archived as http://www.stealthskater.com/Documents/Fractals_03.doc
more of this topic at http://www.stealthskater.com/Science.htm#Fractals note: because important web-sites are frequently "here today but gone tomorrow", the following was archived from http://www.newscientist.com/article/dn18171-the-mandelbulb-first-true-3d- image-of-famous-fractal.html on November 18, 2008. This is NOT an attempt to divert readers from the aforementioned web-site. Indeed, the reader should only read this back-up copy if the updated original cannot be found at the original author's site.
The "Mandelbulb": first 'true' 3D image of famous fractal by Jacob Aron New Scientist, November 18, 2009
It may look like a piece of virtuoso knitting. But the makers of an image they call the Mandelbulb (see right) claim it is most accurate 3-dimensional representation to date of the most famous fractal equation: the Mandelbrot set.
Fractal figures are generated by an "iterative" procedure. You apply an equation to a number … apply the same equation to the result … and repeat that process over-and-over again. When the results are translated into a geometric shape, they can produce striking "self-similar" images -- forms that contain the same shapes at different scales. For instance, some look uncannily like a snowflake. The tricky part is finding an equation that produces an interesting image.
The most famous fractal equation is the 2-D Mandelbrot set -- named after the mathematician Benoît Mandelbrot of Yale University who coined the name "fractals" for the resulting shapes in 1975.
But there are many other types of fractal both in 2 and 3 dimensions. The "Menger sponge" is one of the simplest 3-D examples.
Fake fractal
There have been previous attempts at a 3-D Mandelbrot image. But they do not display real fractal behavior says Daniel White, an amateur fractal image maker based in Bedford, UK.
Spinning the 2-D Mandelbrot fractal like wood on a lathe, raising and lowering certain points, or invoking higher-dimensional mathematics can all produce apparently 3-dimensional Mandelbrots. Yet none of these techniques offer the detail and self-similar shapes that White believes represent a true 3-D fractal image.
2 years ago, he decided to find a "true" 3-D version of the Mandelbrot.
1 The next dimension
"I was trying to see how the original 2-D Mandelbrot worked and translate that to the third dimension," he explains. "You can use complex maths. But you can also look at things geometrically."
This approach works thanks to the properties of the "complex plane" -- a mathematical landscape where ordinary numbers run from "East" to "West" while "imaginary" numbers (i.e., based on the square root of -1) run from "South" to "North". Multiplying numbers on the complex plane is the same as rotating it. And addition is like shifting the plane in a particular direction.
To create the Mandelbrot set, you just repeat these geometrical actions for every point in the plane. Some will balloon to infinity escaping the set entirely while others shrink down to zero. T he different colors on a typical image reflect the number of iterations before each point hits zero.
White wondered if performing these same rotations and shifts in a 3-D space would capture the essence of the Mandelbrot set without using complex numbers (i.e., real numbers plus imaginary numbers) which do not apply in three dimensions because they are on only two axes. In November 2007, White published a formula for a shape that came pretty close.
Higher power
The formula published by White gave good results but still lacked true fractal detail. Collaborating with the members of Fractal Forums (a website for fractal admirers), he continued his search. It was another member -- Paul Nylander -- who eventually realized that raising White's formula to a higher power (i.e., equivalent to increasing the number of rotations) would produce what they were looking for.
White's search isn't over, though. He admits the Mandelbulb is not quite the "real" 3-D Mandelbrot. "There are still 'whipped cream' sections where there isn't detail," he explains. "If the real thing does exist – and I'm not saying 100 percent that it does – one would expect even more variety than we are currently seeing."
Part of the problem is that extending the Mandelbrot set to 3-D requires many subjective choices that influence the outcome. For example, you could extend a flat plane to 3-D by stretching it to form a box, but you could also turn it into a sphere.
"It's an interesting academic exercise to think what you should get," says Martin Turner, a computer scientist specializing in fractal images at the University of Manchester, UK. "But it all depends on what properties you want to keep in the third dimension."
The equations White used may get the job done. But the system of algebra used is not applicable to all 3-D mathematics. "The next stage is mathematical rigor," says Turner.
2
2-D fractal
Mathematicians have studied self-similar shapes like this one since the turn of the last century. But it was Benoît Mandelbrot who named them "fractals" in 1975. Fractals are more than pretty pictures, though, as the mathematics behind them can explain natural phenomena from the length of coastlines to the structure of the Universe.
There are many types of fractals, both in 2 and 3 dimensions. But the most famous is the 2-D Mandelbrot set shown here. Past attempts to extend it to the third dimension have met with difficulty. (image: Wolfgang Beyer)
Spun
One method for generating a 3-D Mandelbrot image is to spin the classic 2-D Mandelbrot around a central axis like a block of wood in a lathe. You end up with a 3-D shape. But it's not much different to the original. There are a few ways to produce this shape but none of them add any extra detail. (image: Paul Bourke)
Raised
The traditional Mandelbrot image is produced through an iterative process with different colors indicating the number of iterations at that point. Another method of making a 3-D Mandelbrot (shown here) keeps the 2-D version but swaps the different colors for changes in height.
With some parts higher and some lower, the effect is similar to the hills and valleys of a relief map. But it's still essentially 2-dimensional. (image: Aexion)
3 Whipped
This tasty-looking 3-D Mandelbrot is actually a projection from 4 dimensions just as a square is a 2-D projection of a cube. It is generated using a 4-D version of complex numbers called quaternions. But despite that complexity, its smooth appearance means there is no interesting fractal detail when you zoom in. Even higher-dimensional maths can be used to create 3-D Mandelbrots but with no better results. (image: Paul Bourke)
Mandelbulb
Daniel White's "Mandelbulb" takes a different approach. He took the geometrical properties of the "complex plane" where multiplication becomes rotation and addition becomes movement of the plane in a particular direction and applied them to a 3- dimensional space.
(image: Daniel White)
Mandelbulb close-up
It took 2 years to perfect the formula which contains incredible detail at even 1000x magnification.
(image: Daniel White)
4
Mandelbulb close-up 2 Here is a close-up of a spiral section of the Mandelbulb juxtaposed with a 2-D Mandelbrot image. (image: Daniel White)
Mandelbulb close-up 3 A close-up of the Mandelbulb "spine" and a similar 2-D Mandelbrot. (image: Daniel White)
5
Mandelbulb close-up 4
The whole Mandelbulb again from a different viewpoint.
(image: Daniel White)
Readers' Comments
1. Gilding The Lilly? by Tom / Wed Nov 18, 2009 13:48:34 GMT Interesting. But adding another dimension to an infinitely complex fractal will only reveal things that were there anyway if you had looked closely enough.
2. A Little Help by Geraint / Wed Nov 18, 2009 13:53:41 GMT Now admittedly I've scanned over this briefly before I go to work so likely I've skipped an explanation somewhere. But as interesting as this is, what are the practical applications of mapping fractal equations in this way? It mentioned that the patterns are reminiscent of certain elements in Nature. Is this related to things like weather predictions and such like?
I'd be grateful if someone could reply to a layman here. Mathematics has never been my most intuitive subject! :)
3. A Little Help by Tony Coleby / Wed Nov 18, 2009 14:45:41 GMT Hey, I'm a layman too. But one obvious real world application -- at least once computers are fast enough to do these kind of equations on the fly (2 years?!) -- is the generation of realistic objects in computer games and simulations which keep their complexity at infinite scales.
Apart from that, it's just fascinating and is -- I believe -- a work of Art!
4. re: "A Little Help" by soylent / Wed Nov 18, 2009 15:50:22 GMT
"Hey, I'm a layman too. But one obvious real world application -- at least once computers are fast enough to do these kind of equations on the fly (2 years?!) …""
6 Not two years, not 20 years. If you can extend Moore's law (and the algorithm is embarrassingly parallelizable), that's only a factor 1000 performance improvement. That doesn't even get you close!
"is the generation of realistic objects in computer games and simulations which keep their complexity at infinite scales."
The market for games centered on abstract art is pretty negligible.
Twiddle with the parameters all you like. All you're going to get is a series of abstract and weird-looking shapes. You're not going to get an infinitely-detailed and accurately-textured mountain all of a sudden. The kind of fractals that Nature employs are more like 3-D versions of Koch's snowflake.
5. re: "A Little Help" by jamie / Wed Nov 18, 2009 17:19:18 GMT It's fundamentally possible today. Double-precision support via dx11 and the use of compute shaders will put this kind of thing in "real-time" dependant on resolution today! That is if MS can get their dx11 support squared away anytime soon.
6. Urgh by Brokk / Wed Nov 18, 2009 15:02:42 GMT I have a background in maths and find fractals useful and interesting. But there are some images like this one which produce a visceral reaction in me. Others (on this site and elsewhere) are saying the Mandelbulb is "beautiful" and "awesome". I find it disturbing. I know this says more about me than about the maths. But I'm curious to know am I the only one?
7. re: "Urgh" by Anonymous / Wed Nov 18, 2009 17:29:11 GMT At first I thought it looked cool. But upon closer inspection, I find it mildly creepy. Not really disturbing but something just "feels off" about it. So nope, you're not the only one! ;)
8. Complex 3-D by Fritz / Wed Nov 18, 2009 15:20:38 GMT Maybe White could expand the complex plane into quaternions and use the imaginary part for the 3 dimensions? Beautiful!
9. No Mysticism, Please by Samuel Monnier / Wed Nov 18, 2009 15:36:58 GMT http://www.algorithmic-worlds.net
Though I enjoy a lot the results obtained by these guys, "finding the 3-D Mandelbrot set" is completely meaningless to me. The Mandelbrot set is obtained by looking at the asymptotic behavior of the iteration z → z2 + c where z and c are complex number.
7 Starting with z = 0 and an arbitrary value of c, iterate this equation. If the norm of z diverges to infinity, you're outside the Mandelbrot set. If the norm of z converges to zero, you are inside. The norm of z might stay bounded and yet not converge. In which case you're on the boundary.
Now, how can we generalize this by seeing z and c as more general objects than complex number (in particular, we would like z and c to be elements of a 3-D vector space to get a "3-D Mandelbrot set")? We need a vector space for which we are able to multiply 2 vectors and obtain another vector. That's what is called an algebra. Moreover, we need a norm for the bailout condition so we need a normed algebra (also called a Banach algebra). http://en.wikipedia.org/wiki/Banach_algebra
That's it. I doubt there are so many of 3-D Banach algebras. A trivial one appears in the 3rd example of the Wikipedia article with n=3. But I doubt that it will give you anything interesting. Except from this, I'd be tempted to conjecture that there are no other 3-D Banach algebras. But I didn't try to check it.
And yes, the "whipped cream" quaternions are 3-D slices of a generalization of the Mandelbrot set to 4 dimensions even if it doesn't satisfies the aesthetic taste of Mr. Nylander.
Let me be clear. I really enjoy that people are finding algorithms to produce new amazing 3-D fractals. I just find it pretty stupid to turn this into some mystical and meaningless quest about what is the true 3-D Mandelbrot set. These are just words. They have no more meaning that the one we attribute to them.
10. re. " No Mysticism, Please" by Brokk / Wed Nov 18, 2009 16:10:40 GMT As I understand it, it's a "3-D Mandelbrot" in the sense that it does in 3-D what the complex number operations do in 2-D. The operation z2+c does a magnification and rotation (the squaring of z) followed by a translation (c). From what I've read elsewhere, the Mandelbulb is made by combining rotations and translations in 3-D. But I don't know exactly what these are.
So if I'm right about that, the Mandelbulb is geometrically analogous to the Mandelbrot set. But it doesn't use complex arithmetic.
11. Been There, Done That. by Allen / Wed Nov 18, 2009 16:17:58 GMT When I saw the "Mandelbulb" images, I was reminded of similar landscapes from the late 60s, early 70s (if you catch my drift). I haven't seen anything reproduce that as well until these images. Ring a bell with anyone else out there?
12. The Mandelbrot Set Is 3-D Already by David / Wed Nov 18, 2009 16:38:44 GMT The third dimension is usually represented as a color gradient expressing how quickly the equation diverges.
8 13. So What … by alphachapmtl / Wed Nov 18, 2009 18:12:52 GMT They seem to be doing some random meaningless tweaks to some improvised formula until they get something 3-D that looks cute. But is there any meaning to it?
This was not the way to the Mandelbrot set. The 2-D Mandelbrot set was not designed. It was the inescapable consequence of iterating the very simplest non trivial second-degree equation over the complex plane.
14. Wrong Definition by Mark / Wed Nov 18, 2009 18:39:51 GMT "Some will balloon to infinity escaping the set entirely while others shrink down to zero. The different colors on a typical image reflect the number of iterations before each point hits zero."
When calculating the points in the set, a "size" is calculated after each iteration. If the "size" grows to 2, it will continue to infinity. These points are outside the Mandelbrot set. The number of iterations it takes to get to size 2 are used to determine the color of that pixel. For a fully zoomed-out image, the iterations are usually less that 200 per point. The Mandelbrot set is actually the uninteresting dark bits at the middle of the image. To my knowledge, none of the points ever reaches zero.
The Mandelbrot set is the set of all Julia sets. For the Mandlebulb to be a true fractal, perhaps it should be the set of all Juliabulbs.
Fractals are objects that have fractional dimensions. Mandelbrot later regretted using the term "self-similarity" to describe them. It's not the defining factor.
15. Fracrove Did It Years Ago by Roy Taylor / Wed Nov 18, 2009 19:52:08 GMT In the early 90s, I wrote a DOS program called "FracRove" that showed the Mandelbrot Set was a 4-D object and showed how to slice through various 3-D views of it. That program continues in its Windows incarnation on my website (http://www.roytay.com/software.htm ).
The technique of multiplying, rotating, etc. doesn't produce a true 3-D image but is simply creating something out of the 2-D conventional view. I don't deny their method produces spectacular new fractal regions to explore. But I don't believe it to be the true 3-D structure.
16. re: "Fracrove Did It Years Ago" by Tom / Thu Nov 19, 2009 00:15:14 GMT Since you didn't explain in your program the other 2 axes are of the Julia set at each 2-D Mandelbrot coordinate. This gives a "whipped cream" effect. Julia sets are not Mandelbrots so it is not a higher dimension Mandelbrot-like fractal.
Mandelbrots are significantly different from other fractals and have much more variety than Julia sets.
Even if this Mandelbulb isn't the unique simplest 3-D Mandelbrot, it still will probably contain most of the shapes and patterns. Just like an order '4' Mandelbrot contains the same sort of patterns and swirls as the regular Mandelbrot. 9
17. re: "Fracrove Did It Years Ago" by fransiss / Thu Nov 19, 2009 15:22:23 GMT Have you seen the stuff on Daniel's website? Does anybody know what app can produce this thing and what formulae to paste in? I tried POVRAY yesterday but I can't even get an image from that thing.
I find their stuff stunning but have not much clue how they manage to get the 3-D feel and lightning and shadowing as they do.
18. Mandelbrot=2? by Rodney / Wed Nov 18, 2009 22:34:31 GMT Nice to see the attempt. But the Mandelbrot isn't 2-D. It is a solution on the complex plane (that is, singular complex numbers). You can't do 3-D on its own. That's why quaternions were discovered. You need to use two pairs of complex numbers, giving 4 values when calculating or a real and 3 imaginary of the quaternion. As soon as you do this, you then have to ask what is the octonion Mandelbrot? And is the square the value 2 because that's the solution you're looking for, the attractor.
What if we look for W=W^(Z(1))+(Z(2)). Would it give the solution for the point Z(1), complex, quaternion, octonion etc. given Z(1)=2.0 real for the Mandelbrot set and Z(2) is the scan over the complete set of points of that function, for that solution?
Or would you end up with something only explorable in the LHC?
19. Error by james / Thu Nov 19, 2009 13:13:57 GMT The colors of the Mandelbrot image represent the number of iterations before the value escapes a certain limit -- not when it reaches zero.
Only points in the centre of the set ever reach zero. Points close to the boundary (i.e., the fractal's edge) never reach either, often behaving chaotically.
20. Convert Any 3-D Object Into A Fractal by nick / Thu Nov 19, 2009 13:23:33 GMT After seeing this figure, i think any 3-D object can be turned into some fractal object. For example, take a human skull and let it be made out of repeating skulls which are made from skulls to (or parts of skulls). Just like they can make Mona Lisa from tiny totally different pictures in the right color and fit them in.
Besides, I wonder if this person has ever seen snow crystals of ideal quantum shapes from Nature. :))
10 21. Could One Construct 3-D Mandelbrots By Using Analytic Functions Depending On One Complex And One Real Parameter? by Matti Pitkänen / Thu Nov 19, 2009 15:53:47 GMT http://tgd.wippiespace.com/public_html/index.html
The 2-D character of Mandelbrot set is due to the fact that the function f: z→z2+c contains only the single complex parameter c. A Mandelbrot set is the set of values of c for which the iteration converges to a finite value.
Has anyone tried to construct 3-D Mandelbrot sets by using complex functions depending on two parameters? For instance, one complex parameter and one real parameter. For instance, z→z2+bz3+c would give 3-D Mandelbrot set. If c is complex, one would obtain 1-parameter family of 3-D Mandelbrot sets: kind of dynamical fractal.
I looked at the 3-D analogs of Mandelbrot fractals obtained by introducing an additional real parameter b besides complex c (say f(z)= bz3+z2+c) so that parameter space becomes 3-D. I wrote a summary about basic features of these fractals (see below).
22. Bulb by dirk alan / Fri Nov 20, 2009 00:21:22 GMT Reminds me of coral reef. Question: is it solid or hollow?
23. re: "Bulb" by Tom / Fri Nov 20, 2009 05:49:13 GMT Mostly solid (much like the 2-D Mandelbrot set). But there are infinitely many caves and crevices. And some areas are more like dust or sponge or fluff than solid. In fact, quite a lot like a coral reef! It almost definitely doesn't contain any air-tight holes in it.
I'm basing these on extrapolating from the 2-D Mandelbrot and from what I've seen. (So I might be wrong.)
http://matpitka.blogspot.com/2009/11/how-to-define-3-d-mandelbrot-sets.html
How to define 3-D analogs of Mandelbrot fractals by Matti Pitkanen / November 20, 2009 Postal address: Puutarhurinkatu 9, 10960 Hanko, Finland e-mail: [email protected] URL-address: http://tgd.wippiespace.com/public_html/index.html
In New Scientist, there was an article about 3-D counterparts of Mandelbrot fractals. It is not at all obvious how to define them. Quite impressive analogs of Mandelbrot set have been found using so called hypercomplex numbers (which can have any dimension but do not define number field but only
11 ring) and replacing the canonical map z→ z2 +c with a more general map (see this). c must be restricted to a 3-D hyperplane to obtain a 3-D Mandelbrot set.
It occurred to me that there exists an amazingly simple manner to generate analogs of the Mandelbrot sets in 3 dimensions. One still considers maps of the complex plane to itself but assumes that the analytic function depends on one complex parameter c and one real parameter b so that the parameter space spanned by pairs (c,b) is 3-dimensional. Consider two examples …
f: z→bz3 + z2+ c (b real) for which b=0 cross-section gives Mandelbrot fractal. The first 2 iteration steps are z=0→c→ bc3 + c2+ c →...
f: z→z3 + bz2+ c (b real) suggested by Thom's catastrophe theory. The first 2 iteration steps are z=0→c→ c3 +bc2+ c →...
Both options might produce something interesting. One could also construct dynamical 3-D Mandelbrots by allowing b to be complex and interpreting the real or imaginary part of b of the modulus of b as time coordinate.
One can deduce some general features of these fractals.
1. These examples represent 3-D generalization of Multibrot fractals defined by z→ zd (this video demonstrates how Multibrot fractals vary as the integer d varies). Obviously these sets look qualitatively very much like Mandelbrot fractal. Hence each b=constant cross section of these 3-D fractals is expected must have the qualitative look of Mandelbrot fractal.
2. The boundary of Mandelbrot fractal is now a 2-D surface of 3-D space spanned by points (b,c) and corresponds to the points (b,c) for which the iteration of f applied to z=0 does not lead to infinity. The line c=0 belongs in both cases to the fractal since z=0 is fixed point of iteration in this case.
As the value of b grows, the cross-section must get increasingly thinner for both negative and positive values of b with large enough magnitude rougly as c < b-1/3 for the first case and c < b-1/2 by looking what happens in first iterations. The direction of b is clearly in a special position reflecting the 2-D character of the basic iteration process. This holds true for any analytic functions.
3. To guess what the boundaries of these fractals could look like, one can try to imagine what one obtains as one builds a small pile of slightly deformed 2-D Multibrot fractals above the b=0 Multibrot fractal or any piece of it (see the images here). One expects complex caves inside a caves structure. The caves are expected to be high.
Also a fractal hierarchy of stalagmite-like structures is expected. Their tips would reflect a disappearence or appearance of a new structure in 2-D Multibrot fractal as the height coordinate b varies. Being inside b>0 part of the fractal might bring to mind the same spiritual feelings that one experiences in a Gothic Cathedral.
4. It is good to hear what the pessistic has to say. In high enough b-resolution, the variation of the Multibrot curve as a function of b could be so slow that the fractal looks like a cylinder having Multibrot as intersection and there would be no fractality in the b-direction. Therefore, the
12 question is whether the fractality in the c-plane implies fractality in the b-direction. In other words, is the fractal curve for a fixed value of b critical against variations of b?
Reality could, of course, look different. 3-D graphics skills would be needed to disentangle this huge complexity. Probably this is far from a trivial challenge. In any case, the visualization as a pile of 2-D Multibrots could be used in the construction of these fractals and would make possible discretization in b-direction and the use of existing 2-D algoriths as such. Maybe some Mandelbrot artist might try to investigate what these fractals look like.
Paul Nylander kindly reproduced a picture of an az3+bz2+c type 3-D fractal. As you see, it is from outside and the local cylinder likeness is obvious from the picture which suggests that there is no genuine fractality in the b-direction. This hides the complexity of the boundary of Mandelbrot which can be seen only by going inside.
He also told me that Rudy Rucker has suggested a similar approach.
Also Janne (see in the 'Comments' below) worked with 2-D sections (see the discussion section and his graph) and thinks that fractality in the b-dimension is not true. He sent also a nice animation about the quaternionic Julia set which also has a local cylinder-like structure.
This animation demonstrates the 3-D complexity below the surface hidden by the views from outside by using 2-D cross-sections. I would guess that this animation catches much about the (b,c)-fractals. The graphics allowing to see the fractal from the point of view of observer at floor of infinitely high fractal Gothic Cathedral might be a fascinating challenge;-).
Readers' Comments
1. At 11:10 PM, janne said... I tried this out quickly on Matlab. While the approach obviously works (bz3+z2+c mapping), it's clear that you need a software with better graphics capabilities. The best looking results were with merging different b slices into one matrix and forming a isosurface out of that one. But even then it looks very crude.
I guess it takes some sort of 3-D raycasting method.
2. At 12:24 AM, Matti Pitkanen said... The basic question is whether there is fractality in the b-direction or not. If not, you get just the Mandelbrot cylinder in high enough resolution for b. This means that you do not have a genuine 3-D fractal.
The best manner to kill the idea is to look at how the 2-D Multribrot varies in interesting manner in various scales as b is varied.
13 3. At 8:32 AM, janne said... Yes, you are right. I only tried a very rough overall discretization of the whole Mandelbrot set. The resulting isosurface looked more like a 3-D Lyapunov fractal. But that's just because there wasn't any real detail to it.
I'll try out some more detailed b direction slicing tonight.
4. At 11:26 AM, janne said... A couple more detailed runs reveal that it's more like a projection similar to Julia sets. There's no fractality involved in the fine structure of the different b-slices.
The most interesting results come from b close to 0 with large |b| the radius of convergence just shrinks down to nothing.
I don't think real value as the third dimension cuts it for 'true' 3-D fractality since the iteration will still always converge to the '2-D' complex plane.
This makes me think about those quaternion approaches. Isn't this basically the same idea with a few more dimensions added to b?
5. At 6:45 PM, janne said... http://www.youtube.com/watch?v=rSTAjSymJAY
6. At 8:51 PM, Matti Pitkanen said... Each slice should look very Mandelbrotian since iteration functions of type zd+c and more generally polyonomial of z must give rather similar structure. Wikipedia contains animation about this.
My guess is that also things changes slowly in b-direction and you get something like the Mandelbrot cylinder.
3-D quaternion approach and hypercomplex approaches are different since z is replaced with quaternion. If one considers only Mandelbrot fractals, one can use hypercomplex numbers of any dimension and just take the space of c-parameters to be 3-D.
7. At 9:34 PM, Matti Pitkanen said... Thank you for a nice picture. You have f(z)= zb+c with b varying. I had in mind f(z)= bz3+z3+c or f(z)= z3+bz2+c. Your choice is more general than the choices based on polynomials since zb is discontinuous along x-axis (exp(i2πb) is not equal to 1 for non-integer b).
Regarding your question "This makes me think about those quaternion approaches. Isn't this basically the same idea with a few more dimensions added to b?".
In standard approaches, the parameter c corresponds to a shift in complex plane, quaternion plane, or hypercomplex plane (Wikipedia contains a page about hypercomplex numbers). In my own case, b does not correspond to a shift. Maybe acting as shift is what is essential for fractality.
14 One manner to kill the idea that a genuinely 3-D fractal is in question is to draw some vertical slices of the fractal with various zooms for b. If these are featureless, one can forget it.
Let us for definiteness restrict the consideration to f(z) = bz3+z2+c.
*Allow b vary freely.
*Constrain the values of c on ray (c= x*exp(i*phi_0), x varies) or a circle around origin (|c|=constant that is c = rexp(i*phi), r=constant). The c-plane would be replaced with (b,x) plane or (b,phi) cylinder in the fractal plot. The cylinder can be replaced with plane for plotting purposes.
*This 2-D slice as would give the intersection of the 3-D fractal with the plane defined by ray or with cylinder |c|=constant.
8. At 11:46 AM, janne said... I see what you're trying to say here. I'll give it a go when I have some spare time and energy.
Consider the hypercomplex case (http://www.youtube.com/watch?v=lwB7KzG9awk): Even there, the surface of the set rotated thru the dimensions isn't fractal thru these dimensions. The Multibrot 3-D approach isosurface (z→bz3+z2+c mapping to be precise) I got looked very similar to these without the extra rotations.
if on the Internet, Press
15