[HN Gopher] The quest to decode the Mandelbrot set
___________________________________________________________________
The quest to decode the Mandelbrot set
Author : limbicsystem
Score : 192 points
Date : 2024-01-26 17:54 UTC (2 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| QuadmasterXLII wrote:
| If you enjoyed learning about the Hashlife algorithm for Conway's
| Game of Life, the bilinear approximation algorithm for computing
| whether a point is in the mandelbrot set has the same je ne sais
| quoi. No one has done a really accessible writeup of it yet, but
| this blog post and the linked forum thread are a good start:
| https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_pra...
| RBerenguel wrote:
| I did some research on this on the side for my dissertation,
| but never published it. The fact centers approximate the
| boundary generalises to almost any point in the plane as a
| consequence of normality of some sequences, and generalises to
| most families of complex iteration under very mild conditions.
| I've had a preprint that I never felt like finishing for
| something like 15 years lying around.
| QuadmasterXLII wrote:
| If you ever decide to put it out as-is, I'd love to read it!
| keepamovin wrote:
| Maybe Mandelbrot shape represents a state space or set of
| possible transformations, configurations or relationships of
| certain solvable or equilibrium dynamical systems, so maybe MLC
| is true if there's a certain structure-preserving relationship
| over sets of these dynamical systems.
| sliken wrote:
| Decades ago two mathematicians argued about the area of the
| mandelbrot set. Both argued an asymtoically approaching different
| numbers. They started a distributed project to calculate the
| area.
|
| I donated time on PA-risc workstations to the effort and was
| surprised to hear that the 2 machines contributed more to the
| final answer then 100s of other contributors. Something about how
| HP's compiler/chip preserved more accurate in the intermediate
| results than others. That surprised me since AFAIK the PA-risc is
| just a normal 64 bit floating point unit, which doesn't every
| have more precision for intermediate results. I believe PCs at
| the time often used the x86, which has 80 bits of precision for
| the intermediate results.
|
| I believe the project was a success, but I don't remember the
| conclusion.
| shrx wrote:
| More context here (alt.fractals discussion from February 1991)
| [0]:
|
| _[...] by computing the area of the M-set using lots of terms
| in a series (Laurent Series?), the upper bound of the area
| seems to converge about at 1.72 (the graph gets quite flat, and
| seems to have an asymptote there), and by counting pixals more
| and more accurately, you seem to get a lower bound of very
| close to 1.52. Both these bounds are close to the values the
| methods would produce in the limit - that is, it is NOT the
| case that these numbers would get closer if a finer grid were
| used, or more terms were taken in the series. So, why the
| difference of 10% or so? No one knows._
|
| [0] https://ics.uci.edu/~eppstein/junkyard/mand-area.html
| denton-scratch wrote:
| > the word evoked the notion of a new kind of geometry --
| something fragmented, fractional or broken.
|
| But that's not what "fractal" means; it means "fractional
| dimension". To say the word "fractal" evoked something is
| subjective - evoked it for whom?
| pohl wrote:
| I suppose the dimension is the something, to be charitable.
|
| Compare to the etymology section here
|
| https://en.wikipedia.org/wiki/Fractal
| pvg wrote:
| For people who know similar-sounding words. Words can evoke
| things well outside their etymology or denotation, one of the
| many reasons people like and use words.
| TheOtherHobbes wrote:
| For people who don't know what a fractional dimension is. And
| mostly don't care.
|
| Fractal art of all kinds was part of a certain trend in 80s/90s
| culture, which also influenced the look of the early Internet.
| And electronic dance music, clubs, and raves.
|
| It was maximalist, psychedelic, colourful, busy, inclusive,
| recursive, and complex.
|
| Whatever the math was doing, it was a very popular signifier of
| certain kinds of experience.
|
| I suspect it's not a surprise that if faded into the background
| when the Internet began to commercialise and blandify in the
| later 90s.
| infogulch wrote:
| Making a program to render a view of the Mandelbrot set is a fun
| exercise, I recommend it.
|
| I found a viewer that works in the browser:
|
| > Mandelbrot Viewer is a universal (desktop and mobile) vanilla
| JS implementation of a plain Mandelbrot set renderer - supporting
| mouse, touch and keyboard interaction.
|
| https://mandelbrot.silversky.dev/
| brazzy wrote:
| Here's my attempt done a long time ago, not very polished:
|
| https://brazzy.de/en/Mandelbrot.php
|
| Even for that, the actual Mandelbrot calculations were the
| smaller part. It's really amazing how such a trivially simple
| formula spawns such endless complexity.
| rnentjes wrote:
| I'll add my julia attempt that animates and works with webgl:
|
| http://julia.perses.games/
| swayvil wrote:
| Complexity is trivial. That's the lesson here. It's human
| perception that's special. It's special in that it has an
| uncommonly low bar on what's considered impressively complex.
|
| Which implies something important, no doubt.
| QuadmasterXLII wrote:
| If you want to keep zooming in the browser and have snappy gpu
| accelerated performance while basically never hitting "reached
| limit of numerical precision," check out
| https://mandeljs.hgreer.com . The math to make this possible
| gets pretty funky: I go over the tricks I used at
| https://www.hgreer.com/JavascriptMandelbrot/
| sapiogram wrote:
| No scrolling with the scroll wheel though :(
| QuadmasterXLII wrote:
| The whole thing is a tower of spaghetti bit hacks in the
| name of depth and speed. Embarassingly, it actually can't
| actually render zoom levels that aren't powers of two and
| I'm not sure how to change that.
| LoganDark wrote:
| Can you make it so that the position you clicked stays at
| the same position after each zoom, rather than centering
| it on your cursor? That way you can click multiple times
| to zoom in on the same point.
| tda wrote:
| Thanks for creating this, I had some fun exploring the set.
| What does the scale parameter stand for?
| QuadmasterXLII wrote:
| The color of each pixel is fragColor =
| vec4(vec3(cos(c), cos(1.1214 * c) , cos(.8 * c)) / 2. + .5,
| 1.);
|
| Where c is the number of iterations divided by the scale.
| tweetle_beetle wrote:
| That really is very fast!
|
| (I don't know if this is a known compromise of your technique
| (I couldn't find it mentioned), but I occassionally get a
| ring of varying thickness centred in the render with an
| inverted coluring to the rest of the pixels, or sometimes a
| solid colour. It varies is size and comes and goes without an
| obvious correlation to zoom level.)
| QuadmasterXLII wrote:
| Yeah, I've been chasing that for a while. I've traced it to
| subnormal float handling. Here's the scenario: if you
| multiply the 32 bit float float 1 x 2^-127 by 1/8, you
| can't get 1x2^-130 because 130 isn't representable in a 8
| bit exponent. On the CPU or a modern GPU, this is special
| cased to result in a "subnormal" float: .125 x 2^-127.
| However, this adds a ton of extra complexitity to the FPU.
| Old GPUs just gave up and say "It's zero." WebGL
| standardized on the old GPU behavior, so a modern GPU
| running webgl sets a flag to behave like the old GPUs.
|
| I'm representing complex numbers as (real_mantissa + i *
| imag_mantissa) * 2^exp where real_mantissa and
| imag_mantissa are themselves 32 bit floats, and I can
| reduce the frequency of the colored rings in my shader by
| keeping the mantissas around ~1000 instead of ~1 (and
| reducing the exponent to keep the value the same) so that
| they are less likely to go subnormal, but I can't seem to
| get rid of the colored rings entirely.
|
| I don't actually know the pathway through the code from
| subnormal underflow -> colored rings, but if you run a CUDA
| renderer with the same algorithm, the rings appear when you
| set the flag for "round subnormal floats to zero."
| crazygringo wrote:
| I made quite a sophisticated viewer as a teen, but boy was it
| slow to run back then, back on a 486.
|
| Now people write viewers that run lighting fast in a browser
| thanks to WebGPU! Like:
|
| https://www.reddit.com/r/fractals/comments/o7l4bm/please_try...
| IronBacon wrote:
| I remember as a teen I did mine in 68000 assembly on a
| Commodore Amiga to get it reasonable fast considering it was
| running on a 16 MHz CPU -- i would say a few seconds to draw
| the canonical image -- but IIRC it reached pretty fast the
| limit of math precision.
|
| At that time I didn't know what was a complex number, but was
| fascinated by the whole concept of fractals and how complex
| structures could be created with a relative simple program.
| Baeocystin wrote:
| A few seconds? Wow! I did a science fair project on the
| Mandelbrot set using my Amiga 2000, and it took me a good
| 45+ minutes to generate a single 320x200 color image. IIRC,
| I wrote the generator in some variant of Pascal, and was so
| happy with the performance increase over Basic on my
| C=128...
|
| I ran in to the precision limit pretty quickly, same as
| you. I didn't understand computers well enough to know
| that's what the problem was, and I remember spending hours
| pouring over my code, trying to figure out where the bug
| was. Good times. :D
| vardump wrote:
| The difference is probably you used floats and he/she
| used fixed point (integers). Of course that way you do
| run out of precision very quickly.
|
| Software emulated floats on Amiga 2000 were really,
| really slow.
| IronBacon wrote:
| If I'm not confusing it with something else, I seem to
| recall that when zooming on the set the calculation was
| nearly instantaneous. As you said, good times! ^__^
|
| I still have that A500 but who knows where I put those
| floppies, I'm tempted to turn it on but I'm scared the
| PSU will blow itself...
| BeetleB wrote:
| I did it in QuickBasic (also on a 486).
|
| Now _that 's_ slow.
| chihuahua wrote:
| I typed in a BASIC program for a C-64 from a computer
| magazine. That was a popular mechanism for software
| distribution back then.
|
| I thought "this will never work", but I was amazed that it
| did in fact work. I remember letting it run overnight to find
| a complete 160x200 4-color image the next morning. Just
| clearing the screen using a loop to fill 8192 bytes with 0s
| took over a minute.
|
| Everything about the computers at the time was so terrible, I
| don't feel any nostalgia for those days or that technology.
| dekhn wrote:
| The one I wrote in BASIC on my Apple IIe never finished, it
| hadn't even got the interesting bits of the set after a day
| or so and I needed to play games.
|
| FRACTINT was great because I could use my 286 and it was
| fairly fast because used integer whenever possible.
| martyvis wrote:
| '"We've got to try to train a neural network to zoom around the
| Mandelbrot set," Kapiamba joked.'
|
| This actually sounds to me to be fine goal. An AI that sounds out
| unexplored depths to reveal interesting sights that maybe
| resemble what we see in the world at our level or cool patterns
| that potentially are a delight to the eye would be pretty cool.
| hnfong wrote:
| The Mandelbrot set is quite well known.
|
| Yet something I learned recently blew my mind. It's about the
| uncanny resemblance between the images generated by the
| Mandelbrot set, and among all things, the popular image of
| Buddha.
|
| For example: https://en.wikipedia.org/wiki/Buddhabrot
|
| Even when looking at the 2D Mandelbrot set renderings, I can't
| help but wonder whether the similarity of the "bulbs" to the
| rather unique Buddha "hairstyle" (of allegedly funny lumps of
| hair) was just a coincidence.
|
| Also, the tower-like makuta headdress in some Buddhist traditions
| look exactly like the thin threads that connect the bulbs at
| around (-2, 0).
|
| I'm not saying they mean anything, but just something uncanny,
| and once I learned about the resemblance, it's hard to unsee
| it...
| flohofwoe wrote:
| Also see the Mandelbrot monk (before you get too excited, it
| was a hoax)
|
| https://abcnews.go.com/Technology/WhosCounting/story?id=9861...
| ggambetta wrote:
| > I can't help but wonder whether the similarity of the "bulbs"
| to the rather unique Buddha "hairstyle" (of allegedly funny
| lumps of hair) was just a coincidence.
|
| Honest question: what _else_ do you think it could be, if not a
| coincidence?
| hnfong wrote:
| Honest answer: some early Buddhist followers taking some form
| of acid/mushrooms and saw geometric visions of the Mandelbrot
| set and thought it was a divine manifestation of Buddha?
|
| I mean, I don't think this is likely, but that's the best I
| got.
|
| I hear the brain likes to go into geometry mode when
| hallucinogens are ingested, and I suppose the brain is
| theoretically powerful enough to compute the Mandelbrot
| sets...
| armchairdweller wrote:
| I have had similar thoughts.
|
| The prominent modes and paths of this 2D probability
| distribution also show some resemblance to the kabbalistic
| tree of life, which is its own, but fairly related topic of
| study. DMT use within a connected strand of this "inner
| science" has been suspected.
|
| Drawing more of these far-fetching connections: The complex
| plane is related to several areas of physics, which might
| somehow find expression in electromagnetic brain dynamics.
|
| In any case, the buddhabrot distribution seems quite
| understudied both from a scientific / mathematical PoV, and
| from the perspectives of the occluded study of the "inner
| realms".
| fractaltemp wrote:
| I've done a wide variety of hallucinogens. While I've seen
| things that could be described as fractal (in that they are
| nested and self similar), I've never seen a literal time
| escape fractal. In my experience, hallucinations look more
| like turbulent flow, like smoke. Deepdream isn't a terrible
| first approximation, but may give you the impression that
| the hallucinations are more dramatic or total than is
| typical.
|
| I can't rule this hypothesis out entirely, but I'm very
| skeptical. Pareidolia seems more likely to me.
| AlanYx wrote:
| There are a few others well-known fractals apart from the
| Buddhabrot that resemble real-world objects. My favourite is
| the burning ship fractal:
| https://en.wikipedia.org/wiki/Burning_Ship_fractal
|
| I found these sorts of things really helpful in getting my kids
| interested in fractals. They love the idea that there are
| "things" they can find that are only viewable through math.
| Y_Y wrote:
| The burning ship is still my favourite object in mathematics
| despite about twenty years of study since I first came across
| it. In an echo of the famous story about Tamm and the Taylor
| series error, I once had to explain the burning ship to
| border control in an impromptu test of my credentials (though
| I'm not sure if I'd have been shot if I got it wrong).
| thanatos519 wrote:
| My favourite way to think about this shape is
| https://en.m.wikipedia.org/wiki/File:Unrolled_main_cardioid_...
|
| ... It's all elephants. The -R spike at the main disc is period
| 2, the fork around +/-i is period 3, and so on to infinity at
| 0.25+0i.
|
| MLC might just be one of those facts that is true but unprovable!
| pronoiac wrote:
| For some fractals on the regular, Benoit Mandelbot on Mastodon -
| https://botsin.space/@benoitmandelbot
| DanielleMolloy wrote:
| This view on the Mandelbrot iteration seems much more interesting
| than the original set:
|
| https://en.wikipedia.org/wiki/Buddhabrot
|
| It is the probability distribution (i.e. the most frequent
| locations visited) over the trajectory of the points that escape
| the plane.
| DanielleMolloy wrote:
| Since the Mandelbrot iteration happens on the complex plane, is
| there any recommended reading / research about its relation to
| scientific fields where the complex plane is applied, like
| mechanical oscillatory systems, quantum mechanics and
| electromagnetism?
| asplake wrote:
| I had a couple of related questions. To what extent does
| complex dynamics map to physical phenomena? And in the opposite
| direction, how is renormalisation used outside of quantum
| physics?
| markisus wrote:
| A very interesting cast of underdog characters appears in the
| article.
|
| You've got one guy with a relentless spirit to continue with
| mathematics in is spare time after being blacklisted from
| mainstream academia because of antisemitism.
|
| Another is a childhood prodigy, who set the record for the
| youngest American IMO team member, but got burned out as an adult
| and went into finance but found his way back through the
| mentorship of another mathematician.
|
| And a third was a biology major. After graduating, he worked as a
| baker. But he wanted a career change so he entered a master's
| program in math and proved an impressive result.
| dylan604 wrote:
| Sounds like a couple of people decided to take an extended gap
| year, and then came back more focused and in a better place for
| the number crunching
| fweimer wrote:
| Note that the article refers to Soviet antisemitism: Jews were
| denied academic jobs, and they couldn't move abroad to work in
| their field, either. Not really related to mathematics as such
| (and definitely not about the mathematical community rejecting
| an antisemite).
| eps wrote:
| > they couldn't move abroad to work in their field
|
| Nobody could do that, Jews or not.
| OscarCunningham wrote:
| What is the conjectured topology of the Mandelbrot set if MLC is
| true?
|
| My understanding is that there's a certain number of bulbs, each
| centred around a point which becomes periodic with period p after
| k steps. But how do they all stick together?
| bongodongobob wrote:
| The entire set is connected iirc.
| qazxcvbnm wrote:
| But what would be its homology, for instance?
| OscarCunningham wrote:
| It's known that it's connected and simply connected. So if
| it's locally connected then I think it has to be
| contractable.
| clintonc wrote:
| MLC stands for "Mandelbrot Locally Connected". It's not
| obvious, but this is equivalent to the bulbs of the Mandelbrot
| set (the domains of parameters where almost all points get
| attracted toward periodic orbits) are dense in the Mandelbrot
| set. Everyone believes it to be true.
| OscarCunningham wrote:
| Yes, but how exactly are the bulbs arranged? Wikipedia says
| 'Not every hyperbolic component can be reached by a sequence
| of direct bifurcations from the main cardioid of the
| Mandelbrot set. Such a component can be reached by a sequence
| of direct bifurcations from the main cardioid of a little
| Mandelbrot copy'. Which sequences of bulbs have little copies
| at the end of them? And how do the little copies attach?
| clintonc wrote:
| The combinatorics of how the Mandelbrot set is put together
| is well-studied, and rather independent of MLC. The
| arrangement of the bulbs on the boundary of the "main
| cardiod" (which is where there is an attracting fixed
| point) is described here: https://en.wikipedia.org/wiki/Man
| delbrot_set#Main_cardioid_a.... Generally, the patterns are
| given by something called Lavaur's Algorithm; see https://e
| n.wikibooks.org/wiki/Fractals/Iterations_in_the_com... for
| some explanation. Attachment points are always at the
| "root" of the Mandelbrot set, which is the cusp of the main
| cardioid.
|
| A consequence of MLC is that the combinatorial picture
| given by Lavaur's algorithm and related analyses is
| "complete" -- all dynamical information is available from
| the combinatorial models.
| OscarCunningham wrote:
| Thank you, that's very helpful!
| derbOac wrote:
| Does anyone know of any good resources on Kolmogorov complexity
| and fractals such as the Mandelbrot set? Or even on information
| theory and fractals?
|
| For some reason reading this article is making me wonder about
| the difference between the information required to generate
| something like a mandelbrot, knowing the underlying rule, and the
| information required to represent it as it is, without following
| the rule. Or e.g., the difference between the information of the
| generating rule and the information implicitly represented
| through the time or number of operations needed to generate it.
|
| It seems like there's some analogy between potential and kinetic
| energy, and kolmogorov complexity and something else, that I'm
| having trouble putting my finger on. Even if you have a simple
| generating algorithm that might be small in a kolmogorov
| complexity sense, if that algorithm entails a repeating something
| over a large number of operations, the resulting object would be
| complex, so there's an implied total complexity as well as an
| "generating" one.
|
| Maybe this is some basic computational complexity concept but if
| so I'm not recalling this, or am being dense. E.g., I'm used to
| discussions of "compressibility" but not of the "generating
| representation information cost" versus "execution cost".
| dcow wrote:
| I think you're forgetting that there's no definitive way to
| represent something, compressed or raw. So while it's
| interesting to acknowledge that we can compress data and
| sometimes rather efficiently, I'm not sure it maps to anything
| physical beyond the fact that decompressing data creates
| entropy.
|
| Perhaps you'd be interested in
| https://en.wikipedia.org/wiki/Landauer%27s_principle. Turns out
| there may be a minimum energy required to _decrease_ entropy.
| Jade has a really good overview https://youtu.be/XY-mbr-
| aAZE?si=7DvSs2DMudsh6gk8
| kjqgqkejbfefn wrote:
| > so there's an implied total complexity as well as an
| "generating" one.
|
| Dessalles's algorithmic simplicity theory of (cognitive)
| relevance is formulated in these terms.
|
| >Situations are relevant to human beings when they appear
| simpler to describe than to generate
|
| The discrepancy between generation complexity
|
| >the complexity (minimal description) of all parameters that
| have to be set for the situation s to exist in the "world"
|
| i.e, the "pixels"
|
| and description complexity
|
| > the length of the shortest available description of s (that
| makes s unique)
|
| i.e. the mandelbrot formula
|
| is named Unexpectedness in this framework.
|
| https://telecom-paris.hal.science/hal-03814119/document
|
| https://simplicitytheory.telecom-paris.fr/
|
| Dessalles published a paper in 2022, Unexpectedness and Bayes'
| Rule
|
| https://cifma.github.io/Papers-2021/CIFMA_2021_paper_13.pdf
|
| >A great number of methods and of accounts of rationality
| consider at their foundations some form of Bayesian inference.
| Yet, Bayes' rule, because it relies upon probability theory,
| requires specific axioms to hold (e.g. a measurable space of
| events). This short document hypothesizes that Bayes' rule can
| be seen as a specific instance of a more general inferential
| template, that can be expressed also in terms of algorithmic
| complexities, namely through the measure of unexpectedness
| proposed by Simplicity Theory.
|
| Maybe there is a way to plug this into the
| https://en.wikipedia.org/wiki/Buddhabrot fractal someone
| mentioned above.
| GuB-42 wrote:
| If you are interested in rendering the Mandelbrot set in a
| variety of ways, as well as its 3D extensions, look here:
| https://iquilezles.org/articles/
|
| The last part is about fractals, especially the Mandelbrot set.
| With some theoretical and some practical articles.
| auroralimon wrote:
| i've often wondered if mandelbrot is what you get when you do a
| simple quadratic iterator in complex numbers, what are the
| comparable sets for quaternions and octonions??
| bunabhucan wrote:
| https://en.m.wikibooks.org/wiki/Pictures_of_Julia_and_Mandel...
|
| There's a whole world of this as well as iterating different
| functions.
| hermitcrab wrote:
| I don't think I understand what 'locally connected' means. You
| can easily choose a rectanglular area that contains 2 areas of
| the set that are not joined.
| nonsensikal wrote:
| You don't get to choose a rectangle, you choose a point.
| hermitcrab wrote:
| The doesn't seem to fit with the comb analogy.
|
| I still don't understand.
| returningfory2 wrote:
| I think this part of the article is incorrect.
| matt-noonan wrote:
| It's almost correct, but misses the point in an annoying
| way that kind of ruins the example. What does work is
| something like the subset of the plane given by { (x, y)
| | x real, y rational } U { (0, y) | y real }. This is
| connected, because you can walk from any point (x,y) to
| any other point (x',y') by traveling horizontally to the
| Y axis at (0,y), vertically to (0,y'), then horizontally
| to (x',y'). But it isn't _locally_ connected away from
| the Y axis because for a tiny enough open set S around a
| point (x,y), there are other points in S that you can 't
| get to from (x,y) without leaving S.
| nhatcher wrote:
| Non locally connected spaces are a bit pathological. Means
| that given a point there is always a neighborhood of the
| point (might be very small) that is connected. An example
| of a connected but not locally connected is:
| https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve
| From (0, O) any neighborhood, no matter how small contains
| points that belong to the curve but cannot reach (0, 0) and
| stay in the neighborhood.
| G3rn0ti wrote:
| It means you can choose two points inside the Mandelbrot set
| and always find a curve that connects the two without you ever
| needing to lift the pencil.
| hermitcrab wrote:
| Isn't that "connected" rather than "locally connected"?
| crazygringo wrote:
| I'm aware that "locally connected" has a very specific
| meaning in math:
|
| https://en.wikipedia.org/wiki/Locally_connected_space
|
| Unfortunately I don't have the slightest idea what it
| actually means... that article does not have any ELI5
| sentence within it.
| hermitcrab wrote:
| >Unfortunately I don't have the slightest idea what it
| actually means
|
| Not just me then? ;0)
| neilkk wrote:
| Suppose you choose such a rectangle, let's say it incorporates
| the 'fringe' of the main cardioid and the fringe of the biggest
| circle. Within that rectangle, color green all the pieces which
| connect to the main cardioid and blue all the pieces which
| connect to the circle. Local connectedness means that there
| won't be any points in that rectangle which have both blue and
| green points arbitrarily close. So there are places where
| 'locally non-connected parts' of the set can be close together,
| but there must be a border between them, rather than them being
| hopelessly entangled.
|
| Strictly speaking, you would do this coloring with all
| connected components of the intersection of your rectangle and
| M. (And the rectangle could be any region.)
|
| The example messes this up, although it is a correct example,
| the square on the diagram showing the local piece containing
| non-connected parts is wrong. The comb has more and more teeth,
| infinitely many in a bounded space, on the left side. Only a
| rectangle which includes the left edge properly shows why the
| set isn't locally connected. The rectangle pictured includes
| finitely many teeth which have a separation between them. A
| rectangle overlapping the left edge of the comb would include
| separate components which get arbitrarily close to that left
| edge and so can't be separated by a border.
| hermitcrab wrote:
| Thanks for taking the time to explain that. So, in terms I
| find easier to understand, MLC would mean that if I:
|
| -take any rectanglular section of the complex plain that
| includes part or all of the Mandelbrot set
|
| -draw the Mandelbrot set in black
|
| -pick an arbitary black point and colour it red
|
| -recursively colour every black point touching a red point
| (flood fill)
|
| Then every black point would be recoloured red. And this
| would work with a pixel based image of the mandelbrot if the
| image had a high enough resolution. Is that right?
| neilkk wrote:
| No, that's not right.
|
| Do those first four steps. You wouldn't (necessarily) cover
| every black point in your rectangle. Choose a remaining
| black point and flood fill from that, say green. Keep on
| doing this with different colours until you've covered
| every black point in your rectangle. You have a bunch of
| regions of different colours.
|
| Now, if the different coloured regions are all nicely
| separate, then your set is locally connected. Because each
| point is either cleanly in one component or cleanly in the
| other.
|
| If on the other hand your drawing looks like https://common
| s.m.wikimedia.org/wiki/File:Julia_set_for_the_... with
| mixed up boundaries where some points are infinitesimally
| close to more than one colour, then it's not locally
| connected.
|
| The difficulty with intuition is that in our intuition,
| coloured regions always have reasonable boundaries (think
| countries in a map: the border can be wiggly but there's
| never infinitely many tiny bits of one country mixed up in
| the boundary of two others). In fractal geometry, things
| like the Newton fractal picture above are quite usual.
| hermitcrab wrote:
| I think I understand now. Much appreciated! 'Locally
| connected' seems like quite poor terminology.
| bilsbie wrote:
| I've always wondered if you could make fractals into some sort of
| game.
| netmare wrote:
| There's MMCE+, the spiritual successor of Marble Marcher. I've
| only played the original some years ago and it was pretty
| awesome. You simply have to move a ball across a 3D fractal
| surface that's constantly evolving in real-time!
|
| Sadly, I can't run MMCE to test, since I'm using a PC and gfx
| card from 2009.
|
| +: https://michaelmoroz.itch.io/mmce
| peter_d_sherman wrote:
| >"When computers revealed all those smaller copies of the
| Mandelbrot set within itself, Douady and Hubbard wanted to
| explain their presence. They ended up turning to what's known as
| _renormalization theory_ , a technique that physicists use to
| tame infinities in the study of quantum field theories, and to
| connect different scales in the study of phase transitions."
|
| https://en.wikipedia.org/wiki/Renormalization
|
| Rampant conjecture/speculation: In the future, perhaps some
| Mathematician might discover a link between _Renormalization
| Theory_ -- and the _Digits Of Pi_... since they seem related...
|
| More specifically, between Renormalization Theory -- and
| algorithms for the Digits of Pi.
|
| Of which, one notable one is The Chudnovsky algorithm:
|
| https://en.wikipedia.org/wiki/Chudnovsky_algorithm
|
| Which leads to Binary Splitting:
|
| https://en.wikipedia.org/wiki/Binary_splitting
|
| Which leads to Hypergeometric Series:
|
| https://en.wikipedia.org/wiki/Hypergeometric_function#The_hy...
|
| Which leads to Gauss' continued fraction:
|
| https://wikimedia.org/api/rest_v1/media/math/render/svg/4d54...
|
| https://en.wikipedia.org/wiki/Hypergeometric_function#:~:tex...
|
| https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction
|
| Which leads to Analytic continuation of 3F2, 4F3 and higher
| functions:
|
| https://fredrikj.net/blog/2009/12/analytic-continuation-of-3...
|
| Which leads to my brain hurting ("Put down that Math book and
| step away from the Math!" <g>) -- because I can't handle all of
| this Math for now! :-) <g> :-)
|
| But there is this very cool picture there:
|
| https://3.bp.blogspot.com/_rh0QblLk0C0/SzEG9q5FxaI/AAAAAAAAA...
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