While the Mandelbrot set as usually defined is 2D, each point has an associated Julia set, where instead of the additive constant, the starting point is varied (the original Mandelbrot set always uses zero as starting point). Together, they give a 4-dimensional set, where two dimensions are given by the starting point (zr, zi), and the other two by the additive constant (cr, ci). The original Mandelbrot set is a cut through this 4D set at the plane zr=zi=0, while the Julia sets are cuts orthogonal to theat, at planes with constant cr and ci.
You can find a picture of a "4-D" Mandlebrot set in a mid/late 80's issue of Scientific American. I was generating pictures of this on a 286 pc. (with EGA graphics) 15 years ago, and the pictures in TFA of z^2 look *nothing* like that did.
You can find a picture of a "4-D" Mandlebrot set in a mid/late 80's issue of Scientific American.
I was generating pictures of this on a 286 pc. (with EGA graphics) 15 years ago, and the pictures
in TFA of z^2 look *nothing* like that did.
Hah, I can beat that! I used a Compaq portable [oldcomputers.net] with an 8088 processor, 256 K of RAM and 2 floppies! I wrote a C program based on that original Scientific American article, and then had a Basic program read the results and display it. I think the C program took a week to run.
The joke, of course, is that the Compaq didn't have a color screen—it had a small grayscale monitor built in. But I still thought it was really cool.
While not a pure mandelbrot, but a buddhabrot rendering: For the curious, here's [archive.org] a nice 2D projection of such a (rotating) 4D fractal I whipped up a while back.
Archive.org offers the full.avi file for download (the AVI version is about 4000 times more awesome than the flash version), and it's in public domain, so you are perfectly within your rights to go do it yourself.
Also, trying to extend the Mandelbrot set to 3D is ill-defined as there is no good 3D algebra equivalent to the complex numbers (two, 1 and i) or quarternions (four, 1 and i, j, k) - hence you can't express the iteration formula in 3D.
I was following the fractalforums thread for a while, and IIRC that is what a lot of the discussion focused on - "how can we define the squaring operation in 3D such that the Mandelbrot iterative equation gives us something like our vague notion of what we want the Mandelbulb to look like?"
Site is down, but I got an email notification from fractalforums a few days ago, and they had some incredible results. The pursuit is at least as much aesthetic as it is mathematical, and in that respect they've succeeded marvelously.
This post needs more +insightful. What a lot of people are missing by getting wound up in the maths is that it is an artistic endeavour. Their definition of "a mandelbrot" (and yes, this broken terminology bugs the pedant in me beyond belief) is nothing to do with z^2+c, and everything to do with "a pretty looking blobby thing that maintains an aesthetically pleasing and visually interesting level of surface detail at all magnifications".
{0,0,1}^2 doesn't seem to be well-defined. Not only isn't the formula well defined at that point (division by zero), it cannot even be continuously extended to that point, because lim_{e->0} {e,0,1}^2 = {-1,0,0} while lim_{e->0} {0,e,1}^2 = {1,0,0} and even lim_{e->0} {e,e,1}^2 = {0,-1,0}
I had missed a lot of interesting aspects of the 4D Julia/Mandelbrot combo when it was discovered, since computers were so much slower. I wrote my first Mandelbrot program on a Kaypro in high school. Used to run it over night just to get a 100x100 or so image, with low iterations.
The Mandelbrot set has those hairlike strands coming off of it, particularly at high resolution near pi radians. Nearby Julia set fragments, so to speak, all connect through those strands. Since the strand is between 1 and 2 dimensional in the Mandelbrot plane (having infinite arc length within a finite area, the strand within the 4-D coordinates is less than 4-D. So you could almost see something interesting in 3-D there. (Projected to 2-D of course. People who say they see 3-D crack me up, since the back of the eye is a 2-D surface.)
By the way, I particularly like the logarithmic spirals.
You don't even need a second eye, or at least, you don't need a parallax between them. Simply focusing on an object gives a good idea of its distance. To bring an object at a certain distance into focus, the eye muscles must contract "just so", allowing an estimation of that distance.
An then of course there is our brains, which interpret what we see. This is the reason why we can still have the illusion of 3D when looking at a truly two dimensional picture or TV screen. Of course, we can also be fooled, for
It's definitely nifty, the pictures are beautiful, and the creator deserves praise, but the author himself says it's probably not a "true" 3D Mandelbrot:
As exquisite as the detail is in our discovery, there's good reason to believe that it isn't the real McCoy....... Evidence it's not the holy grail? Well, the most obvious is that the standard quadratic version isn't anything special. Only higher powers (around after 3-5) seem to capture the detail that one might expect. The original 2D Mandelbrot has organic detail even in the standard power/order 2 version. Even power 8 in the 3D Mandelbulb has smeared 'whipped cream' sections, which are nice in a way as they provide contrast to the more detailed parts, but again, they wouldn't compare to the variety one might expect from a 3D version of Seahorse valley.
So, Slashdot, I know this is asking a lot, but can you PLEASE at least read the article before posting? Thanks.
From my almost 7-digit standpoint, your feuding looks a lot like cyber-mythology! Is there a deeper story here? Were you both swallowed and subsequently regurgitated by a 3-digit UID?
but do you even had computer in the 4 digit era? or was slashdot some sort of paper mail based discussion forum?
Gawd, don't they teach you brats anything in school these days? It was all vacuum tubes back then. Of course, it's all ball bearings, now. We would've _killed_ for ball bearings back in the day!
Professor: "I'm sorry, Fry, but astronomers renamed Uranus in 2620 to end that stupid joke once and for all."
Fry: "Oh. What's it called now?"
Professor: "Urrectum. Here, let me locate it for you."
What are they trying to do, make up some 3D fractal that just looks like the mandelbrot? This mandelbulb seems pretty arbitrary, and the whole point of the story seems to be that they've found a good one, not that they've found any kind of "true" solution.
If that's the case, it's been a sad day since at least 1984. These things teach us interesting things about numbers and are interesting in and of themselves. As a way of making math more visually beautiful they also serve to draw the interest of youth to a field ordinarily seen as dry and boring.
I wonder if we'll ever reach the point where we will be able to define, with equations and rules, a sea slug using the principles of cellular automata [imachination.net]?
I wouldn't doubt it a bit. A sea slug is already defined by known rules and equations, it's just a matter of doing the math. Their genomes aren't terribly extensive compared to other organisms so it should be quite possible to simulate one quite accurately with a few simple equations and basic rules of chemistry and physics.
Has anyone actually done this? With even a ''simple'' organism ( yes, those are air-quotes ), like a paramecium? It sounds easy in theory, but I bet when we actually get down to it, there'll be a few speedbumps and unexpected obstacles in the way.
Things are not even close. Look at vcell [uchc.edu] to see what's close to the state of the art in cell simulation. Right now, it's a matter of trying to get a few reactions and cell compartments working correctly. I don't think anyone has even come close to modeling any type of complete cell.
Remember the film, Jurassic Park? They applied some simple math to make flocking behavior in their dino models look realistic. It works - just about everybody says the dinosaur flocking looks just like real flocking. Of course real biologists who have been trying to find the math behind real flocking have tested those equations the film makers used, and found some trivial little problem like you need to have faster than light telepathic communication between animal brains if you don't want the animals to get into a ridiculous gridlock once you add in some real environment modeling, but it sure looks like it's real flocking.
And I'm sure we'll get paramecium models or mitochondrion models, or whatever, which 'look just like' the real thing, but turn out to be built on math that has fundamental problems with the rest of reality and uses some cheap hack like omitting surface roughness or gravity to gloss over that part, many times before anyone gets an actual model. We'll see 'accurate' models of atomic nuclei that build all 13 stable elements (or all 1047). 'Accurate' models of natural selection that show only plants should evolve eyes will follow. Eventually, your sea slug will act just like a real one does when the liquid it swims in is molten Sodium, (but not, unfortunately, in water).
People will probably work some or most of these out. Accurate computer modeling of some events has happened, and many more will probably happen with advances in technology. Claiming that all of them will definitely work makes about as much sense as claiming all computer based aircraft models can safely skip the wind tunnel test stage of development.
Weird, I definitely saw that thing after taking acid once, in fact I floated though it for quite a while. It may look all pretty on your screen, but that shit put me off drugs for life, man.
I imagine if they included Mandelbrot fractals as something you can roll up in Katamari, then there would no longer be ANY need to experiment with psychedelic drugs ever again.
cool, nice to see my images linked on slashdot:) hopefully we'll have some gpu-accelerated results to show you all soon (and for those with opencl supporting cards, executables).
btw interested parties might like to check out my 3840x2400 resolution render of the 7th degree version here: http://lyc.deviantart.com/art/siebenfach-139038934 [deviantart.com] (it's buried deep in the thread, and fractalforums is creeking a bit)
A very nice open source app, available through the Ubuntu/Debian repositories. The author's page [mimec.org] even got a windows version.
It supports multi-core CPUs, i.e. if you really want to tax each of your CPU's core to the limit, just use the app to browse through the mandelbrot set. It also supports a 3D extrapolation of the 2D set (OpenGL and software).
Strangely enough it doesn't seem all that popular, as the forum [mimec.org] doesn't seem all that populated..
and here I thought I was coming to read a post about Romanesco Broccoli [google.com] (link goes to gis for "romanesco"). Seriously, it's like eating math.
The common Mandelbrot set is really a 2-dimensional slice of a 4-dimensional object identified by both the combination of the complex numbers Z0 and C in the canonical Zn+1 = Zn^2 + C. The mandelbrot set lives in the plane where Z0 = 0 + 0i, while the Julia sets live on infinitely-many-squared orthogonal planes in the remaining two dimensions, each one intersecting Mandelbrot's plane in a single point of complex coordinates C.
Visualizing this hyperspace monster was made easy by POV-Ray [povray.org].
It took my computer two week of computation to render 80 seconds of
animated 3D slices of a the quaternion [sugarlabs.org].
Check out the scene source [sugarlabs.org].
/me looks forward for a real-time Julia4D explorer.
No, Bob Howard at the Laundry already confirmed this one was ok. However, this is perilously close to the Turing-Lovecraft theorem which the public isn't supposed to know ab n34pnt!@!$ *NO CARRIER*
Actually, the Mandelbrot set is already 4D (Score:5, Informative)
While the Mandelbrot set as usually defined is 2D, each point has an associated Julia set, where instead of the additive constant, the starting point is varied (the original Mandelbrot set always uses zero as starting point). Together, they give a 4-dimensional set, where two dimensions are given by the starting point (zr, zi), and the other two by the additive constant (cr, ci). The original Mandelbrot set is a cut through this 4D set at the plane zr=zi=0, while the Julia sets are cuts orthogonal to theat, at planes with constant cr and ci.
Reply to This
Re:Actually, the Mandelbrot set is already 4D (Score:4, Interesting)
This.
You can find a picture of a "4-D" Mandlebrot set in a mid/late 80's issue of Scientific American.
I was generating pictures of this on a 286 pc. (with EGA graphics) 15 years ago, and the pictures
in TFA of z^2 look *nothing* like that did.
Reply to This
Parent
Re: (Score:3, Interesting)
You can find a picture of a "4-D" Mandlebrot set in a mid/late 80's issue of Scientific American. I was generating pictures of this on a 286 pc. (with EGA graphics) 15 years ago, and the pictures in TFA of z^2 look *nothing* like that did.
Hah, I can beat that! I used a Compaq portable [oldcomputers.net] with an 8088 processor, 256 K of RAM and 2 floppies! I wrote a C program based on that original Scientific American article, and then had a Basic program read the results and display it. I think the C program took a week to run.
The joke, of course, is that the Compaq didn't have a color screen—it had a small grayscale monitor built in. But I still thought it was really cool.
Re:Actually, the Mandelbrot set is already 4D (Score:5, Interesting)
While not a pure mandelbrot, but a buddhabrot rendering: For the curious, here's [archive.org] a nice 2D projection of such a (rotating) 4D fractal I whipped up a while back.
Reply to This
Parent
Re: (Score:3, Informative)
Archive.org offers the full .avi file for download (the AVI version is about 4000 times more awesome than the flash version), and it's in public domain, so you are perfectly within your rights to go do it yourself.
Re:Actually, the Mandelbrot set is already 4D (Score:5, Insightful)
Reply to This
Parent
Re:Actually, the Mandelbrot set is already 4D (Score:5, Insightful)
Site is down, but I got an email notification from fractalforums a few days ago, and they had some incredible results. The pursuit is at least as much aesthetic as it is mathematical, and in that respect they've succeeded marvelously.
Reply to This
Parent
Re:Actually, the Mandelbrot set is already 4D (Score:5, Interesting)
Reply to This
Parent
Re: (Score:3, Interesting)
{0,0,1}^2 doesn't seem to be well-defined.
Not only isn't the formula well defined at that point (division by zero), it cannot even be continuously extended to that point, because
lim_{e->0} {e,0,1}^2 = {-1,0,0}
while
lim_{e->0} {0,e,1}^2 = {1,0,0}
and even
lim_{e->0} {e,e,1}^2 = {0,-1,0}
Re:Actually, the Mandelbrot set is already 4D (Score:4, Insightful)
Good point. Hamilton was working on multiplying triples when he discovered the quaternions. Perhaps it can't be done in a sensible way.
Reply to This
Parent
Re:Actually, the Mandelbrot set is already 4D (Score:4, Interesting)
I had missed a lot of interesting aspects of the 4D Julia/Mandelbrot combo when it was discovered, since computers were so much slower. I wrote my first Mandelbrot program on a Kaypro in high school. Used to run it over night just to get a 100x100 or so image, with low iterations.
The Mandelbrot set has those hairlike strands coming off of it, particularly at high resolution near pi radians. Nearby Julia set fragments, so to speak, all connect through those strands. Since the strand is between 1 and 2 dimensional in the Mandelbrot plane (having infinite arc length within a finite area, the strand within the 4-D coordinates is less than 4-D. So you could almost see something interesting in 3-D there. (Projected to 2-D of course. People who say they see 3-D crack me up, since the back of the eye is a 2-D surface.)
By the way, I particularly like the logarithmic spirals.
Reply to This
Parent
Re: (Score:3, Insightful)
Re:Actually, the Mandelbrot set is already 4D (Score:5, Insightful)
But most people have two eyes, and the parallax between them gives the third dimension.
Reply to This
Parent
Re: (Score:3, Insightful)
You don't even need a second eye, or at least, you don't need a parallax between them. Simply focusing on an object gives a good idea of its distance. To bring an object at a certain distance into focus, the eye muscles must contract "just so", allowing an estimation of that distance.
An then of course there is our brains, which interpret what we see. This is the reason why we can still have the illusion of 3D when looking at a truly two dimensional picture or TV screen. Of course, we can also be fooled, for
Re:Actually, the Mandelbrot set is already 4D (Score:4, Funny)
Really? A sphere is 2D? How are you enjoying things in flat world?
Reply to This
Parent
Not a "true" 3D Mandelbrot (Score:5, Informative)
It's definitely nifty, the pictures are beautiful, and the creator deserves praise, but the author himself says it's probably not a "true" 3D Mandelbrot:
http://www.skytopia.com/project/fractal/2mandelbulb.html#epilogue [skytopia.com]
As exquisite as the detail is in our discovery, there's good reason to believe that it isn't the real McCoy. ... ...
Evidence it's not the holy grail? Well, the most obvious is that the standard quadratic version isn't anything special. Only higher powers (around after 3-5) seem to capture the detail that one might expect. The original 2D Mandelbrot has organic detail even in the standard power/order 2 version. Even power 8 in the 3D Mandelbulb has smeared 'whipped cream' sections, which are nice in a way as they provide contrast to the more detailed parts, but again, they wouldn't compare to the variety one might expect from a 3D version of Seahorse valley.
So, Slashdot, I know this is asking a lot, but can you PLEASE at least read the article before posting? Thanks.
Reply to This
Re:Not a "true" 3D Mandelbrot (Score:5, Funny)
Reply to This
Parent
Re: (Score:3, Informative)
Re: (Score:3, Informative)
There is a subtle difference between "a solution" and "the solution".
But yeah, I was selling it a bit because the pictures are so lovely.
Re:Not a "true" 3D Mandelbrot (Score:5, Funny)
So, Slashdot, I know this is asking a lot, but can you PLEASE at least read the article before posting?
No! I hate everything you stand for.
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Parent
Elder feuds reignited? (Score:3, Funny)
UID 3706 replies to UID 6544:
> No! I hate everything you stand for.
From my almost 7-digit standpoint, your feuding looks a lot like cyber-mythology! Is there a deeper story here? Were you both swallowed and subsequently regurgitated by a 3-digit UID?
Re:Elder feuds reignited? (Score:4, Funny)
UID 3706 replies to UID 6544:
I am not a number, you young punk! And get off my damned lawn!
Reply to This
Parent
Re:Elder feuds reignited? (Score:4, Insightful)
but do you even had computer in the 4 digit era? or was slashdot some sort of paper mail based discussion forum?
Gawd, don't they teach you brats anything in school these days? It was all vacuum tubes back then. Of course, it's all ball bearings, now. We would've _killed_ for ball bearings back in the day!
Reply to This
Parent
Re:Elder feuds reignited? (Score:4, Funny)
Reply to This
Parent
Re:Elder feuds reignited? (Score:5, Funny)
*Burp*
And tasty they were, too.
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Parent
Ice Cream From Uranus? (Score:5, Funny)
That ruined it for me.
Reply to This
Re:Ice Cream From Uranus? (Score:5, Funny)
Fry: "Oh. What's it called now?"
Professor: "Urrectum. Here, let me locate it for you."
Reply to This
Parent
That thing looks like all of my nightmares. (Score:5, Funny)
You could put it in a horror movie and make it pulsate.
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Poorly-defined problem (Score:4, Insightful)
What are they trying to do, make up some 3D fractal that just looks like the mandelbrot? This mandelbulb seems pretty arbitrary, and the whole point of the story seems to be that they've found a good one, not that they've found any kind of "true" solution.
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A sad day indeed... (Score:5, Insightful)
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Parent
In nature - I give you, Brassica oleracea! (Score:4, Informative)
Some of it, at least, has already happened: see this fine example of Brassica oleracea [ubcbotanicalgarden.org], for instance.
Then again, you might have been referring to some of the fractal images that call to mind the work of H. R. Giger... < shiver >.
Cheers,
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Parent
Looks like a big sea slug. (Score:5, Interesting)
Reply to This
Re: (Score:3, Interesting)
I wouldn't doubt it a bit. A sea slug is already defined by known rules and equations, it's just a matter of doing the math. Their genomes aren't terribly extensive compared to other organisms so it should be quite possible to simulate one quite accurately with a few simple equations and basic rules of chemistry and physics.
Re:Looks like a big sea slug. (Score:4, Informative)
It's all chemistry, physics and math.
Has anyone actually done this? With even a ''simple'' organism ( yes, those are air-quotes ), like a paramecium? It sounds easy in theory, but I bet when we actually get down to it, there'll be a few speedbumps and unexpected obstacles in the way.
Things are not even close. Look at vcell [uchc.edu] to see what's close to the state of the art in cell simulation. Right now, it's a matter of trying to get a few reactions and cell compartments working correctly. I don't think anyone has even come close to modeling any type of complete cell.
Reply to This
Parent
Re:Looks like a big sea slug. (Score:5, Insightful)
Remember the film, Jurassic Park? They applied some simple math to make flocking behavior in their dino models look realistic. It works - just about everybody says the dinosaur flocking looks just like real flocking. Of course real biologists who have been trying to find the math behind real flocking have tested those equations the film makers used, and found some trivial little problem like you need to have faster than light telepathic communication between animal brains if you don't want the animals to get into a ridiculous gridlock once you add in some real environment modeling, but it sure looks like it's real flocking.
And I'm sure we'll get paramecium models or mitochondrion models, or whatever, which 'look just like' the real thing, but turn out to be built on math that has fundamental problems with the rest of reality and uses some cheap hack like omitting surface roughness or gravity to gloss over that part, many times before anyone gets an actual model. We'll see 'accurate' models of atomic nuclei that build all 13 stable elements (or all 1047). 'Accurate' models of natural selection that show only plants should evolve eyes will follow. Eventually, your sea slug will act just like a real one does when the liquid it swims in is molten Sodium, (but not, unfortunately, in water).
People will probably work some or most of these out. Accurate computer modeling of some events has happened, and many more will probably happen with advances in technology. Claiming that all of them will definitely work makes about as much sense as claiming all computer based aircraft models can safely skip the wind tunnel test stage of development.
Reply to This
Parent
Flashback (Score:4, Funny)
Weird, I definitely saw that thing after taking acid once, in fact I floated though it for quite a while. It may look all pretty on your screen, but that shit put me off drugs for life, man.
Reply to This
Video games need these now (Score:3, Interesting)
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Katamari Mandelrot (Score:3, Insightful)
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Zooming (Score:4, Informative)
Here's a 7500x7500 (56 megapixel) image of the fractal: http://seadragon.com/view/fnr [seadragon.com].
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Re:Zooming (Score:4, Insightful)
I love how ontopic your signature is.
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Parent
Slashdotted (Score:5, Informative)
Seems to be slashdotted, cached version: http://www.skytopia.com.nyud.net:8090/project/fractal/mandelbulb.html [nyud.net]
Reply to This
w00t (Score:5, Informative)
cool, nice to see my images linked on slashdot :) hopefully we'll have some gpu-accelerated results to show you all soon (and for those with opencl supporting cards, executables).
btw interested parties might like to check out my 3840x2400 resolution render of the 7th degree version here: http://lyc.deviantart.com/art/siebenfach-139038934 [deviantart.com] (it's buried deep in the thread, and fractalforums is creeking a bit)
Reply to This
a great leap forward (Score:5, Funny)
for scientific screensaverology
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Fraqtive (Score:5, Informative)
It supports multi-core CPUs, i.e. if you really want to tax each of your CPU's core to the limit, just use the app to browse through the mandelbrot set. It also supports a 3D extrapolation of the 2D set (OpenGL and software).
Strangely enough it doesn't seem all that popular, as the forum [mimec.org] doesn't seem all that populated..
Reply to This
broccoli (Score:3, Interesting)
and here I thought I was coming to read a post about Romanesco Broccoli [google.com] (link goes to gis for "romanesco"). Seriously, it's like eating math.
Reply to This
Animated quaternion (Score:4, Interesting)
The common Mandelbrot set is really a 2-dimensional slice of a 4-dimensional object identified by both the combination of the complex numbers Z0 and C in the canonical Zn+1 = Zn^2 + C. The mandelbrot set lives in the plane where Z0 = 0 + 0i, while the Julia sets live on infinitely-many-squared orthogonal planes in the remaining two dimensions, each one intersecting Mandelbrot's plane in a single point of complex coordinates C.
Visualizing this hyperspace monster was made easy by POV-Ray [povray.org]. It took my computer two week of computation to render 80 seconds of animated 3D slices of a the quaternion [sugarlabs.org]. Check out the scene source [sugarlabs.org].
/me looks forward for a real-time Julia4D explorer.
Reply to This
Or something like... (Score:3, Funny)
Langoliers remake.
Those things already look like they are made of teeth. Endless rows of teeth that devour the world.
Re:Now do 4d and animate it! (Score:5, Funny)
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Parent
Re:All I see is a big white rectangle (Score:5, Funny)
With a message saying Page cannot be displayed. Not that impressive.
Did you try zooming in?
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Parent
Re:All I see is a big white rectangle (Score:5, Funny)
Did you try zooming in?
It's 404s all the way down.
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Parent