the specification one could change symbols (getting a different generated sequence). But once the specification is finished, that is it. No more information exists.
Information content also does not care about computational efficiency, otherwise the information in a sequence would vary as technology changes, which would be a poor definition. You keep confusing these different topics.
Now, if you’ve never studied this topic properly, stop arguing things you don’t understand with those who’ve learned do. It’s foolish. If you’ve studied information theory in depth, then you’d not keep doubling down on this claim. We’ve given you enough places to learn the relevant topics.
... and how would you decode that information? Heisenberg sends his regards.
EDIT: ... and of course the point isn't that it's 1:1 wrt. bits and atoms, but I think the point was that there is obviously some maximum information density -- too much information in "one place" leads to a black hole.
Fun fact: the maximum amount of information you can store in a place is the entropy of a black hole, and it's proportional to the surface area, not the volume.
We can create enough compute and SRAM memory for a few hundred million dollars. If we apply science there are virtually no limits within in a few years.
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.
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.
In 1986 we wrote a parallel Mandelbrot program in assembly instructions in the 2K or 4K on-chip SRAM of 17 x T414 Transputer chips linked together with 4 x 10 Mbps links each into a cheap supercomputer [1]. It drew the 512 x 342 pictures on a Mac 128K as terminal at around 10 seconds per picture.
I later wrote the quadruple-precision floating-point calculations in microcode [2] to speed it up by a factor of 10 and combined 52 x T414 with 20 x T800 Transputers with floating point hardware into a larger supercomputer costing around $50K.
With this cheap 72 core supercomputer it still would have taken years to produce the deep zoom of the Mandelbrot set [4] that took them 6 months with 12 CPU cores running 24/7 in 2010 [3]. In 2024 we can buy a $499 M4 Mac mini (20-36 'cores') and calculate it a a few days. If I link a few M4s together with 3x32 Gbps Thunderbolt links into a cheap supercomputer and write the assembly code for all the 36 cores (CPU+GPU+Neural Engine) I can render the deep zoom Mandelbrot almost in realtime (30 frames per second).
That is Moore's law in practice. The T414 had 900,000 transistors, the M4 has 28 billion transistors at 3nm (31.111 times larger at 50% of the price).
The M2 Ultra with 134 billion transistors and M4 Max (estimate 100 billion transistors) are larger chips than a M4 but they are relatively more expensive then the M4 so it is cheaper and faster to link together 13 x M4 than buy 1 x M2 Ultra or 6 x M4 instead of 1 x M4 Max.
Cerebras or NVidia also make larger chips, but again, not as cheap and fast as the M4. Price/performance/Watt/dollar is what matters, you want the lowest energy (OPEX) to calculate as many floating point numbers as possible at the lowest purchase cost (CAPEX), you do not want the fastest chips.
You will want to rewrite your software to optimize for the hardware. Even better would be to write the optimum software (for example in variable precision floating point and large integers in Squeak Smalltalk) and then design the hardware to execute that program with the lowest cost. To do that I designed my own runtime reconfigurable chips with reconfigurable core and floating point hardware precision.
I designed a 48 trillion transistor Wafer Scale Integration (WSI) at 3nm with almost a million cores and a few hundred gigabyte SRAM on the wafer [5][6]. This unchipped wafer would cost around $30K. It would cost over $130 million to manufacture it at TSMC. This WSI would have 1714 times more transistors but cost only 60 times at much, a 28 times improvement, but it is an Apples and oranges comparison. It would be more like a 100 times improvement because of the larger SRAM, faster on-chip links and lower energy cost of the WSI over the M4.
The largest fastest supercomputers [7] cost $600 million. To match that with a cluster of M4 would cost around $300 million. To match it with my WSI design would cost $140 million total. For $230 million you get a cheap 3000 x WSI = 144 quadrillion transistor supercomputer immersed in a 10mx10mx10m swimming pool that is orders of magnitude faster then the largest fastest supercomputer and it would be at orders of magnitude lower cost, especially if you would run it on solar energy only [8], even if you would buy three 3000 wafer scale integration supercomputers ($410 million) and only run it during daylight hours and space it evenly around the equator in cloudless deserts. Energy cost dominates hardware costs over the lifetime of a supercomputer.
All the numbers I mentioned are rounded off or estimates, to be accurate requires me to first define every part of the floating point math, describe the software calculations exactly, make accurate hardware definitions and would take me several scientific papers and several weeks to write.
Exceedingly interesting! Say, I have a board from back in the 80s that you may know about - nobody else I've asked has any idea. It's a "Parallon" ISA card from a company called Human Devices, that has something like 8 NEC V20s on it. I think it's an early attempt at an accelerator card, maybe for neural networks, not sure.
Yes I've heard of such a thing [1], it is probably worth $50. This PCB board is just a cluster of 8 V20 (Intel 8088) compatible 16 bit processor, nothing to write home about. It is not considered an early attempt at an accelerator card. Depending on your definition many were done earlier [2] going back to the earliest computers 2000 years ago. My favorite would be the 16 processor Alto [3].
In 1989 I build my 4th Transputer supercomputer for a customer who programmed binary neural networks.
In those early days everyone would use Mandelbrot and Neural Networks as simple demo's and benchmarks of any chip or computer, especially supercomputers. So it is not ironical or serendipitous that a magazine would have a Mandelbrot and an article on a microprocessor in the same issue. My Byte Magazine article on Transputer and DIY supercomputers also described both together.
Thanks for answering, appreciated. I suppose I will just hang onto it as an interesting piece of history, though I never thought it would be worth much - more I'm just hoping to find someone out there who wants it for some personal reason so I can "send it home", so to speak.
Is anyone doing anything with transputer technology now? Do you think it has a chance at resurgence?
> the earliest computers 2000 years ago
Typo, exaggeration, or a reference to something like the Antikythera Mechanism?
>Is anyone doing anything with transputer technology now?
Yes, our Morphle Engine Wafer Scale Integration and our earlier SiliconSqueak microprocessor designs and supercomputers borrows many special features of the Transputer designs by David May. Also of the Alto design by Chuck Thacker, SOAR RISC by David Ungar and a few of the B5000-B6500 designs by Bob Barton. Most of all they build on the design of the Smalltalk, Squeak and reflective Squeak VM software designs by Alan Kay and Dan Ingalls.
>reference to Antikythera Mechanism[5]?
Yes. I could have referenced the Jacquard Loom [1], Babbage Analytical Engine or the later Difference engine [2], Alan Turing, Ada Lovelace and dozens of other contestants for 'first' and 'oldest' computational machines, they are all inaccurate, as are these lists[3][4]. Or I could have only referenced the ones from my own country [6].
>Do you think it has a chance at resurgence?
Yes, I hope my chips and WSI is that resurgence of European microprocessor chips. I just need less than a million euros in funding to start production of prototype WSIs. Even just a single customer buying a $130 million supercomputer is enough. YCombinator should fund me. A science grant or a little support from ASML R&D might also be enough to complete our resurgence. It will take 2-3 years from the moment of funding to go into mass production [5].
There are virtually no limits (for Mandelbrot and computing in general) because there are few limits on the growth of knowledge [5].
In a few decades we will have learned to take CO2 (carbon dioxide) molecules out of the air [4] and rearrange the carbon atoms in 3D structures atom by atom [6]. We will be able to grow the transistors and the solar cells virtually for free. Energy will be virtually free, a squandrable abundance of free and clean energy [2]. At that point we will start automatically self-assembling Dyson Swarms constructions of solar cells with transistors on the back [1] on the Quebibyte scale to capture all the solar output of the sun [3] and get near-infinite compute for free. We would finally be able to explore the Mandelbrot space at full depth within our lifetime.
This is the future I want to live in. Bountiful cheap energy is much more attractive than the futures most people write about now, that vary between some sort of managed decline, dystopianism, or other negativity.
I hope to see you or your faithful recreation in the Matrioskha brain one day.
> I hope to see you or your faithful recreation in the Matrioskha brain one day.
That where my earliest thoughts, 11 years old, that got me into computing: upload my brain into a Matrioskha brain or at least study the human brain by building a supercomputer to simulate it. I'm still working on building cheaper supercomputers and cheaper clean energy 50 years later. Deep zoom Mandel calculations are still a good benchmark to measure my progress.
Three Sci-Fi stories that describe faithful recreation of a human in a computer [1][2][3].
>Bountiful cheap energy is much more attractive than the futures most people write about
A squandrable abundance of free and clean energy means we solve the climate change and sixth mass-extinction crises in the next few years! It should be the only future we fund today. Just cheaper solar cells would do that. No new inventions needed.
Cheap clean energy is the most important next step for humanities survival. It will only cost 100 million, maybe a few hundred million dollars at most [1].
It also means we would get a money free economy as depicted in Star Trek The Next Generation. Even interstellar travel by solar laser and solar sail would become possible if energy is nearly free.
[1] Alan Kay, How? When “What Will It Take?” Seems Beyond Possible, We Need To Study How Immense Challenges
Have Been Successfully Dealt With In The Past https://internetat50.com/references/Kay_How.pdf
I'd rather prefer to pursue the path of producing potent phytochemicals to unleash perfect psionic powers, thereby shortcutting the need for all these boring physical procedures, instead persisting mind over matter as an afterthought.
I remember fractal zooms in real time on the Amiga 500 and they were using a trick: they'd only recompute a few horizontal and vertical lines at each frame, with most of the screen just being a copy/blit of the previous frame, shifted. After a few frames all the pixels from n frames back were discarded.
This was a nice optimization and the resulting animation was still looking nice. Good memories.
Has there been any progress on the analysis or formalization of the how or why of the Mandelbrot set's infinite beauty and complexity from such a simple formula? Sorry for the poor framing of my question... Just seems there is something to be learned from the set beyond just that it exists and looks cool.
In practice, the Mandelbulb is usually only computed to a few iterations (e.g. 20) in order to maintain smooth surfaces and prevent a lot of surfaces from dissolving into ~disconnected "froth".
So deep zooms and deep iterations aren't really done for it.
Also, it's generally rendered using signed distance functions which is a little bit more complicated. I haven't looked at the equations though to figure out if perturbation theory is easy to apply -- I'm guessing it would be, as the general principle would seem to apply.
It's so humbling to read how complex such calculations can get. I took a crack at making a JS client side zooming app a while back and it was miserably slow and would run out of memory as the number precision was limited by JS's max float size...
I have yet to see a Mandelbrot explorer written in Javascript that allows infinite zoom without losing detail and a good UI that works on desktop on mobile.
Does anybody know one?
If there is none, I would build one this year. If anyone wants to join forces, let me know.
The real glory of it is the math – it’s using Webassembly to calculate the reference orbit, and then the GPU to calculate all the pixels, but with an enormous amount of fussing to get around the fact that shaders only have 32 bit floats. The interface works on mobile and desktop, but if you have any tips on how to polish it, let me know.
You could polish it by using variable size precision floats or at least quadruple size 128 bit floating point. This requires you to create a programming language compiler or use my parallel Squeak programming language (it is portable) and have that run on Webassembly or WebGL. It would be easier to have it run directly on CPU, GPU and Neural Engine hardware. The cheapest hardware today would be the M4 Mac mini or design your own chips (see my other post in this thread).
An example of this polished solution is [1] but this example does not yet use high precision floating point [2].
> to get around the fact that shaders only have 32 bit floats
I wonder if there are places around the set where rounding through the iterations depending on the number format chosen, materially affects the shape (rather than just changing many pixels a bit so some smoothness or definition is lost).
Have you considered publishing it under an open source license?
Then I could see myself working on some features like: Selectable color palette, drag&drop and pinch-to-zoom on mobile and fractional rendering (so that when you move the position, only the new pixels get calculated).
I have an old "beta" release of sorts that's live: https://fracvizzy.com/. Change the color mode to "histogram", it creates much more interesting pictures imo. Doesn't really work on mobile yet fyi.
I Just got back into the project recently. I'm almost done implementing smooth, "google maps" style continuous zoom. I have lots of ideas for smoother, more efficient exploration as well as expanded coloring approaches and visualization styles. I'm also working on features for posting / sharing what you find, so you can see the beautiful locations that others find and the visualization parameters they chose. As well as making it easy to bookmark your own finds.
Infinite zoom is probably a long ways out (if ever), but with JS numbers you can zoom pretty far before hitting precision issues. There's a lot to explore there. I'd love to get infinite zoom someday though.
Here's a few example locations I found quickly (I have more links saved somewhere....):
Namely people make these Poincare section plots for Hamiltonian systems like
https://mathematica.stackexchange.com/questions/61637/poinca...
That section that looks like a bunch of random dots is where chaotic motion is observed. There's a lot of reason to think that area should have more structure in it because the proof that there are an infinite number of unstable periodic orbits in there starts with knowing there are an infinite number of stable periodic orbits and that there is an unstable orbit on the the separatrix between them. Those plots are probably not accurate at all because the finite numeric precision interacts with the sensitivity to initial conditions. The Byte article suggests that it ought to be possible to use variable precision math, bounding boxes and such to make a better plot but so far as I know it hasn't been done.
(At this point I care less about the science and more about showing people an image they haven't seen before.)
"In 1920, Pierre Fatou expressed the conjecture that -- except for special cases -- all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. ... can be interpreted to mean that for a dense set of parameters 'a', an attracting periodic orbit exists [in the logistic map]."
Friggin' blew my mind at age 20. Still kinda does. :)
An analogous result is true for the number line: between any two rational numbers, there is an irrational number.
But that isn't cause to suspect that, if you look at a random piece of the number line, you should be able to see a lot of rationals. They're there. But you can't see them. The odds of any given number you look at being irrational are 1.
In a plot like that one those groups of circles that you see are resonances where the winding ratio on the torus (how many degrees you go around per turn) is rational. The area between two resonances (or non-resonant behavior) has a little bit of chaos in those regular regions if you look closely: there is a certain chaotic zone. Those areas of pervasive chaos happen when the chaos area around the rationals covers everything, see
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%8...
which involves doing a sum over all the rationals.
In the 2 degree of freedom case those tori are a solid wall so even if you have a chaotic zone around a resonance the motion is constrained by the tori. For N>2 though there are more dimensions and the path can go "around" the tori. You could picture our solar system of 8 planets having 24 degrees of freedom (although the problem is terribly non-generic because the three kinds of motion in a 1/r^2 field all have the same period) It sure seems that we are in a regular regime like one of the circles in that plot but we cannot rule out that over the course of billions of years that Neptune won't get ejected. See
https://en.wikipedia.org/wiki/Arnold_diffusion
which is poorly understood because nobody has found an attack on it. You have the same problem with numerical work that you do in the plot because sensitive dependence on initial conditions magnifies rounding error. Worse than that normal integrators like Runge-Kutta don't preserve all the geometric invariants of
https://en.wikipedia.org/wiki/Symplectic_geometry
so you know you get the wrong results. There is
https://en.wikipedia.org/wiki/Symplectic_integrator
but other than preserving that invariance those perform much worse than normal integrators. This is one of the reasons why the field has been stuck since before I got into it. This
https://en.wikipedia.org/wiki/Interplanetary_Transport_Netwo...
came out of people learning how to find chaotic trajectories to use for transfer orbits and is an exciting development though. Practically though you don't want to take a low energy-long time trajectory to Mars because you'll get your health wrecked by radiation.
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.
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).
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
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
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
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.
The phrase “infinite number of 0’s” does not contain infinite information. It contains at most what it took to describe it.
The smallest representation of Chaitin's constant is "Ω". This matches the smallest representation of pi.
The “information” in a sequence of symbols is a measure of the “surprise” on obtaining the next symbol, and this is given a very precise mathematical definition, satisfying a few important properties. The resulting formula for many cases looks like the formula derived for entropy in statistical mechanics, so is often called symbol entropy (and leads down a lot of deep connections between information and reality, the whole “It from Bit” stuff…).
For a sequence to have infinite information, it must provide nonzero “surprise” for infinitely many symbols. Pi does not do this, since it has a finite specification. After the specification is given, there is zero more surprise. For a sequence to have infinite information, it cannot have a finite specification. End of story.
The specification has the information, since during the specification one could change symbols (getting a different generated sequence). But once the specification is finished, that is it. No more information exists.
Information content also does not care about computational efficiency, otherwise the information in a sequence would vary as technology changes, which would be a poor definition. You keep confusing these different topics.
Now, if you’ve never studied this topic properly, stop arguing things you don’t understand with those who’ve learned do. It’s foolish. If you’ve studied information theory in depth, then you’d not keep doubling down on this claim. We’ve given you enough places to learn the relevant topics.
I’m here to nitpick.
Number of bits is not strictly 1:1 to number of particles. I would propose to use distances between particles to encode information.
EDIT: ... and of course the point isn't that it's 1:1 wrt. bits and atoms, but I think the point was that there is obviously some maximum information density -- too much information in "one place" leads to a black hole.
See my other post in this discussion.
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.
I later wrote the quadruple-precision floating-point calculations in microcode [2] to speed it up by a factor of 10 and combined 52 x T414 with 20 x T800 Transputers with floating point hardware into a larger supercomputer costing around $50K.
With this cheap 72 core supercomputer it still would have taken years to produce the deep zoom of the Mandelbrot set [4] that took them 6 months with 12 CPU cores running 24/7 in 2010 [3]. In 2024 we can buy a $499 M4 Mac mini (20-36 'cores') and calculate it a a few days. If I link a few M4s together with 3x32 Gbps Thunderbolt links into a cheap supercomputer and write the assembly code for all the 36 cores (CPU+GPU+Neural Engine) I can render the deep zoom Mandelbrot almost in realtime (30 frames per second).
That is Moore's law in practice. The T414 had 900,000 transistors, the M4 has 28 billion transistors at 3nm (31.111 times larger at 50% of the price).
The M2 Ultra with 134 billion transistors and M4 Max (estimate 100 billion transistors) are larger chips than a M4 but they are relatively more expensive then the M4 so it is cheaper and faster to link together 13 x M4 than buy 1 x M2 Ultra or 6 x M4 instead of 1 x M4 Max. Cerebras or NVidia also make larger chips, but again, not as cheap and fast as the M4. Price/performance/Watt/dollar is what matters, you want the lowest energy (OPEX) to calculate as many floating point numbers as possible at the lowest purchase cost (CAPEX), you do not want the fastest chips.
You will want to rewrite your software to optimize for the hardware. Even better would be to write the optimum software (for example in variable precision floating point and large integers in Squeak Smalltalk) and then design the hardware to execute that program with the lowest cost. To do that I designed my own runtime reconfigurable chips with reconfigurable core and floating point hardware precision.
I designed a 48 trillion transistor Wafer Scale Integration (WSI) at 3nm with almost a million cores and a few hundred gigabyte SRAM on the wafer [5][6]. This unchipped wafer would cost around $30K. It would cost over $130 million to manufacture it at TSMC. This WSI would have 1714 times more transistors but cost only 60 times at much, a 28 times improvement, but it is an Apples and oranges comparison. It would be more like a 100 times improvement because of the larger SRAM, faster on-chip links and lower energy cost of the WSI over the M4.
The largest fastest supercomputers [7] cost $600 million. To match that with a cluster of M4 would cost around $300 million. To match it with my WSI design would cost $140 million total. For $230 million you get a cheap 3000 x WSI = 144 quadrillion transistor supercomputer immersed in a 10mx10mx10m swimming pool that is orders of magnitude faster then the largest fastest supercomputer and it would be at orders of magnitude lower cost, especially if you would run it on solar energy only [8], even if you would buy three 3000 wafer scale integration supercomputers ($410 million) and only run it during daylight hours and space it evenly around the equator in cloudless deserts. Energy cost dominates hardware costs over the lifetime of a supercomputer.
All the numbers I mentioned are rounded off or estimates, to be accurate requires me to first define every part of the floating point math, describe the software calculations exactly, make accurate hardware definitions and would take me several scientific papers and several weeks to write.
[1] https://www.bighole.nl//pub/mirror/homepage.ntlworld.com/kry...
[2] https://sites.google.com/site/transputeremulator/Home/inmos-...
[3] http://fractaljourney.blogspot.com
[4] https://www.youtube.com/watch?v=0jGaio87u3A
Maybe https://www.youtube.com/watch?v=zXTpASSd9xE took more calculations, it is unclear.
[5] Smalltalk and Self Hardware https://www.youtube.com/watch?v=vbqKClBwFwI
[6] Smalltalk and Self Hardwarehttps://www.youtube.com/watch?v=wDhnjEQyuDk
[7] https://en.wikipedia.org/wiki/El_Capitan_(supercomputer)
[8] https://www.researchgate.net/profile/Merik-Voswinkel/publica...
Some reference about its existence here, in a magazine that (ironically? serendipitously?) features a fractal on the cover: http://www.bitsavers.org/magazines/Micro_Cornucopia/Micro_Co...
Ever heard of such a thing? I think at this point, I'm trying to find someone who wants it, whether for historical purposes or actually to use.
In 1989 I build my 4th Transputer supercomputer for a customer who programmed binary neural networks.
In those early days everyone would use Mandelbrot and Neural Networks as simple demo's and benchmarks of any chip or computer, especially supercomputers. So it is not ironical or serendipitous that a magazine would have a Mandelbrot and an article on a microprocessor in the same issue. My Byte Magazine article on Transputer and DIY supercomputers also described both together.
[1] https://en.wikipedia.org/wiki/NEC_V20
[2] https://en.wikipedia.org/wiki/History_of_supercomputing#:~:t....
[3] https://en.wikipedia.org/wiki/Xerox_Alto
Is anyone doing anything with transputer technology now? Do you think it has a chance at resurgence?
> the earliest computers 2000 years ago
Typo, exaggeration, or a reference to something like the Antikythera Mechanism?
Yes, our Morphle Engine Wafer Scale Integration and our earlier SiliconSqueak microprocessor designs and supercomputers borrows many special features of the Transputer designs by David May. Also of the Alto design by Chuck Thacker, SOAR RISC by David Ungar and a few of the B5000-B6500 designs by Bob Barton. Most of all they build on the design of the Smalltalk, Squeak and reflective Squeak VM software designs by Alan Kay and Dan Ingalls.
>reference to Antikythera Mechanism[5]?
Yes. I could have referenced the Jacquard Loom [1], Babbage Analytical Engine or the later Difference engine [2], Alan Turing, Ada Lovelace and dozens of other contestants for 'first' and 'oldest' computational machines, they are all inaccurate, as are these lists[3][4]. Or I could have only referenced the ones from my own country [6].
>Do you think it has a chance at resurgence?
Yes, I hope my chips and WSI is that resurgence of European microprocessor chips. I just need less than a million euros in funding to start production of prototype WSIs. Even just a single customer buying a $130 million supercomputer is enough. YCombinator should fund me. A science grant or a little support from ASML R&D might also be enough to complete our resurgence. It will take 2-3 years from the moment of funding to go into mass production [5].
[1] https://en.wikipedia.org/wiki/Jacquard_machine
[2] https://en.wikipedia.org/wiki/Difference_engine
[3] https://www.oldest.org/technology/computers/
[4] http://www.computerhistories.org/
[5] https://www.youtube.com/watch?v=wDhnjEQyuDk
[5] https://en.wikipedia.org/wiki/Antikythera_mechanism
[6] Oldest Dutch computers? https://www.cwi.nl/en/about/history/#:~:text=CWI%20developed....
[7] https://www.youtube.com/watch?v=vbqKClBwFwI or https://www.youtube.com/watch?v=wDhnjEQyuDk or https://www.uksmalltalk.org/2022/06/smalltalk-and-self-hardw...
In a few decades we will have learned to take CO2 (carbon dioxide) molecules out of the air [4] and rearrange the carbon atoms in 3D structures atom by atom [6]. We will be able to grow the transistors and the solar cells virtually for free. Energy will be virtually free, a squandrable abundance of free and clean energy [2]. At that point we will start automatically self-assembling Dyson Swarms constructions of solar cells with transistors on the back [1] on the Quebibyte scale to capture all the solar output of the sun [3] and get near-infinite compute for free. We would finally be able to explore the Mandelbrot space at full depth within our lifetime.
[1] https://gwern.net/doc/ai/scaling/hardware/1999-bradbury-matr... and https://en.wikipedia.org/wiki/Matrioshka_brain
[2] Bob Metcalfe Ethernet https://www.youtube.com/watch?v=axfsqdpHVFU
[3] https://en.wikipedia.org/wiki/Kardashev_scale
[4] Richard Feynman Plenty of Room at the Bottom https://en.wikipedia.org/wiki/There%27s_Plenty_of_Room_at_th...
[5] David Deutsch: Chemical scum that dream of distant quasars https://www.youtube.com/watch?v=gQliI_WGaGk
[6] https://www.youtube.com/watch?v=Spr5PWiuRaY and https://www.youtube.com/watch?v=r1ebzezSV6s
I hope to see you or your faithful recreation in the Matrioskha brain one day.
That where my earliest thoughts, 11 years old, that got me into computing: upload my brain into a Matrioskha brain or at least study the human brain by building a supercomputer to simulate it. I'm still working on building cheaper supercomputers and cheaper clean energy 50 years later. Deep zoom Mandel calculations are still a good benchmark to measure my progress.
Three Sci-Fi stories that describe faithful recreation of a human in a computer [1][2][3].
[1] https://en.wikipedia.org/wiki/Accelerando
[2] https://en.wikipedia.org/wiki/The_Annals_of_the_Heechee and the last part of https://en.wikipedia.org/wiki/Heechee_Rendezvous
[3] https://en.wikipedia.org/wiki/3001:_The_Final_Odyssey
A squandrable abundance of free and clean energy means we solve the climate change and sixth mass-extinction crises in the next few years! It should be the only future we fund today. Just cheaper solar cells would do that. No new inventions needed.
Cheap clean energy is the most important next step for humanities survival. It will only cost 100 million, maybe a few hundred million dollars at most [1].
It also means we would get a money free economy as depicted in Star Trek The Next Generation. Even interstellar travel by solar laser and solar sail would become possible if energy is nearly free.
[1] Alan Kay, How? When “What Will It Take?” Seems Beyond Possible, We Need To Study How Immense Challenges Have Been Successfully Dealt With In The Past https://internetat50.com/references/Kay_How.pdf
>We would finally be able to explore the Mandelbrot space at full depth
What does that mean? The Mandelbrot set is infinitely intricate.
This was a nice optimization and the resulting animation was still looking nice. Good memories.
I had just been (re)watching the Numberphile and 3blue1brown fractal videos this morning so this is a great complement.
So deep zooms and deep iterations aren't really done for it.
Also, it's generally rendered using signed distance functions which is a little bit more complicated. I haven't looked at the equations though to figure out if perturbation theory is easy to apply -- I'm guessing it would be, as the general principle would seem to apply.
Here it is nonetheless if anyone's curious :
App : https://yzdbg.github.io/mandelbrotExplorer/
Repo : https://github.com/yzdbg/mandelbrotExplorer
Does anybody know one?
If there is none, I would build one this year. If anyone wants to join forces, let me know.
The real glory of it is the math – it’s using Webassembly to calculate the reference orbit, and then the GPU to calculate all the pixels, but with an enormous amount of fussing to get around the fact that shaders only have 32 bit floats. The interface works on mobile and desktop, but if you have any tips on how to polish it, let me know.
An example of this polished solution is [1] but this example does not yet use high precision floating point [2].
[1] https://tinlizzie.org/~ohshima/shadama2/
[2] https://github.com/yoshikiohshima/Shadama
https://www.hgreer.com/JavascriptMandelbrot/#an_ugly_hack_th...
I wonder if there are places around the set where rounding through the iterations depending on the number format chosen, materially affects the shape (rather than just changing many pixels a bit so some smoothness or definition is lost).
Have you considered publishing it under an open source license?
Then I could see myself working on some features like: Selectable color palette, drag&drop and pinch-to-zoom on mobile and fractional rendering (so that when you move the position, only the new pixels get calculated).
i’ll whack a license on it later today
I have an old "beta" release of sorts that's live: https://fracvizzy.com/. Change the color mode to "histogram", it creates much more interesting pictures imo. Doesn't really work on mobile yet fyi.
I Just got back into the project recently. I'm almost done implementing smooth, "google maps" style continuous zoom. I have lots of ideas for smoother, more efficient exploration as well as expanded coloring approaches and visualization styles. I'm also working on features for posting / sharing what you find, so you can see the beautiful locations that others find and the visualization parameters they chose. As well as making it easy to bookmark your own finds.
Infinite zoom is probably a long ways out (if ever), but with JS numbers you can zoom pretty far before hitting precision issues. There's a lot to explore there. I'd love to get infinite zoom someday though.
Here's a few example locations I found quickly (I have more links saved somewhere....):
* https://fracvizzy.com/?pos[r]=-0.863847413354&pos[i]=-0.2309...
* https://fracvizzy.com/?pos[r]=-1.364501&pos[i]=-0.037646&z=1...
* https://fracvizzy.com/?pos[r]=-0.73801&pos[i]=-0.18899&z=12&...