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This, for now, is a personal survey.  I'll be making some posts of a few peculiar items I've found over my journeys on the parameter plane in the future here, where theory is properly discussed.  But feel free to beat me to the punch...

Later...

>>>>>>>>>>>>>>>>>>>>A LATE NOTE FOR NEWCOMERS TO THIS THREAD, 2/3/2012:

I encourage you to examine the relatively SIMPLE mathematics of my own offerings here, which chronicle the discovery and early evolution of a new and significant fractal type, now called Multipowerbrot.

There is, I sense, a battle for the heart and soul of fractal geometry, with those who favor graphic sophistication and polish on one side, and simplicity of math and presentation on the other.  I am on the latter side.  For the mathematically oriented, studying my entries in detail, in a forensic sort of way, should be a fairly rewarding experience for those interested in the workings of the Mandelbrot set and its closest relatives.

Enjoy!

I remember when Cliff Pickover asked essentially the same thing back in the late '80s/early '90s.  To me, yes, there's not much big novel stuff going on or left to do, but, being that fractals are math, there are always new math ideas to incorporate.  So, I hope that there will always be novel things going on with even the "lowly" 2D M-set.

Lowly?  It still impresses ME.  Last year, I went on this journey that BEGAN w. f(z)->zc+1/zc.  Did index sets for f(z)->(za+b) + (1/(za+b)) where b is a nonzero constant.  On the complex *a* param. plane, interesting M-like things emerge outside the unit circle.  Stuck for pix right now, on a public computer w. disabled drives & time limits.  *b* plane for fixed *a* is kinda neat, too.  Will post pix under better circumstances.

There's definitely a lot more to be done with M-type 2D fractals. Even the good old Mandelbrot has more to be coaxed out of it! For one thing, there's the matter of optimization--I for one desperately want a faster algorithm for computing the Mandelbrot. From a more picturesque point of view, there's always a new way of coloring the Mandelbrot that reveals new details.

One big question I've been pondering lately is strongly applicable to M-type 2D fractals, though also to 3D ones--how does the shape arise from the formula? How well can we "steer" the shape?

That was a very long-winded way to say: YES, there is definitely something novel to do.

reesej2- On "steering" the shape, oh yeah, there's lots of things I've called Mandelbrot gymnastics, involving alternative funcs., but I could never really predict what I'd get. Later, I'd see the results and say: "yeah, that makes sense..."

On 3d, I don't know if you're familiar with my* (let's call it) "notquat" commutative 4d space.  (under 3d on this site).  I'm starting to get a clearer idea of the shape there myself, despite some horrendous errors on the way...

>I need to add about my last post that the params a and b in that formula are both COMPLEX ones.<

Maybe they will look familiar.  I thought I've seen such things out on the web in the past.  I'll show examples when I can.

Happy Computing!


*--As if it doesn't belong to everyone.  I just FOUND it...

The shape of the main cardioid arises from describing the region containing orbits that settle down to a final fixed point:  Require z² + c = z to get the interior of the region and also require |2z| = 1 to get the boundary.  The second condition is the neutral stability criterion: |df/dz| < 1 for stable orbits (interior of the cardioid) > 1 for unstable orbits (Misiurewicz points on dendrites) and = 1 for the boundary of the cardioid.

How the other shapes arise is left as an exercise for the reader.   :)

@Ikmitch: Yes, I'd found that information, but it's not as helpful as it seems. That allows us to, given a formula, compute the general shape it yields. What it most certainly doesn't allow us to do is backsolve--given a certain general shape, how can we construct a formula that will produce it? I've been working on finding a way to control the relative positions and shapes of the period-1 and period-2 bulbs independently, but that's much easier said than done. The results I'm getting involve solving nonlinear systems with dozens of variables, under the modest assumption that the formula is quadratic in both coordinates.

If you look at the M-set with a simple point of view, yes, there's nothing new...

But! As I have been steadfastly studying the M-set, I am finding value in prediction and analysis; patterns that exist within patterns of zooming.  I find predictions in what is to come, based on what I zoom in on.  I find new ways to create patterns using different zooming patterns.  I have been logging all my work privately since not many care about the process of personal research of a probably over-researched fractal algorithm; yet nonetheless, I go on and spurt out a little thought once and a while about a fascinating bug called the M-set.

I have not been here long enough to know if this has been discussed already, but there is probably a real 2d mandelbrot analog to the mandelbulb of the form where p and q are real or integer multipliers of the rotation angle.

Here are some examples: http://kernsholler.net/KHJS/RealMandel/RealMandelbrot.html

Tater

And it would have a three and four dimensional version by extending the parameters p and q as new dimensions.

Tater

Reesej2-  Years ago, Barnsley claimed to have a program that could backsolve a formula, but I had my doubts about it being able to come up with more than an approximation of the original one.  It was related to image compression in general.

Have to take other responses back for a read.  Later!

(via bunneemail) -pix were posted by my "agent" in advance on a machine that allows uploads, unlike the one I use now...

1st is of f(z)->(za+b)+(1/(za+b)), where b=0 and the view is of a param. plane.  Critical pts are z=2, z=-2.

2nd has -2 @ its ctr. for b=-7. b<-7 results in separation into 2 objects.

3rd is of b param. plane, where a=1, ctr. is b=-1, & starting z=2.

4th: a=-1, & ctr. is b=2  This pic borders on multiple connectedness.
Since division often involves wild gyrations, I got in the habit of setting hi escape values, which FractInt often seems to reject arbitrarily.  Here, is was 127.9.  Still there is "glare" of quickly escaping points about what would normally be terminating Misiurewicz (sp?) pts.

Many of you HAD to have seen this stuff before.

Thought of posting a teaser suggesting where this story leads, but I don't want to give away the ending.
In this tale, it's important that the journey is retraced.

I agree with you Thunderwave. We have barely scratched the surface of the Mandelbrot set. We will only ever be able to scratch about on it's surface, no matter how powerful our computers are and the ever expanding depths they allow us to zoom into. Infinity is infinite!

If you take this view, that we have barely scratched the surface, then you can imagine that beyond the depths we don't have patience to compute, there could very well be new patterns of zooming to explore. Take for instance zooming into embedded Julia Sets. There are certain steps you must take to be able to perform certain zooms.

At this moment in time, I've reached another plateau in my knowledge of methods to zoom into the Mandelbrot Set.

At the same time, still regarding ways of zooming, it's not that I'm finding more ways of zooming into the Mandelbrot Set, it's more like I'm integrating the many into the few.

(via bunneemail)

With the previous pix, you get a squaring effect without squaring due to the difference between highest (1)and lowest (-1) exponents.

After a zoom into the a param. plane while b=-2, you arrive at the 1st pic below.
The next is of the julia set for the ctr. of the last.  It's ctrd. on the critical point z=2, and this point is not visually obvious in the assymmetrical julia set.  The critical point is magnified greatly in the last 2 pix.  V. noisy, this spot...

->Please excuse the amount of time it may take to get to the next page of this thread, as it runs rather long, with many pictures.
(Really good stuff though, I think...)   (1-11-2011)

(via bunnee mail)

The 1st pic below squares the original function seen in the very first pic in this thread.  You may have seen things like it in McMullen's work.

The 2nd is *new*, I think, and significant.  I couldn't believe my eyes when I first saw it.  Nowhere in the literature have I found a suggestion that multiple power shapes could coexist on the same plane.  There was only various multibrot generalizations by themselves.  Take the function used in the first pic, square it, as in pic 1, and make b=1 real.  So simple, and the essence of fractals is (simplest formula)+(iteration)=(complicated results).  Kind of a philosophy of mine, so that my part is no more complicated than it needs to be.  This example is multiply connected.  Others are not, and instead simply connected, also as single objects.

Over a year ago, I first began developing these expressions and doing some decent documentation, along with many hi-res pix, that unfortunately won't fit under the rules about posting on this site.

For now, I'll just let it soak in a little, and show more later.

Happy Computing!

I still want to see the Newton formula with power varying by one of the axis, as well as Supermandelbrot and Superjulia versions of the Nova and Phoenix formulas...

(via bunnee mail)

I'm reposting b1d1M.gif in 320x200 below just in case some of you have trouble w. the 640x480 (true aspect ratio) format in my last post.  If you do get such problems, please let me know, ok?

The last pic in my last post began a zoom series that contines with the last 3 below, keeping the target coordinates in the ctr. of each.  Minis in the main object are dominated in size and quantity by 4th power shapes (M4). These are usually in the ctrs. of embedded julia shapes that also suggest 4th power (J4).  An infinite amount of the more familar 2nd power minis (M2), generally smaller, are found similarly at the centers of J2 embedded shapes.  All are found in a netlike structure, whose character, as a background for them, reflects its locality in the set.

If anyone knows this object to be previously discovered, or any that do some similar things, please let me know.  Structurally, this set is of a class more complicated than standard M.  I, for one, would like to see it more widely known.

With some parens., I've clarified the formula in previous posts.  FractInt is tempermental enough doing division as it is...

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The first 3 pix below more or less finish the zoom series.  The last picture is an almost whole version of the Julia set for the center of the zoom, with z=0 at the ctr. of the pic.  Julias under this function are symmetrical but hollow and off-center on the plane.  There is another critical point, z=4, but I'm getting ahead of myself.  More pix to come...

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A closer look at z=0 is shown in pic 1 below, mag. 4x.  Notice the shadow embedment in it, due to its proximity to a larger mini (see b1m3.gif) shown early in the previous zoom series.  Next is mag 1000x.  The 3rd shows z=4 at its ctr., mag. 10000000x.  Notice that its central feature is a bit off-center.  Also consider that z=0 is always a preimage of z=4.  There are also many J4 features within it, as in the last pic.

An alternative function, f(z)->((z^2)c+1)+(1/((z^2)c+1)) yields an IDENTICAL index set to this one, but the Julia sets for IT are arranged differently.  More on that next time.

(Your input on this is very welcome.  Feel free at all times!)

Later... 

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For the alternative function mentioned in my last post, the whole Julia set looks like the 1st pic below.  A closer look at the J4 feature @ ctr. is shown in the next one.  The last pic shows z=2 @ ctr., greatly magnified.  In both functions, nonzero critical points do not center the features there.

I see things we can do with complex numbers iterated on the plane as infinite geographies to explore, and this one is quite rich.
Next time, there are a few more details to cover before moving on.  This story is still nearer its beginning than its end, so hold on tight...there are surprises.

Don't miss the formula files available below, from which all the pix in this thread were produced.  If you're an old hand with FractInt, you'll know what to do to look around at this object.  Helpful comments precede the list.  To read, use any primitive text editor.

Gotta take time out to thank my Bunny for so graciously uploading for me.  It never hurts to have someone who believes in you!

Our Best to you for the holidays!  Later.

(via bunnee mail)

The 1st pic below is of a greatly magnified M2 mini, and the 2nd is a detail of the corresponding Julia set for the coordinates at the ctr. of the 1st pic, using the type 1 formula.  Also magnified, it has at its ctr. z=0, which shows the quadratic (J2) shape that can always be found there.  So what happens when the ctr. coordinates are within a quartic (M4) form, as in the 3rd pic (same mag as 1st pic)?  In the 4th pic, you can see what I call a "hybrid" quadratic, for lack of better terminology (same mag as 2nd pic).  Note that such shapes do not exist in Julia sets of standard M, notated for our purposes here as M2.

For easier study, I was determined to find a formula for the same effect (quadratic and quartic M shapes in the same object on the parameter plane) in a simply connected form, but I was totally unsure if I had sufficient skill to do so.  I fooled with it algebraically, and will show the results of that in the next episode of this continuing saga...   

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The oddity of M4 shapes in an M2 general structure (1st pic below) is obtained with the formula f(z)->((z^4)c)+1.  For any M generalization in this form, f(z)->((z^n)c)+1, I like to stick w. integer n.

The 2nd pic was found in my algebraic searching, getting close, and worth showing: f(z)->(((zc+1)^2)-1)^2.  Notable for Julia-like features in an index set.

The 3rd pic has the formula f(z)->((((zc+1)^2)-1)^2)-1, and was exactly what I was searching for!  M4 & M2 in a single simply connected object!  I was enormously satisfied to have found it, in March of 2009.  Was I first to do so?  (Please advise!)  Notice the repetitious nature of the formula, which aids in further inclusive generalization.  More about that next time...

DO NOT MISS part 5 of the documentation I was making of it some months later, also below.  It's an easy read, but contains references to hi-res pix that go w. it, but will not fit in this site's format.

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The 1st pic below contains M8, M4, & M2 shapes and is obtained w. the formula f(z)->((((((z^2)c)+1)^2)-1)^2)-1.  The shapes may be discerned better in the left (major) antenna detail in pic 2.  Zooming in on the antenna ends revealed an interesting number x=2.55377397403003...  which satisfies x(x-2)=sqrt(2).  In fact, practically ALL the most well-known mathematical constants can be somehow found at work in this particular set, including pi (the M2 bulbs), e, and the golden mean.

I had long wondered why these functions that inclusively generalized powers of 2 in these sets all shared w. standard M the exact same (famous "San Marco") Julia set for c=-1, as in pic 3 for the 8-4-2 formula.

But then, JUST YESTERDAY, while gathering pix for this entry today, I reexamined the nesting of the formula within itself that produces these inclusive generalizations. I had long thought of the formula structure itself as a fractal.  I then wrote more "iterations" into it, enough to generate a 512-degree index set shown in the last pic.

Consider the calculation of pi to billions and trillions of digits for world records, when so few digits of precision are required to describe a circle as wide as the known universe to the accuracy of a human hair.  Here, it is equally absurd to claim "the most complicated object in mathematics" when the record could be broken daily.

But WHOAAA!!!  I realized, as you can see in the last 2 pics, that as these index sets increase in degree, they more and more resemble San Marco!

So I could confidently conjecture that San Marco is the equivalent of an index set with this formula structure taken to infinite degree.

You never know what you might find...so keep looking!

Happy New Year!!!

(via bunny express)

Shown below are minis from the antennae of the 512-degree inclusive generalization function at various magnifications: M2, M4, M8, &M16.  If you have already explored M-generalizations, you already know that the higher the degree, the larger and more crowded the minis appear.  Here, higher-power minis seem to dominate lower-power minis in both size and number in the space.  Still, there is an infinitude of every power shape, from 2^1 to 2^9 powers, and their arrangement is complicatedly hierarchical.

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The progression of minis I began in the last post can be continued up to M512, but it's pointlessly tedious (for me, anyway) to count all those tiny lobes...

Below instead begins a zoom series into it that keeps the destination coordinates at the center of each picture.  Currently, my machine is grinding away at a hi-res version of the last of them.

Does anyone know of a formula parser that goes beyond floating-point precision?  Too often, I'd like to go a bunch deeper, and hit the wall.  Have you ever run into formulae that you feel deserve a deeper look, but you just can't go there?

I speculate that it would be a programming nightmare to try to provide such a thing.  Does anyone know, with any specifics, why I haven't heard of it yet?

(via bunny express)

The zoom series continues below, augering into a spiral that doesn't end...but not for long, since it uses up a lot of that precision I spoke of in the last post.  I found a destination point that had 9-digit precision, since fixed variables in the FractInt formula parser cannot be specified beyond that, and later I will show the Julia for the center coordinates there as well.

I'm belaboring the study of these (pretty randomly picked) coordinates because as an example, it will help show the dynamic structural details of this entire class of inclusive generaliations.  I hope to demonstrate their importance as a new find. Hope you like things even more intricately complicated than M...

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The zoom series reaches its end with the 3rd picture below, in a lobe of an M16 mini viewable whole in the 1st picture.  The coordinates remain at center, and are non-escaping.

The whole Julia set for those coordinates is seen in the last pic, w. a J32 feature at its center, z=0.  I'm looking for a pattern that can predict the Jn at the center of each respective set of maximum degree, and haven't worked it out yet.  (As of 1-11:) For any of these functions of degree 2^n, Julia set center features are of power shape 2^(n+1)/2.

I use an older machine that might be slow by current standards.  The hi-res version of the 3rd pic here, mentioned while in progress in a previous post, took over 30 hours!  The detail is rewarding, however.  Though I forget the exact procedure, so long ago, I had to retrieve a 4-line file from the XP OS collection and install it to get 32-bit DOS emulation.  I notice that "improvements" imposed by MS only make it harder to do good fractal pix as time goes on...lo res pix like these at least get done faster.

Any ambitious viewers doing any of this stuff on their own now?

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The series below begins a zoom into z=-1 in the Julia set, a critical point for which z=0 is always a preimage.  Once again, we see that nonzero critical points are in no way visually obvious.  Scale-dependent features seen in the index zoom are again encountered here in the Julia set at appropriate depths, as will be seen in the continuation, next.

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At the end of this series, we find that z=-1 is inside a lobe in a J16 feature, not unexpectedly.  The feature is off-center, as are all @ non-zero critical points.

When the degree goes very high, those higher-power minis tend to eclipse the lower-power ones both in number and size.  Next time, ways to track the lower-power ones down "from afar".

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I should point out here that no matter what the degree, functions of this kind have Julia sets w. overall 2-symmetry, and so far, I haven't found critical points for any of them that don't turn out to be preimages of z=-1.

Generally, lower-power minis are more elaborately framed by dendritic structures than the higher-power ones in proportion to their size.  The framing reflects the local symmetry, which usually improves as one zooms in.

The first pic below shows a hybrid J4, and zooming out from it in the 2nd pic shows how surrounding 4-symmetric dendritic arrangements help to locate it, echoing its focus.  The 3rd pic magnifies the left tip of pic 2, showing how the J2 in the last pic was found, by its surrounding 2-symmetric echoes.

For this class, all minis of powers less than that local to the index set coordinates will be hybridized shapes in the Julia set, as are the ones shown here.  Not shown in this bunch is J8, but the same rule applies.

From here, I would rather return to the 8-4-2 combination, which I am somehow fondest of, as was partially explained previously, and for aesthetic reasons.  Next time.  Later.

(via bunny mail)

The centers of the 1st 3 pics below are for coordinates inside a canyon that bores its way toward the origin in the index set.  The left side of the canyon has M8 features, and the right has M4 ones.  I like to call it the "continental divide". Index sets of all degrees in this class have them.

The last pic is of the Julia set for these coordinates, for a bulb on a mini there.  Values for c close to the origin naturally produce large Julia sets.  c=0 would cover the entire plane. (!) 

Nice renders fracmonk! Way to go!

(via bunny mail)

Wonderful to hear from you again, J, & THANX for the encouragement.  Have you read this whole thing?  I've many questions, but for decidedly vain reasons, I wonder if this stuff is truly *new*.  Only lately, I saw a paper in PDF by Wolf Jung, published sometime in 2009, that had a picture that somewhat resembled the M4-M2 combination seen here in post 22 here, but with a more complicated, less obvious formula.  Who got there first?  I know at least SOME of the things I put up here are original, I far as I can tell.  If you know, let me know...spent so much time doing this stuff, I haven't had a decent chance to look around. (hence the screen name...)

The reason why I've concentrated on on minis here (& their edges, at that) should now be obvious.  "Continental" coordinates yield Julia sets that are monolithic, & resist revealing their inner structure without something like a BoF60 treatment.

The 1st pic below shows a detail of the Julia set in the last post, with the same center as the more magnified 2nd pic, that shows the J2 feature, @ 300x.  The 3rd, M4 @ a mere 7.5, from the origin, & the last, M8 @ a mere 2.5x, also visible in the 1st pic, lower left.  M8 in the 8-4-2 combination is found largest @ z=1/sqrt(c).

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Another instance in which B0F60-type interior studies would be useful is in verifying "zero points" of minis.  For lack of better terminology, these are counterparts of c=0 in standard M, only in minis.  The Julia sets below, from the major antenna, *appear* to have the following zero points:  1. M2, c=-2.  2. M4, c=-1-sqrt(2).  3. M8 c=-e+1.  Other estimations distort the circles more, while they already exhibit "tidal" eccentricities.  Circles are pulled both horizontally and vertically within each set.

But these values demonstrate how 8-4-2 seems to be packed with the most well-known constants interacting.  That's why I find it the most interesting of the bunch.  As a single object, it seems to unify them.

At the time I found these objects, I liked to try to nail down such points, a habit which hid the non-self-similar structural aspects (shown in previous posts) within many of the Julias for a while.

The last pic is an artifact, a common error for FractInt when it calculates Julia details.  While the index set is relatively well-behaved, some of its coordinates will defy study.  This one is found in the same antenna, so it is assumed to be non-escaping.  Any width is beyond floating point's precision, but calc rebels at relatively low magnification.

Happy Computing (anyway!).

No fracmonk, not yet.

I can see that your labours though hard have been rewarding, mine also.

I have not done research on your choice of roots of unity, but have a general idea now of how they fit in the general picture. Your choice of roots of unity are a natural extension which i could not evaluate before, but now can make some sense of, in comparison to quaternions.

I think Hamilton found only one of the combination of solutions to the roots of unity for this level of dynamic equilibrium. He may have found others but i have not determined that. Your set of solutions are another variant. I think that there should be a solution that consists of triples with a mod( 8 ) ring structure.

Although this sounds complex it is only the terminology, it is better explained by a geometric model, 8 cubes stacked in a cube form.

Hamilton was a whizz at spherical trigonometry, so he may have visualised the roots, but not tested them all! Your set of roots would lie on two orthogonal great circles, even though you choose one at π/4 in the so called complex plane.

We are very fortunate to be able to animate these magnitudes and their relations , but i note you are old school and hold pictures in your head of what they should look like. That makes you an expert in my book!

Keep up the good work.

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Jehovajah- I would be very glad if this stuff can help you in your researches.  It's exactly the kind of thing I'd hoped would happen.

When I first encountered the figure shown in post 15 here, I was then only trying to get lucky finding a simply connected counterpart for it and did.  (!)  I thought that that might offer more considerations to those trying to prove MLC.  In later posts, I will affirm that there's something unique to unity (no pun intended...) in yielding multiple power shapes.

The pix below continue the last post.  The 1st is a fairly small M8 mini whose ctr. is well approximated in 9 digits only.  The whole 2-symmetric Julia set follows, along w. the J4 ctr. detail in the next, and the a J2 in the last.  Since the local features are roughly circular, one can only tell the types by the arrangement of secondary features surrounding them.

Later.

To clarify the formula structure I've outlined, an alterative function yielding the standard M shape is:

f(z)->(z^2)c+1.  Changing the + sign to - only reverses its left-right orientation, so let p=(z^2)c-1.

f(z)->(((p^2)-1)^2)-1  then yields degree 8, by nesting p thusly.  Let the expression after the arrow = q, and

f(z)->(((q^2)-1)^2)-1  then yields degree 32, (etc).

And the calculation is as efficient as it gets, as well.  Is that cool, or what?

Standard M, for all its intricacy, stays simply connected, in one piece, tenaciously so, at any scale, a property of its function on the complex plane, and nothing about that is short of miraculous.  These sets I've shown do exactly the same thing, only for MULTIPLE "multibrots".  This would have been very unexpected if it weren't for the encounter with the object in post 15.

It is now for those who do complex analysis to explain why this formula structure yields the results it does, especially in light of post 23 here.  I like to believe that such things help rather than hurt!

Next time, substitutions for -1 in the formulae.

...& new pix, a job for...BUNNY EXPRESS!

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The 1st pic below completes the series from post 36, & shows the J8, with its immediate surroundings.

For squared z in this formula structure  f(z)->(((((z^2)c+d)^2)+d)^2)+d, there is apparently ONLY one other value besides d=-1 that yields a one-piece simply connected index set.  d=-2 is shown in pic 2.  A while back, I came up w. a formula that elongated the M-set, nicknaming it "Mandelsword"(not shown).  This object might deserve such a name more than that one.  A look at the 3rd pic shows the leftmost cardioid, the only one with no "tail", which then must be considered the "main" one in the set therefore.  Its dendritic "hairiness" proportionally exceeds that of the main cardioid of standard M, so it is M-like, but not a distorted version of M itself.  In this object, all shapes are uniformly M2, however, not uniformly M8, as the degree might suggest, when uniformity would have been expected.  This also happens in the simply connected sets of higher degree where d=-2, as in the degree 32 nesting, shown in the last pic.  I noticed that as degree increased, the set shape looks more and more like the Julia set for standard M for c=-2.  The "line" often gets too thin to even show, but it seems to be there.

With z squared in the formula, all Julia sets center at the origin.  Non-integer d yields non-connected index sets.  More about that later.

More generally, it seems that an index set must be connected in one piece itself, to reliably indicate the same property in its Julia sets.  That's been my experience.  Yours, too?

Does anyone know about this for sure?

For some reason, only one-piece index sets attract me.  There's a certain perfection to them...

(pix via bunny express)

To detail the left end of the object in the last pic of my last post, the 1st pic below has a 35x mag. & the 2nd is at 100x.  Notice the even fernlike quality of the dendrites when the degree increases.  The 3rd pic is only mag 10x, from further right, and shows a sort of evenness in the spacing between the larger minis in an object the comes off as "all antenna" overall.

For extremists, the last pic is again of the left end, this time in a degree 2048 nesting of the formula, near c=0



Now may be a good time for free advice, and you know what they say about free advice...

The objects in these last 2 posts have never been seen by human eyes before about a week ago, so far as I know.
If anyone asks you what experimental mathematics is all about, tell them that.

That's why I do what I do.  Usually, you don't find new things so frequently, and that's made things pretty hectic for me lately, trying not to make mistakes, etc.  If you do this stuff, it's for the love of it, and, as the saying goes, you "don't quit your day job".

Does that make me "old school"?  If so, I'm honored by the description.

'Nuff said, I think...

Happy computing!

(Via bunny express)

For the formula in post 38, initial z=0 is critical for d=-1 and d=-2, and in those cases, will yield connected index sets.

The 1st pic below is an index map of non-escaping points for init. z=0 for d=-1.5, and is NOT connected.  The second shows a Julia set from it, for c=.25+.029i, d=-1.5, which is PROBABLY connected.

The 3rd pic is the index for d=.25 for init. z=0, also NOT connected.  The last pic shows the Julia set for c=6+6i, in which the origin point also does NOT escape, but the set is NOT connected.

In these cases, z=0 apparently can ONLY indicate the presence of non-escaping points in a Julia set for fractional d, but NOT its connectedness.

My skills give me no innate ability to find critical points, so I'm a little lost there...in degree 8, there are supposedly 14 of them.

Any suggestions?*

Obviously, I've been looking at correspondences between values for d in this formula and those for c in f(z)->z^2+c.
Next time, a look at d=-1.75 (a strange case, so you'll find me there...).


__________________________
* In a worst-case scenario, it could be that an intersection of ALL index sets resulting from ALL 14 critical points for a given fractional d may be needed to determine simple connectedness in their Julia sets.

(pix via bunny express)

In the M function, c=-1.75 is @ the cusp of its largest mini.  W. the function I've been discussing here, staying w. init z=0 the index set for d=-1.75 is notable for these properties:  1. Its relatively large size, given its distance from the origin, compared to the index sets for many lesser d distances from the origin.  2. Its largest M2 shape points left, as opposed to the "polarity" of the index set for d=-2.  3. It perfectly preserves the shape of standard M in every detail.  4. It has an infinite collection of smaller separate satellites that themselves share these properties 3, 4, 5, & 6.  5. Its escape set is one-piece multiply connected.  6. Julia sets for non-escaping z=0 have their own corresponding set of similar properties.

(3,4,5,&6 are a stab at a kind of set theory poetry...)

The 1st pic below shows the whole index set at .03x magnification.  The next 3 sample a zoom series, keeping the destination coordinates at the center of each, that being (just barely) inside a (distant) satellite M2.  Mags. are .1x, 100x, & 1000000x, respectively.

Next time, a look at the Julia set for the coordinates.  

(pix via bunny express)

The 1st 3 pix in this Julia series zoom in at the origin from the whole set in pic 1.  Notice how locally, the shape is specific to the coordinate location shown in the last pic in my last post.  The last pic below shows this again in a randomly chosen satellite away from the origin.

For 3d fans:
Considering d as strictly real, stacking its incremental 2d index sets for complex c with initial z=0 constructs a 3d object whose limits are not finite:  d=0 covers the entire plane; and any large d still has a prisoner set, however tiny.  This set should be connected as a solid only over those 3 dimensions. In other words, for it to be connected would require all 3.  Similarly, complex d would then yield a 4d index set.

Next time, just one more look at another satellite, this time, closer to the "continent".  See ya...

(pix via bunny express)

On the date of the post, yesterday, the library was closed due to a storm, but bunny express got thru.  The pix were right, but the labeling was wrong, my fault.  The pix were corrected.

The 1st is mag. 1x, showing the large size of the continent for d=-1.75 again.  Mag. 10000x in the second pic shows a 'cloud' of satellites that betrays its location in the 1st pic in the way embedded Julias might in other functions.  The 3rd pic, mag. 100000000x, is from the approximate middle of the formation, as pic 4 attests.  Note the clouds at every scale.

(pix via bunny express)

In the Julia set pix below, the 1st shows the whole Julia for the coords in my previous post.  Not surprising is the cloud shape of J satellites surrounding the origin in the 2nd pic, mag. 20x.  In the 3rd pic, mag. 1000000x, the local feature at the origin is an intact Julia one would expect from the last pic of the previous post.  The center of the last pic, mag. 100, of the cloud in the 2nd pic here, contains a local feature similar to the one shown in the 3rd here.

Though I prefer connected sets, as stated before, there is still one more weirdie you should see.  It's kind of a "jackelope".  For those unfamiliar, a rabbit with antlers, a favorite old joke among taxidermists...and maybe a new one for a certain very large, very litigious genetic engineering company that shall remain nameless.  Hint: they steal DNA, splice it into things they probably shouldn't, patent it as their own, and sue, for instance, farmers when their fields get polluted by the pollen from it, when it shows up in THEIR crops, as if they WANTED that somehow.  Nice guys...

...got good genes?  Maybe you'll need to pay them a licensing fee if you decide to have children...

Anyway, jackelope, next time, be good...

(pix via bunny express)

I don't know what came over me in my last post, but sorry for that- it's sort of off-topic.  For years now, we've been expected to keep a sense of humor about such things.  Or else.

What?  That's not funny?

The 1st pic below shows the whole mapping for d=-i, with init. z=0, clearly not connected.  At a glance, the 2nd pic, showing the largest contiguous fragments, is an oddity, but upon closer examination, is not meant to be connected, probably, ever...and note the beginnings of a second tail as well as the period-3-looking antlers...but it is what it is, as they say...a jackelope.

The 3rd pic cycles the M8,4,2 nesting expression back into the function from the very beginning of this thread, so it appears in multiply connected form. There are many, many other approaches to generalization...

The last pic was a test to see how a hi-res 2048x1536 pic would show up in a post here.  In this view of the major antenna, the 'polarity' is reversed from prior renderings, so that all signs before d are consistent.

And TALK 2 me!

Problems, clarifications (if I can help), your own insights, criticisms, accolades (if applicable), (so-called) genius grants, your 2 cents, funny stories, explaining local connectedness in plain English somehow, musings on what Euler might have done had he had a computer, whatever...yes, all these are welcomed.

Later.

I do like these images. I also wonder what 3D version of these would be like...how about even simple Quaternion versions of them?

miner49er-  If you understand the formulae well enough and can translate them into the particular syntax of the formula parser of your favorite generator confidently, knock yourself out!  Not sure if there's any advantage to quaternion renderings, though.

Lately, I've been following a "less is better" philosophy regarding dimensions.  The complex nos. are a closed set, so for a fixed d, there's no reason to depart from the complex plane.  In post 42, I was thinking of c=a+bi,  a by bi by d(real), for init. z=0, and particularly in the interval -2<d<.25.  I thought that would be ambitious enough for now.

You know that d=-1 and d=-2 offer connected results.  Lately I'm searching for a critical relation between c and d that yields connected sets for fractional d.  I'm probably chasing phantoms, at least, that's what I suspect as an outcome.  All the same, I'll try for a while, unless someone can show me that I'm wasting my time.

To continue last post, I don't think that the most important development in this thread should get lost:  that is, the multiple multibrot effect of d=-1 in 2d in its formula structure, within a simply connected one-piece set.

d=-2 turning out simply connected as well was a welcome up-to-the-minute bonus discovery.

The fact that both remain exact period 2 integers in the standard M-set is what informs me most with regard to comments in the last post about the unlikelihood of connected sets in the formula structure for fractional d.

When the going gets tough, the agile go elsewhere...

Later.

(pix via bunny express)

I turned my attention to odd powers, which had been in the back of my mind lately.  Prior to the posting of the pix of another new object below, only 2 sets of eyes have ever seen it before, most likely.  After posting yesterday, I went home and wrote the formula for it.

The 1st pic shows it whole.  It has M3, 9, & 27 multibrots, and is generated by the function: f(z)-> (((((z^3)c+i)^3)+i)^3)+i, with critical points z=0, z=i.  It is simply connected, one-piece, and follows nesting rules like those discussed in previous posts for successive squaring, only that in this case, it's cubed instead.  Theoretically, as degree 27, it should have (2d-2) 52 critical points in all in its locus of connectedness (and good luck w. that!).

Maybe then you will excuse the facile observation I ended my last post with.  Apologies for that.  It's NOT advice, but it was 'moving on' that produced these pix...

The 2nd pic shows the left end larger, where the 'main body' of the set is, dominated by M9 shapes, dwarfed by its largest bulb to right, which is dominated by M27 shapes.  The 3rd pic shows the "continental divide" these sets always have.

The last pic shows a smaller M3 mini than the 2 on the main antennae in pic 1, arrayed with both M9 & M27 minis.  The highest power shape always dominates in overall area and number. (maybe 'frequency' describes the situation better than 'number')

(via bunny express)

These pix below are given to show how the nesting of cubing expressions has similarity to squared ones previously shown.

The first is f(z)-> (z^3)c+i, which yields a standard M-like overall index structure, only with M3 bulbs, simply connected.  Pic ctr.:c=0.

The 2nd is f(z)->(((z^3)c+i)^3)+i, and yields a connected index set with julia-like features, as a sort of intermediate step, with M3 at its heart.  Pic ctr.: c=1.

The 3rd is the degree 243 nesting, where one can begin to see the resemblance (albeit rotated) to the Julia set for z^3+i, shown in the last pic.  i, of course, is an integer distance from the origin, as are d=-1 and d=-2 in the squared formula structuring you've seen here before.  Here too, if such nesting was taken to infinite degree, the 'index set' would be of identical shape to the Julia in the last pic.

So there are plenty of new toys to play with now.

Got any good ones yourself?

Later.

(via bunny express)

As long as I live, I will always be looking for more things like these.  For the sheer fascination they offer me, I've worked in isolation, but working that way has few advantages.  While I don't NEED support or feedback to do what I do, these things are appreciated.

What I think most miss about these objects is their cohesiveness.  These sets are determined to be in one piece, as if they possessed a will to do so, with each location clinging to an unbreakable whole.  And as in the original M-set, their detail is never exhausted at any scale.

At first, there was M, and its 'multibrot' generalizations, and most of us were left to believe or assume that those were the only M-like objects possible that were simply connected.

The class of objects containing multiple multibrots in this thread show clearly that this is not so.  In my view, they will very significantly complicate questions of connectedness applied to M-like objects, in both topological and number theory terms, and pose many new and worthy questions.

My original aim here was to ignite discussion of these issues, but discussion of theory requires dialogue.  So, any comments? (Those meant to be UNDERSTOOD are preferred...more about THAT later).

In the meantime, below begins a zoom series into the odd-power object first presented in post 49 here, into a valley dominated by M27 bulbs.  Once again, the destination coordinates are centered in each successive magnification.

(via bunny express)

I hope that the descriptions I've given here so far have been understandable!  If not, let me know...

In my experience, I've learned that it's smarter to ask when you don't know, and tragic when you feel you need to look like you know, when you don't know, and don't ask.

I've noticed a lot of people who SOUND like they they know what they're talking about.  Sometimes, they do.  But if no one can understand what they're saying, it's useless.  And then, there are some that just don't really WANT to be understood.  They just want people to think they're smart.  That's off-topic, but useful here (and everywhere).

Always ask questions...

The zoom continues in the pix below, in a dendrite, into the vicinity of an M3 mini.

(pix via bunny express)

There's a lot of info out on the web on M basics, most of it helpful.  Then there's "advanced" theory, coming from what amounts to a priesthood, which may not be terribly effective when it comes to helping beginners jump the huge rift we could call "intermediate".  Practically speaking, "intermediate" really isn't there.  I seldom understand what complex analysts say about M-like sets mostly due to notation with which I have no acquaintance.  I prefer more verbalistic explanations.  There's not a lot of help for us there, so I think we should help each other, having little other choice in that.

In the zoom continuation below, the 1st pic brings the focus close to an M27 mini above one of the canyons of one of the 2 largest bulbs of the M3 seen in at the end of the last post.  This vicinity influences the features found there surrounding the M27 as seen in the 2nd pic.  The last pics hints at the nature of the destination locality, to be seen in the next post.

Happy computing!  Later.

(pix via bunny express)

I think I need a website for this stuff, especially to show hi-res pix, and to make more extensive documentation available.  But then, I'd have to MAINTAIN it, as if I had the time...

But having only recently found all this new stuff, I have to catch up on that documentation as well, while my studies of it have only scratched the surface.

A special msg. to lurkers:  Much as you try, you won't suck it dry.  But keep sucking!

The zoom ends with the 1st pic below, inside an M3 mini.  The 2nd pic is of the whole Julia set for the point in the ctr. of the 1st pic.  The 3 largest contiguous groupings of non-escaping points in this 3-symmetric set are J27.  At the center is always found a J9 for this function.  In the last pic, another zoom begins with z=i at center, mag. 10x, for comparison with the index zoom just finished, to be continued...

Later.

(pix via bunny express)

I think that the subset of complex functions that, when iterated from critical points that allow it, can produce a one-piece connected index set of non-escaping points, should be classified topologically as such and given a name.  M is probably the 'simplest' in this category.

In fact, wouldn't it be helpful if ALL complex functions could be categorized topologically?  Then, one could say: "This function is known to belong to this topological category.", and know what to expect from it.  Maybe then too, patterns would emerge that could indicate WHY they would fall into the categories they do, once defined.

What do you think?

Right here in this thread is as good a place as any to speak your mind.

I believe there is a painful lag in the community's discovery of the rich significance of the material in this thread.  I'll be patient...

Again, what do you think?

As the Julia zoom into z=i begun in the last post continues, the 1st pic visits a dendrite of a J27.  The 2nd shows the neighborhood of the last pic, entering the fringes surrounding a J3.

(pix via bunny express)

The transformation of St.Valentine's Day into what it is today was only a contrivance of a greeting card company, to enhance its profitability, of course.  When you have no thoughts or feelings of your own (a growing trend), they'll sell you some, readymade.  How romantic...zombies in love...but my BEST advice, for now, is to throw away that fancy new cellphone you felt so compelled to get.  It'll make you still more stupid, if that's possible.

I've probably just pissed off nearly everybody, but I've done it from a safe distance, LOL. (Lots Of Loathing?)
And that's the point: "social networks" are only furthering alienation.  Still, send a card to all your "friends".  Maybe you'll get some strange.

Love is something else entirely.  May you find it.

Long since this holiday took on a life of its own, I do wish fractal fans cardioids on cardioids on cardioids on...(M3 does that, whether on its own or in the company of other M types, as in this function).

The Julia zoom continues below into a field of J27's each surrounded by dendritic forms suggesting its locality, as seen in the last pic.  Next time (if there is one), the destination of the series.

The sun still shines, the wind still blows, and I think I should be a bit less fatalistic, don't you?  

To be honest, I didn't read and study the contents of this thread yet, I'm about to do it later when I have some more free time. Anyway I'm a novice and I have to learn a lot from fractals and mathematics to fully understand most of the things publicated in the forum by researchers like you, but I encourage you and all the comunity to continue researching on mandelbrot-like fractals because my intuition says that there are a lot of things to be discovered yet, and maybe really revolutionary stuff. As for the "community lag" you mentioned, I want to say that I see the community of fractalforums more interested on the eye-candy side of fractals than any other stuff. I think fractal art is great, I'm also a little on it, and I have seen in this forum the most beatiful and amazing pictures and animations made out of fractals ever. But my motivation for being here is more transcendental, and I'd love to see more people also interested on researching and looking for answers because I'm truly convinced that there is A LOT of novel left to do.

Ok, I did a quick review of all of your posts, and I'm truly amazed! This is definitely something new and great!
Multiple powers shapes in connected sets, smoothly combined! and what about z=(z*a+b)+(1/(z*a+b))+c, no powers involved??
I'm at work now, I'll take a deeper look back at home, and try it in Ultrafractal.
I'm not sure why you didn't get almost any feedback...
Why don't you use imageshack.com or other site for uploading better resolution images and then link them in your posts?
I'll render some of them in a few hours and I'll post here to see if we can get more attention by using some "eye candy" like I mentioned before  :dink:

Kali-  Thank you, thank you, thank you.  You get it!

I rendered some pics and uploaded to imageshack, for the formula (z*z*a+b)+(1/(z*z*a+b)) drawing the plane a
The first is for b=0 and then the rest for b=1

There are only two showing the multiple powers minibrots, because you did already a lot pics of that. I'm pretty sure the "multipowerbrots" are the more important aspect of your discovery, but I find the sets also very beautiful and with a great complexity of shapes.

I'm VERY interested on what you've done, and I will be researching more on your ideas when I have more time.

Thanks for sharing your findings.

<OFFTOPIC>I liked your thoughts about St. Valentine's Day, I totally agree with that.</OFFTOPIC>

(http://img51.imageshack.us/img51/6976/mpowers1.jpg)
(http://img222.imageshack.us/img222/3439/mpowers2.jpg)
(http://img402.imageshack.us/img402/5229/mpowers3.jpg)
(http://img576.imageshack.us/img576/5480/mpowers4.jpg)
(http://img585.imageshack.us/img585/320/mpowers5.jpg)

(pix via bunny express)

To hopefully wrap up yesterday's theme, I'm obviously not a complete Luddite, as this fractal research requires super-fast calculation.  My concern is that we're overautomated, making us weaker, less able and more dependent, and that's very unappealing to me.  We must carry too much crud as it is, to get through a day in a practical way.  Again, sometimes, less is better.  I'd rather travel lighter, and maybe stave off early dementia...

At the end of the Julia zoom series, the J3 is dwarfed by 3 nearby J27's that surround it.  The critical point z=i is found well within it in the last pic.  Next, same object, different spot.

Kali- Those pix are very familiar to me, and I can't begin to tell you how reassuring it is to have you showing up here when you did.  Did you notice that certain parts will get cut off unless the bailout's really high?



Here's more of a philosophical question I've always wanted to bang around, but I'm not really sure if it belongs here:

Did these things already exist before they were found?

(pix via bunny express)

A new journey into the M27-9-3 object begins in the pix below, in the left cusp, where M9 continental features predominate.  As usual, each view is 10x larger magnification than the previous one, unless otherwise noted.

(pix via bunny express)

In the pix below, the zoom continues deeply into a spiral...


I'm constantly thinking of ways to get more eyes on this stuff.  I thought of becoming a celebrity, since they're the only ones people pay any mind to these days.  But, you don't get the microphone unless you have little to say worth saying, a catch-22.

So...no.

Oh well, back to the drawing board...

(pix via bunny express)

At the hairy edge of the spiral in the pix in the last post, the zoom continues in the 1st pic below.  In the 2nd, the features that surround an M3 mini are seen.  In the last, the destination will be in one of the 3 M27's there that dwarf it.  Notice the 9 arms that emanate from the M3.


For technocrats:

Very generally, I seldom encounter trouble using periodicity checking in index sets, but it often wreaks havoc with Julias, especially at lower magnifications.  Can anyone explain why that's so?


Some like it hot:

Solar flare activity is high lately, but I don't need to know what's going on at the surface of the sun right now.  I'm content to wait the customary 8 minutes or so...aren't you?

So much for instant gratification...does away=ago?

Have a cool weekend!

(pix via bunny express)

In the pix below, the zoom ends inside, but near the edge of the M27 mini.  For this location, the whole Julia set is seen in the last pic.  The 3 largest groups of non-escaping points are J27, and surround the J9 at center, which should be considered hybrid.

Earth is more accurately 8 mins. 19 secs. at light speed from the sun.  No real human has ever departed from that distance from this hot fractal by more than a couple seconds.  By the weak anthropic principle, it's the perfect place for us to evolve as we had.

Too bad it's run by ruthless parasites.

IMPORTANT:  We are to believe that wealth is the only measure of success, and obtainable only by our degree of collaboration with them.  Often for ridiculously little, most collaborate, and their hold gets stronger.  The first rule is not to anger them, and having your own thoughts does that more than anything else.

You must save yourselves from such subversion!  Take a pill and apply to some large, famous computer graphics company.  There must be some texture sub-subroutine you could be working on, as part of the Borg.  When you find out whether their main objective is to cartoonize humanity or to humanize cartoons, let me know.

Too alarmist?  Want more?  Just keep those parking tickets coming.  It takes that little to set me off lately.

For today, at least,

"That's all, folks!"

(pix via bunny express)

The 1st pic below is a detail of the whole Julia set in my last post.  In it, the center is z=i and the origin is at the midpoint of the bottom edge of the pic, and about half the central J9 is seen there, mag. 1.   As coordinates near to the origin in the index set, the Julia sets for them tend larger.  Zooming in on z=i, the scenery in the 2nd pic includes spirals, and J27 and J9 minis large enough to identify.  V. pretty in hi-res, not shown*.  In the last, the zoom into the nonzero critical point reflects things seen in the index zoom previously shown.  In these functions, as you may already have noticed, the nonzero critical point in the Julia set is more informative about the local features in the index set than others.

__________________________________
* I am forced to GUESS, by number of views indicated on pix posted, that some of you may be having problems viewing higher-res. pix for reasons I can only speculate on.  I have therefore avoided posting such.  Again, if you had difficulties with any of the few I had, let me know.  Some take a lot of space & presumably slow things down, an effect unwanted by the moderators, I imagine.

As far as they are part of numbers nature, and numbers are reflection of real world laws, I think yes.

(pix via bunny express)

As the Julia zoom continues in the pix below, so does the correspondence with the earlier index zoom.

makc-    I agree wholeheartedly.  My theory is that if numbers govern nature, we should understand better the nature of the numbers themselves.

I've never been very good at application, a condition which has drawn me into pure study, for pure love of it.  Many criticize "solutions in search of a problem", as so much in pure mathematics can be, but I think everything there finds usefulness eventually.

I truly delight in those who CAN easily find applications for new results.  "It takes all kinds.", my grandmother said.

But I myself am restricted to an intuition, at best, that the things you see here in this thread will relate somehow* to how nature does things, as I think Dr. M. had, with regard to the M-set.  On a spiritual level, it is an article of faith for me.  So am I preaching to the choir?  I would like to think that we fractalists are midwives for a new way of looking at the world, a necessary one, given our current state as a species, which I too often view with pessimism, and Rays Of Hope detectable only with state-of-the-art sensors**.

(I cringe, awaiting the arrival of high-velocity stones!)

__________________________
* The nastier question is exactly how.
** Available from ROH Technologies, $29.99+tax, each.  HA!  Made you laugh!

No stones?

To my great sadness, Bunny Express is out of commision temporarily, and pix planned for today will be postponed until Monday at the earliest.

Those will wrap up that series, and a new perspective involving an earlier object, previously neglected, will be explored later.

Have another cool weekend...

Is it possible to have a simply connected set that has both z^2 and z^3 msets?

(pix via bunny express)

kjknohw-  I haven't managed it yet, and I'm thinking that maybe odd+even power shapes may remain incompatible.  But I've been wrong before...you know, I've tried to make that happen quite a lot- no success.  It may be subset condition involving multiples. 


The 1st pic below shows clearly the surroundings of a J3, seen more clearly to right in the 2nd pic.  It must be presumed to be a hybrid, since the coordinates for it are found inside an M27 in the index zoom.  Here, the nonzero critical point at the centers of these pix is within one of 3 J27's that surround the J3, as seen in the last pic here.  In Julia sets of this class, all prisoner point groupings of lesser degree shapes than that of the coordinates locality are hybrid.

Something a little different next time...later.

(pix via bunny express)

The pix below show an object first seen in post 22 in this thread.  It was a trial-and-error precursor to M4-2, also first seen there.  The formula for this object is the same, except for a change of sign (for consistency, reversing its orientation), and in leaving off the last (-1), so that f(z)->(((zc-1)^2)-1)^2.  Since it wasn't what I sought when I first saw it, I never gave it much attention until very lately.  The more I consider it, the less able I am to explain it, but I'll try...

The largest contiguous area in it is Julia-like, and there is a hierarchical distribution of M2-like miniatures bristling its edges.  The largest of them is obvious in pic 1.  The M2 minis in turn bristle with more Julia-like blobs, which in turn bristle with...
Left (c=1-sqrt(2)) and right (c=1+sqrt(2)) real axis limits are greatly magnified in the other pix.

More next time...

(pix via bunny express)

The 1st pic below shows the antenna of the object in the last post, centered at c=2.  The 2nd pic is of the Julia set for c=2, shown centered at z=1/c, as all Julia sets for this function are.

The point c=.25 in stardard M, called the "cusp", I prefer to think of as "navel" instead.  In M, it's the only one without an "umbilical cord", as I like to think of dendritic connections to all the miniatures.

The equivalent point for the largest M2 in the set currently discussed here is c=>.84375... (approx.) and not shown.  If you want to kill your machine for a fairly uninteresting pic, hunting for the exact location is a good choice.  The last pic shows the Julia set for the value I gave.  The shape makes sense, right?

More later...



NEWS:

Behavioral psychologists now believe they can control the entire universe, upon learning that it revolves around the actor Charlie Sheen.

Who knew?

Can't you just FEEL that manipulation?

-or-

HEADLINE:

NORTH AMERICA DESTROYED BY METEOR

Charlie Sheen OK, found clinging to large white rock in Bahamas

(I got a million of 'em...)

(pix via bunny express)

Does no one get my jokes?

The 1st pic below is of the Julia set for the real axis left limit c=1-sqrt(2).  The 2nd is for the approximate "zero point" of the largest M2, c=.91, and the last is for the center of its main bulb, which appears to be c=sqrt(2).  Not shown is c=1, which is identical in shape to the San Marco Julia for c=-1 in standard M.  Here, the location c=1 is found within the cardioid body of the largest M2 in this index set, oddly.  As to why, your guess is probably better than any I might come up with.  I just find 'em and show 'em...

  Back in the day I used to use XAOS.  I can't replicate it in ChaosPro, although I can still do it in XAOS... can't zoom in enough in CP (bugs out when you try to zoom in enough).

  Anyways.. you can find these awesome... "arcs/electric field lines" if you SUPER deep zoom in certain formulas.

 I'd generally use formulas like z^2-z^3-p*c-c+sqrt(p*z) and set the plane to 1/(mu+.25) to do this (p is previous iterations z value, c is the pixel value, and z is.. well it's z).  I can get close to the mu plane.. .ahhhh, duhh..... just figured out what I'm doing wrong (hopefully)..

  Following are some images.  The thing is, I haven't seen people do stuff like this with standard mset formulas, so I don't know if it's done often (or if it's a bug in Xaos?).  I'd like to know the name of the phenomena so I can look it up if anyone knows?

  Click to biggify:
(https://lh5.googleusercontent.com/_gbC_B2NkUEo/TXCF1g1qy1I/AAAAAAAABEQ/sdjzhEvK1rA/s144/line%20drawing%202%20smooth%20mode%20awesome%20a.jpg) (https://lh5.googleusercontent.com/_gbC_B2NkUEo/TXCF1g1qy1I/AAAAAAAABEQ/sdjzhEvK1rA/line%20drawing%202%20smooth%20mode%20awesome%20a.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TXCF13R7IqI/AAAAAAAABEU/s66y5NvqRzo/s144/2%20line%20antipure%20form%202chalice.png) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TXCF13R7IqI/AAAAAAAABEU/s66y5NvqRzo/2%20line%20antipure%20form%202chalice.png) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TXCF2DGriRI/AAAAAAAABEY/wvElHBCEm0o/s144/2%20line%20anti%20purp.png) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TXCF2DGriRI/AAAAAAAABEY/wvElHBCEm0o/2%20line%20anti%20purp.png) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TXCF2DK1Z0I/AAAAAAAABEc/JoeMt0MA1dk/s144/2%20line%20ghosinverset.png) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TXCF2DK1Z0I/AAAAAAAABEc/JoeMt0MA1dk/2%20line%20ghosinverset.png) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TXCF2ppttlI/AAAAAAAABEg/kFMCv4AHkKE/s144/2%20line%20green%20bud.png) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TXCF2ppttlI/AAAAAAAABEg/kFMCv4AHkKE/2%20line%20green%20bud.png) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TXCF3GU9R-I/AAAAAAAABEk/TOVeXw0CtfU/s144/fract0.png) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TXCF3GU9R-I/AAAAAAAABEk/TOVeXw0CtfU/fract0.png) (https://lh5.googleusercontent.com/_gbC_B2NkUEo/TXCF3I540pI/AAAAAAAABEo/d_sAPQfdpoE/s144/line%20drawing%202%20green%20palate%202.png) (https://lh5.googleusercontent.com/_gbC_B2NkUEo/TXCF3I540pI/AAAAAAAABEo/d_sAPQfdpoE/line%20drawing%202%20green%20palate%202.png) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TXCF3WByLNI/AAAAAAAABEs/Uivq2rBfMmo/s144/line%20drawing%202%20green%20palatec.png) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TXCF3WByLNI/AAAAAAAABEs/Uivq2rBfMmo/line%20drawing%202%20green%20palatec.png) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TXCF3uhnfeI/AAAAAAAABEw/-mq2LrlvUFs/s144/3%20line%20grail.png) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TXCF3uhnfeI/AAAAAAAABEw/-mq2LrlvUFs/3%20line%20grail.png) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TXCF3pA8lkI/AAAAAAAABE0/4t0YveAmB7s/s144/line%20drawing%202%20smooth%20mode%20above%20original%20beautiful%203.png) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TXCF3pA8lkI/AAAAAAAABE0/4t0YveAmB7s/line%20drawing%202%20smooth%20mode%20above%20original%20beautiful%203.png)

(pix via bunny express)

MB- I think the lines are purposefully rendered under some option to show orbital relationships or something, but I've never been attracted to that sort of thing, and could be extremely wrong in guessing about it.

In the 1st pic below, an equivalent of the Fiegenbaum point from M is shown from the index set I'd been discussing.  It's followed by the whole Julia set, and then a magnification of its center, z=1/c, in both cases.

For MY explanation of why the c=1 Julia set takes the San Marco shape, take the formula and "play computer" with it.  It mimicks the behavior in M.  Of its surroundings, more controversial, maybe the cardioid should be viewed as a distorted bulb.



Real world events will probably put some uncertainty into the availability of my next post.  Please be patient if I can't respond in a timely way for a while.  And feel free to keep things interesting yourselves...

Later!

 :rotfl:

  Hrmm.  I don't recall seeing that option in Xaos...  but will definitely check it out.  I tend to think that it isn't something that simple:  I only found the lines at extreme zooms in certain areas of specific fractal formulas, usually very close to the x or y axis, and if I recall correctly, on the 1/(mu+.25) plane.  I'll have to try it in a non-Xaos thing like fractint...

(pix via bunny express)

MB-   I still don't know what they represent exactly.  A 1992 book, Fractals: The Patterns of Chaos, by John Briggs did a broad overview of everything remotely fractal at the time.  A caption on p.74 had C. Pickover calling them "Mandelbrot stalks", without further explanation.

Strange that back then, my very first rendering of M was a decomposition plot, that traced field lines.  It came from a primitive roll-your-own BASIC program made from a recipe in one of those early books.  It seems that in these pix, however, the stalks trace something interior to the prisoner sets.


A zoom series from the object I'd been describing begins in the pix below.

Again, timing of my future posts remains unreliable.  All things must pass.  Some just can't get behind you fast enough...

(pix via bunny express)

Below, the zoom continues into what I like to call a "super-paisley zone", where Julia-like blobs become less "shapeless" than in other regions of the set.

This set is probably the simplest expression (this one contains only M2 minis) of its type (differing from multibrots and the "multiple multibrots" seen first in this thread) as a simply connected set.  It has me totally stumped.  A working name for them like "Julia-strong index sets" might be appropriate.  Your thoughts?

Later!

(pix via bunny express)

The zoom continues below, into a spiral that is part tail-of-paisley-shaped blob, and part parent-M2 mini-bristling-with-more-paisley-blobs.  Fractals are known to be complicated, and I guess this one does not disappoint...

In the last of these pix, one of the mini's minis begins to become visible.

The journey concludes next time, but when that will be again remains uncertain...

Later!

Introducing slit-scan fractalography :D the ideais to vary some parameter in a way that certain lines will have it constant. here, power p in z = z^p + c is varied according to p = 3 + abs (atan2 (c.y, c.x)) / PI and 4 - abs (atan2 (c.y, c.x)) / PI, i.e we have left-to-right and right-to-left mixes of 4th and 3rd power mandelbrots.

same as in prev post, p = 2 + abs (...) i.e. 3rd to 2nd power mix as you go left to right.

(pix via bunny express)

makc- pretty neat, but it has discontinuities, as when doing z^.5, etc.



In the 1st pic below, the zoom series begun in previous posts ends.  The 2nd shows the whole Julia set for the center coords. of pic 1, itself centered at approximately z=1/c.  I guess z=2/c should behave like z=0 does then, and also be critical.  The last pic enlarges z=1/c, and the detail corresponds to the location in the index set.  Next time, a few more views.


TOTALLY off-topic:
On siting nuclear powerplants in earthquake zones ANYWAY in a fear-based economy:

Situation:  Seismic warnings indicate a large earthquake is imminent, in about a minute.  Should I throw the control rods all the way in without PERMISSION, and get fired for causing a regional blackout after the earthquake doesn't actually happen?

Long-range air quality forecast:  airborne radioactive isotope levels higher.  To avoid lung cancer, stop breathing.

I have met the enemy, and he is us.   -Pogo

In a relatively minor imposition, my jury duty obligation is met, and a CERTIFICATE was issued to that effect, so there should be less restriction on my future posts.

Need to laugh badly?  Select ANY newspaper picture, and substitute the caption with this one:  "Don't let this happen to you!"  It works in almost ALL cases.

Later!

using Perlin noise for p in z = z^p + c (edit: now online (http://wonderfl.net/c/Adky))

Short animation of last one:
http://www.youtube.com/watch?v=M1rmgSDEJ-8

(pix via bunny express)

makc- watched your "burning heart" animation.  Pretty cool, but I miss the connection with M...


The 1st 2 pix below show subsequent enlargements of z=1/c.  The last pic was an almost randomly chosen view, and that finishes what I wanted to show about that object.

I've been working on 2 things lately.  One is a coherent collection of the formulae used in this thread in one place, as opposed to "try this, try that, try to remember the file name, etc.", one that would be easier to get a behavioral overview of.

The other is a 4d with 2 real & 2 imag params.  I do all my looking in 2d, taking slice views, so I don't have to get involved with 3d programs that probably can't display what I want to show with any fidelity.  So I figure on showing some slice views of this very familiar M-like object for you to play with as you will.  (Soon as I can)

Later.

(pix via bunny express)

However slow it might be, here's another example of the utility of a true, non-trig.-based 3d generator.  For all I know, such a thing may exist without my awareness.  I can't think of a simpler example:

As an approach to viewing M in 3d (& 4d), we have f(z)->(z+c)(z+d).  c and d are equivalently-behaving complex constants, and the critical point is (initial) z=(-c-d)/2.  The pix below can be considered as a storyboard plan for an animation.  Each has the origin at center, and they show the c parameter plane for fixed d=-2.5, d=-2.25, and d=-2 respectively.  You may as well know in advance that for the interval -2.25<d<0, the object is connected in 1 piece, and otherwise split in 2 when d is real, as in the 1st pic.  The second shows them beginning to touch.  Future posts will continue the series in .25 increments, while a "smooth" animation would require significantly smaller ones, say, .01 per frame.  Notice the relationships between values in the interval -2<c<.25 in standard M and values d+.25 here.  These are sketchy, hurried renderings, and the last of them is of a pinch point.

More next time.

the connection is rather straightforward, each frame is actually M-set, but its power varies from pixel to pixel by Perlin noise. if you zoomed into some frame you would see distortions "smoothed out" more as you go deeper, as eventually all pixels on the screen would be calculated with almost same power.

(pix via bunny express)

makc-  I apologize sincerely!  For days, I'd been thinking that I had not been fair with you.  Maybe I've let my obsession with simple and multiple connectedness in M-like sets narrow my notion of "M-like" too much.  Universality of M is very strong in the complex parameter plane.  Too often, when I search for critical points, I'll miss, and YET, still see M all over...what you often see is still M-like even if its not what you're looking for.

I don't know if anyone has ever fully explained why there are discontinuities in objects made from functions with non-integer exponents of z.  It would be nice to know.

Also, is there a simple way to summarize what Perlin noise is, how it works, and what it can tell us?



The storyboard series continues below with d=-1.75, then d=-1.5, another pinch point, and d=-1.25, which corresponds to the center of the main bulb in M.  More next time...

Later!

(pix via bunny express)

The progression continues below with d=-1, another pinch point, d=-.75, and d=-.5, the last 2 corresponding to the cardioid interior.


Got potassium iodide?

I'm going with that old standby, the tin-foil hat.  I can make a spiffy one in less than a minute, so...

I'm good.


Have a non-ionizing weekend, if you can manage it.

Later!

it's just to make random but continuous function. Perlin was the 1st guy who published an algorithm for that, IIRC. check out wikipedia if you're interested.

(pix via bunny express)

makc-  I did, recently.  Originally though, my first reaction was instinctual, that a CG overlay of pseudorandom noise on M would only serve to OBSCURE discussion of theory here.  M-like sets are exacting and deterministic, and they're about how numbers behave.  Why smear up the view?

Perlin noise is a different KIND of fractal phenomena, and I don't think that imposing it on M will enhance understanding of either.

Further, if one defines "M-like" too broadly, the concept becomes meaningless, especially for our purposes here, which is to discuss how they work.

CG visual effects imposed on M, if they make you happy, can always be discussed in their own thread.  I don't think they help here, really, but that's just my opinion, of course.

If you want to figure out what qualifies as M-like, however, this is the place.  THAT may need consensus...


The exploration into real d in the series concludes in the pix below.  The 1st, d=-.25, still corresponds to the interior of the cardioid.  When d=0, as in the 2nd pic, you should recognize the lambda function.  As d increases from zero, as in the last pic, d=.25, the split widens, w. the resulting halves becoming ever smaller.

Next time, a small peek into imaginary d.

Later.

Why not calculate at .01 and make a movie? It's fairly easy to do with a tool like virtualdub (http://virtualdub.org/) once you get your images numbered (000.jpg, 001.jpg, 002.jpg and so on). Would save you a lot of posts :) Also, what's with putting text some time after the post? This way only people who come at right time are able to read it.

makc-  My bunny posts in advance because I cannot where I am.  It was explained earlier.  Uploads and downloads are restricted too.

the idea was to be able to see images of multiple exponents M-sets simultaneously, but it did not work out like I expected. so consider that as a dead-end xperiment branch.

(pix via bunny express)

The lambda 3d series shown recently can be extended to 4d when complex d is used.  Below are pure imaginary values for d=.25i, d=.5i, and d=.75i, respectively.  Separation of continental features takes place at approx. |d imag|>0+,-.569946625...i.  There may be a greater imaginary value for d that connects dendritically, but I'll never find it...


M has distinct and well-documented properties.  I propose that M-like sets, to be considered that, must have these, in approximate order of importance:

1. well-defined prisoner and escape sets
2. connectedness
3. geometric symmetry
4. infinite perimeter
5. finite area (in 2d)
6. repeated fractal shapes
7. a hierarchical structure
8. scale-dependent features (so-called theme & variation)

I think all of those may be necessary.  Any to add or subtract?  Your critique very welcome.


makc- Dead ends- I'm not sticking my neck out too far to say WE get a lot of those.  I know I do.  But you keep looking.  There's a lot of dry times, but I gotta say I've been hot lately- at least in terms of what I'd been looking for, -for AGES...

Happy hunting!

Later!

I've been working on that overview of nesting formula permutations, and it's been a very fruitful exercise, as some interesting behavioral patterns have revealed themselves.  I think it promises better insights, so that's what I'm doing.  There is also a bunch of documentation I have to catch up with.  This stuff seriously needs its own website, or maybe a few papers or a book, but none of that promises to happen very soon.  As the season here warms, other more personal responsibilities will allow me less time for this work.  "Time.", he moans sadly...

But I'll be in touch, and report things that seemingly just can't wait.

Later.

The work is going well, but unfortunately, not fast enough for the hordes of eager, brilliant fellow researchers clamoring to discuss the theories behind it...

I forgot, it's naptime.  It's always naptime now.

Later.

I research many things besides fractal math, as you might have guessed.  Some are even of a serious and sensitive nature, but I digress...

While incredibly off-topic, the following is loosely based on a true story (and somehow connected):

In 1960, it was decided that men would be sent to the moon and returned safely to earth within the decade.  Assuming a high probability that it actually happened as reported, one expert on space exploration more recently stated that if we chose to repeat this exercise today, it would cost much, much more and take about 20 years to accomplish.  (...with more advanced technology...)

When the 1st Earth Day was declared, it immediately occurred to me that every day was Earth Day, unless, of course, you were doing lots and lots of drugs.  I often feel the same way about April Fools' Day (today), especially lately, when the late departure of winter where I live was attributed to global warming.

So today is as good as any to make predictions:
The international standard for "normal background radiation" will have to be increased to take into account pollution from recently damaged reactors in Japan, by which all additional fallout from future nuclear disasters will be measured.  Monitoring of new cancer cases close to such sites will end sooner than the 10 years after the 3 Mile Island accident, after which it gets a little embarrassing for plant operators.

And let's hear it for international banking, always a day late and 60 trillion dollars short.

Oil commodity speculators attempting to falsify Mandelbrot's price curve theory are reminded to send Col. Khaddaffi a thank-you card.

It's only 54 days until Towel Day is celebrated again.  Don't panic.

Thank you for your indulgence.

Later.

have you seen this video btw

http://www.youtube.com/watch?v=9mTSdAtlhRY

it's along the lines of what you were doing here

In the aftermath of Fools' Day, my "Violent Torpedo of Publicity Tour" continues, in truth, with due deference to C.S.

Wish You Were Here.


makc- Saw the animation.  I figure that the exponent changes incrementally, but did stigomaster specify whether init. z does too?
Do you know the formula?  Also, did you notice how the disconnected (dendritic) bits at some point can connect in its 3rd dimension? 

makc-  Thought I'd remind you that I'm almost always accessing via library computer.  So there's no sound, if the animation had any.  To discuss theory, explanation to accompany pix, etc., is very desirable, and best as text.

from video description:

p.s. "I'm almost always accessing via library computer" - that's too bad. Obama has his own computer, and you should, too

http://www.youtube.com/watch?v=miaAC3d4RDU

(pix via bunny express)

makc-  Saw the Obama video you posted, but again, I get NO SOUND from here, so again, I'm at a loss.  I forego home internet by choice, however.  Cell phones, credit cards, GPS, OnStar, etc. can double as surveillance tools, and when I'm in a more paranoid mood, I think they were developed PRIMARILY for that.  If I was REALLY paranoid, you'd find me in a corner in a fetal position, sucking my thumb.  If nobody bothered, someone would eventually find the mummified version.

Thanks for the formula, though.  Using the same animation method in your post 101, it wouldn't be hard to apply my post 88 specs. to it, if you have command of the program.  Why not just do that?

In that object, there are always 2 complete M perimeters, and 1 or 2 distinct connected groupings of non-escaping points, for every d value, in every 2d frame.

The 1st pic below is for d=-.25+i in the same view as pix in the lambda series I showed.  The 2nd pic shows the center of that object, c=.75+i, slightly enlarged.  The 3rd shows a plot of c=d for the same formula, essentially then f(z)->(z+c)^2 using critical point (init.) z=-c, identical to M.

Later.

There are SO MANY MORE permutations available in the nesting formulae than I thought there would be that can yield one-piece connected sets, because I've found so many approaches to alter them that work that way.  The trick is to get an overview of what changes to make and what can be expected from them...in other words, I'm looking for and finding some interesting behavioral patterns.  There's a kind of recombinance of available power shapes.  It's gonna take awhile...a really long while...

(!)

Later!

I heared tinfoil hats help :sad1: or wet towels, as in "total recall". or just towels, as in "hitchhicker's guide".

Hey, fracmonk, this image I posted for the Spring Competition, it's using a formula "inspired" by your posts. Should I share the prize with you if I won?  :embarrass:

(http://nocache-nocookies.digitalgott.com/gallery/6/3869_18_03_11_3_04_49_0.jpeg)

One more thing: I really like your acid humor and criticism about our goverments and society. I don't think is a problem to have your comments inserted here from time to time, but you should open a blog for this, don't you think? I will be your first follower  :)

makc-  If you could actually benefit from potassium iodide pills, (or tin foil hats) you've got much bigger problems.  And you might...I bet you "know where your towel is" at all times!

Kali-  A very generous gesture on your part, and I'm honored.  But I think the contest prize is a cup with the winning pic printed on it.  While we debate over whether it's half empty or half full, some would get a few laughs from sawing it in half.  Still, I like to think old Solomon was just kidding about the baby and all...

I wouldn't blame anyone for getting angry at me for expressing myself beyond this topic here.  This is probably not its proper place, though I think it's the right time.  You could count the rules that really matter on one hand, and yet, we have so many more.  Governments are only subsystems of a more global effort of a very few to turn the rest of us into cannibalistic rats in a maze.  You can't let that happen and still call yourself human.

Keep the faith!

Later.

I actually never listened to hitchhicker series beyond 1st 30 minuts. Much more verbose than the movie, and I didn't like the voices, too. So the towel joke is kinda over my head. In the movie, they use towels to slap vogons few times (in the end), but that's all.

 :rotfl:

There's another solution: instead of dividing space, dividing time. I'll use the cup for 6 months, then send it to you. But you pay the shipping  :D


I don't think anybody can be such a fool to get angry for that. I was really encouraging you to ALSO write your stuff outside the forum!
As for "cannibalistic rats in a maze", I think we are that already, in a figurative sense, since a long time ago... But maybe we are turning into a more literal version of it!

"cannibalistic rats in a fractal maze" sounds better (so we are not off-topic anymore ;))

Keep the resistance!

Later.

makc- Try the Wikipedia entry about "Towel Day".  Explains all.

Kali-  Don't forget that you still have to win it first.  If you do, I'll be content if you posted a picture of a picture of...a picture of the cup, here.

I come from an art background, and once was Art Director of a mag. devoted to avant-garde art, until what they called "art" just offended me nonstop.  The same people who might think that gold-plating dog poo was somehow genius could never bring themselves to consider, for example, science as art.  In math, I would consider Euler's Identity as art, or the M formula, now responsible for millions of beautiful pix.  Not to obscure such beauty, I always present pix here with the same palette, never rotated.  And I'm not even a minimalist!  But much fractal art does obscure its own math.  To each's own, I suppose.

And thanx bunches for helping with the Rat Problem.  I was beginning to think it was just ME...



Returning to topic, so far, I've pictured over 100 connected sets using nesting formula variations, most of which were described previously in this thread.  As degree increases, more variations are available.  I see a sort of periodic table coming from relations between them.  What a puzzle...

Later.

(pix via bunny express)

Almost inadvertently, I found that there lives a programmer in Pennsylvania whose name is rare and yet identical to mine.  I figure he deserves no less than to be undisturbed concerning any matter I've discussed here.  Thank you for that.

The pix below are a few degree 8 variations of the nesting formulae I've been working on lately.  Each of these remains one-piece connected.  The 1st index set is notable for the absence of M2 features, the 2nd for the absence of M4 features, and the last for the presence of Julia-like features, with M4 shapes in it otherwise.  Figuring the rhyme and reason of them, so that their patterns can be understood, again, will take a while.

Later!

As you may have gathered, I've taken on the task of making sense of the nesting formulae, and also have some ambition to review this whole thread for a list of key posts, highlights, if you will.

The "real world" tugs harder at me lately, and that's why my posts have been intermittent.  My bunny is fond of occasionally chirping musically: "Nobody cares!", and there is much evidence to support that.

Are we to view the world as fallen and task ourselves to leave it behind?  Some think it can be saved, so I'm going off to do that singlehandedly.  (Sure I am...)  Either way, what will be will be.

For the foreseeable future, I'll be checking in here, looking for the random contributions of others, and, for the time being, only responding to specific questions, if any, when I can.

Also, I'm joining the "Save the Humans" campaign that the whales were kind enough to organize.

Later.

it would indeed be helpful to summarize your findings. it iis nice experience to take random poly formula, tweak variables and see what kind of picture you get, but really useful outcome would be to scetch the picture of what you want to get and calculate formula for that. I dont think you're anywhere close to that  point, but feel free to correct me if you are.

makc-  Sorry for not getting back earlier.  Things are very hectic for me now.
You've got to crawl before you can walk sometimes.  Very early on in this thread, posts 3 & 4 addressed that question.  Not only am I nowhere close to doing that, I'm not currently pursuing it.  My ambition here is far more humble- only to discover functions producing M-like shapes that are connected.  If I could devote myself to this full time (and I can't), maybe then I would be able to anticipate outcomes better.  I'm still at a stage of amazement at the COHESIVENESS of non-escaping point sets produced by given functions, like the ones that produce the M-set, the lambda set, or some others I've given here.  I consider this "property" (or behavior, whatever) alone to be truly miraculous, and if I ever found out WHY it occurs in various functions where it does, and not in others, that alone would be a very satisfying personal accomplishment.

With that, I would greatly appreciate any help anyone could give me.

And yet, the cost of such things is often to lose a good portion of that sense of wonder one has while such things remain mysterious.  Sometimes too, we take seemingly obvious things for granted, and fail to wonder at things we should be looking at more closely.  New questions inevitably come up in both cases, and we only wish we could get better answers sooner.

Later!

Some guy here posted interesting observation (http://www.wolframalpha.com/input/?i=plot+roots+of+%28%28%28%28%28%28z%5E2%29%2Bz%29%5E2%2Bz%29%5E2%2Bz%29%5E2%2Bz%29%5E2%2Bz%29%5E2) (result attached). Doing this in other direction might be the way - from graph of roots we could calculate the polygon in a form of (z-z0)*(z-z1)*...(z-zn) and if someone figures out how to rearrange that into "iterative" form, we got ourselves nice fractal designer app :)

Makc-  Got your msg. secondhand by telephone, and came up w. this, w. scant details:
I took the progression ( (((z^2)+z)^2)+...)as a longer way of saying: f(z)->((z^2)+z+c, where c=0.  Then you get the Julia below, the familiar "brain" fractal.  Its real interval is -1<z<0, so it centers at z=-.5.  It turns out all Julias for the function center there, it turns out to be the critical point for the index set, which, in turn, is like standard M, only shifted to left (negative) .25 real on the parameter plane.  I often find critical points by testing some values for Julia sets first.

Unless I completely misunderstood, which is always possible...

Have been looking to make an entry into the Spring 2d Contest, but, dunce that I am, it took a long while to figure the procedure.  Then it rejected my first pic submission as oversize, while its storage value is quite small. NOW, I'll have to drive all the way back to my place and make up a medium version one larger than the 320x200 puny one with annoyingly wrong aspect ratio, that I left to hold its place, and return to the library with it before it closes, as if it's got a Koch snowflake's chance in hell of winning...  User freaking friendly...

So I went back in a driving rain, made the 3-second picture, brought it back, had another exasperating time in this hall-of mirrors, and made a new submission only by wiping out the old one, cutting and pasting the text, etc., and then, I do not see it appear in the gallery.  So, maybe it's there, and maybe it isn't.  Now, I have to get back before the river crests and the road washes out, Lassie.  (No freaking kidding-  I JUST DON'T NEED THIS!!!)  But I felt I should let you know what was going on...

Have Floyd, will travel.  Seriously, it keeps me calm...

Later.

Yeah you did, just look at the picture of roots, they are all sitting on standard Mandelbrot border.

My contest entry appears at the bottom of the Annual Fractal Art Contest 2011 page (where nobody goes) just beneath the categories, along with another pic, as if the 2 were highlighted.  It seems there are several avenues in which the contest thread can be navigated and therefore, it is easily bypassed.  I would have expected it to go in the list of other entries, and still can't figure why it didn't just go there.  So is it in or not?  If anyone can advise or assist...

Lately, I've been staying in a place where when nature bites, it bites HARD.  Eastern U.S. weather has been generally severe lately, enough to make the news.  Here, there have been high winds, torrential rains, minor (so far) flooding below, heat's been out for days now, and physically gaining net access from there is a hardship.  Hence the reference to the old TV show, where a really incredibly talented dog routinely rescues the rest of the cast from falling down wells, etc.  Lassie seemed to be able to bark in English too!

If you need to contact me, you can leave email @    fracmonk@mail.com   if it comes to that, and I will get back to you when I can.  There are reasons for the alternative, which I don't want to get into here and now.


makc-  Given my situation as described, I'm too tied up to give a fair look to your example, but I'm very interested in how it's different from what I said.  Please give details-  what YOU get from the citation you made, how it works, from your p.o.v....  You may want to take a leading role in this thread, if you care to, as I clearly can't right now.  It's foundation is good.  Build!


Thanx 4 your understanding.

Later!

well the difference is obvious, what you got is four-leaf clover when you were supposed to get mandelbrot (on which border red points sit). so the application idea is this: user clicks along the "imaginary" oundary of fractal he wants, and we create the poly with roots in these points. nex (at this step is something I am not sure how to do) we find iteration rule corresponding to this poly in a way of that mandelbrot example. if this turns out to be impossible for random poly, I guess one could fnd some sort of "closest match". finally, you use iteration rule to render the image.

makc-  There was a notable absence of a constant in your original notation, so I hope you can see why I thought what I did.  Then it sounded as if you were proposing inverse iteration, or backwards iteration, which is also not new.  It requires randomly choosing between pos. and neg. roots in each step, and has the notorious weakness of not being able to reach into cusps, particularly.  It does demonstrate the same shapes to be available under iteration in either direction.  Forward is vastly superior in ease and accuracy, which is probably why the other method all but died years ago.  But you may still be talking about something else again, in which case, when it comes down to "Does it bother you that much?", it becomes a DIY proposition.

Lately, I've been wondering what bifurcation diagrams of real axes of the functions I've shown that produce multiple power shapes might look like, but it hasn't bothered me enough yet.  Anybody familiar with bifurcation diagrams can do that, if it bothers them that much.

Judging by history, I don't think it will.

Later.

I did a formula in UltraFractal for bifurcation maps some months ago, so I'm here to change history.

The formula I used was: z=(z*z*c+1)+1/(z*z*c+1) - 50 iterations

It's weird, as I expected... or a little more. It consist in two parts, one above and a lower one.
I did a full version, then a zoom on both parts.

Here is the full size to show you how the map is arranged:

(http://img820.imageshack.us/img820/6504/bifurc1.jpg)

link for better view:
http://img820.imageshack.us/img820/6504/bifurc1.jpg

And the zooms:

Upper:
(http://img163.imageshack.us/img163/1662/bifurc2.jpg)

Lower:
(http://img204.imageshack.us/img204/3332/bifurc3.jpg)


The black parts are actually very compacted vertical lines, as they show up with less iterations and zoom, but I forgot to upload and I'm too lazy now for uploading that to show you, sorry :)

Kali-  U R the best!  Any chance of changing the aspect ratio so it's less stretched vertically?  (These pains, they always want more...)

Thanx.

Later.

Oh, no more boring bifurcations... look at this beauty made by your formula, and colored by exponential smoothing (http://en.wikipedia.org/wiki/Exponential_smoothing). I'm very intrigued by results of this coloring method, which I use all the time. I don't fully understand how it works, however, as I'm very basic at math. With more iterations, the shapes becomes more precise and are much like the ones of escapetime coloring... but perhaps this technique, used for smoothing and forecasts of timeseries data, can tell us something more about this fractals made of iterative functions. As for now, it was for me a fun way to explore them:

(http://img850.imageshack.us/img850/7112/expsmoothing02762.jpg)

(http://img833.imageshack.us/img833/6720/lava02762.jpg)

Having read your post of your image for the contest, and the post 112 of this thread, among some other stuff you wrote here, It seems to me that you are a bit dissapointed about the path and protagonism that fractal art is taking lately, but correct me if I'm wrong. Also you said "we the fractalist", and I think you are mixing up things. Fractalists and Fractal Artists are not the same. I mean, they can be, or not. I'm myself a more a fractalist than a fractal artist (or at least I'm trying to be, I'm really new at this). I'm, like you, more interested in researching, understanding and in the quest of new types or methods of obtaining complex fractal images, always using the most simple formula possible. It's even a philosophical way of thinking, because I'm convinced the entire universe works much as this fractals, with all the inmense complexity coming out from the absolute simplicity. But I also like to admire fractals as art, looking for the aesthetically pleasing, but that's something not always related to fractalists. Many fractal artists, just play around with parameters, formula combining, colors, lights, etc. etc., without having much idea on what they are doing really, mathematical speaking. But, as artists, they have an special eye for visuals and makes stunning wonderful pieces of art without the need of understanding even what the hell is the meaning of the word "fractal". And is nothing wrong with that... I see this forum has become a bit more into Fractal Art than researching, but that's not bad at all... the fact is that the works are getting more and more amazing, and most people just like to see this spectacular fractals, without getting much involved into understanding or researching or whatever... I don't think there's a problem on that, do you? Anyway I see that there's some people doing research, sharing knowledge, making new softwares, colaborating with each other and so on... and the other people who are mostly "watchers", encouraging with their positive comments, so it's a really nice community... At first I was a little upset with some few things, like pictures uploaded here that are not even a fractal at all, or have very little fractal properties... i.e. an sphere with some wires dancing around... cool render, cool lights, cool colors, a lot of positive feedback... but maybe more suited for a 3D studio forum, don't you think? But is not big deal, each one has their own reasons for being here, and that's ok. I think the true reason for being upset was my lack of capacity for doing such cool 3D rendering  :embarrass:

Well, I have to say that I really enjoy this periodically updated thread of yours, it's very interesting, I'm always looking forward for new findings you make. I even had been exploring your formulas, and I based on one in my latest findings, as you can see in my latest post of the thread "Very simple formula for fractal patterns". Take a look at the whole thread if you didn't already, you have to like this, because is a really simple formula (and variations of it) that produces nice patterns. And they turned to be M-related, something I had no idea when I got the first version of the formula. This is the thread: http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/
I'd like to have your feedback on it... get out of this thread for a while!!! :D

Well, sorry for the long post, and I hope I was able to express my ideas using this poor english  :embarrass:

Regards,

Kali-  You're going to transform me into a man of few words!  Rarely does it happen that I would agree on all points with so many points made.  I take it that you prefer more monochromatic renderings than I.  I appreciate that, too, when I see it, but I don't do it, preferring CRAZY colors...most of the time.  As for exponential smoothing, I've been shy of such methods as continuous potential, true color, etc., because I'm more of a map freak.  I'm sure, however, you'll have many takers.  The rationale is usually to eliminate detail in general overview graphics, otherwise, where it can distract from a point being made.  There's room for all...

-except when there isn't, and then the guns usually come out...

And your thread is truly OUTSTANDING!

Thanx.  Later.

Kali-  Just back from your thread- It looks like a perky little formula I must find a way to play with.  Pretty sure it's not M-like, though.  Still, very impressed, and it looks like you've been v. busy w. it, too.

I still think that bifurcation diagrams of these functions, the ones that produce simply connected (as opposed to multiply connected) would be more interesting than you'd think at first consideration.  I'm going into serious fractal withdrawal, having nearly zero time for it.  Too involved w. low tech real world cave man stuff...

But GO, baby GO!

Later.

Fracmonk, thanks for the good feedback about my work, and I'm glad you agree with me on what I wrote in my last post here.

About my formula, keep in mind that you must use inner coloring for seeing the patterns, as the values doesn't diverge.
I don't know what coloring methods Fractint includes, but In my new thread I explain how 'exponential smoothing' works, so you can implement it as an add-on to the formula part: http://www.fractalforums.com/new-theories-and-research/a-new-look-into-escapetime-fractals-using-abs-and-inner-coloring/

Off course what I said about bifurcations diagrams was a joke, I'll try to get more decent ones and posting here. I wrote a vb.net application for plotting the regular M bifurcation, with an option to save the output to a sound file (yeah, a pretty strange feature), but I must add the possibility for going up and down the plot, in order to make it usable for the atipical diagram that your formula produces. I'm a bit busy now, but I promise I'll do it in the next days. Take a look at this thread to see what I'm talking about: http://www.fractalforums.com/new-theories-and-research/sound-wave-generation-from-1d-mandelbrot-formula/

Looking forward for your researchs, keep on the good work!

Kali,   My bunny reports that she has printed out the linx U supplied here, for me to study.  Thanx.

Fleshing out an earlier observation more fully, post 38 mentioned that in M, integer c is echoed by integer d in these multipower M-like sets,the simplest of them being: f(z)->((((zc-d)^2)-d)^2)-d.  The shape of d=1 is found in post 22, and the sword shape of d=2 can be seen in post 38.  The only other integer possibility for c within M is the origin, but that makes the entire plane non-escaping with the above formula.  But since any nonzero value for d results in a finite index set, I thought I'd test a d value ARBITRARILY CLOSE TO zero.  Pic 1 shows d=.0001, and curiously, changing its sign gives the same shape, with a 45 degree rotation.  As a non-integer value, it is NOT connected, and enormous, since there is an inverse proportional relationship between |d| and set size.  It does, however, begin to approach the shape of f(z)->(z^4)(c^4)-1 (pic 2), which IS connected and reasonably sized.  By simply squaring z in the first formula given above, one obtains the 8-4-2 formula, and, again using an arbitrarily small d, one similarly approaches the shape given by f(z)->(z^8)(c^4)-1 in pic 3.  These last last 2 pix both only yield the M-like power shapes of highest degree.  I thought some of you might find that interesting...


Irresistably off-topic:

A fool self-destructs, and is easily replaced.  We're the IMF, with global answers to local questions.  We don't want to run the whole world.  We do.  Things are never as they seem.  We make it so.  IMF.

The Wharton School will differ on this, but finance does not equal economy, but is the art of distorting it.

They would prefer that you keep believing that history is accidental...

Later!

Another thing I've been playing with lately is reordering execution of multipower formulae lines.  For certain critical points, one can give what amounts to very different longhand-expression equivalents that yield identical index sets.  It's a little like getting on a merry-go-round at different places, and still getting the same ride...

Elsewhere:

After 25 years of entraining the minds of women into a conformist (and perhaps vaguely anti-male) Borg, and being outrageously compensated for it, Oprah's show is over, finally, as promised so, so, long ago.  Sure it is...apparently, managing the many haystacks of money can wind up taking up all your time.

Some people say "Era" with a long e sound, some say it with a short e.  Anyway, one of her fans said it best, I'm sure without meaning to:  "It's the end of an Error."  Doesn't that say it all?

Maybe you had to be there, but that really made me laugh when it hit my ears.

Later!

Despite the unreliability of my ability to log on & make posts, I take every opportunity to check for new posts by others, when I can, especially if they contain inquiries.  Please don't feel forsaken!

Later...

Update from the real world, where the "war" rages on...that basically means I'm letting the computer sort of catch up.  I'm doing studies from sets I've found, to determine relationships between index sets and their Julias, but w. very time-consuming hi-res pix.  "Set it & forget it.", coming back much later, & maybe it's done, & I'll save it, since that's all I have time for lately.  Found out that FractInt does more than I thought it could, precision-wise.  The status screen (<tab>) will only show 9 digits of a fixed param, but it actually will do up to 14 in 320x200 and 12 in 2048x1536 accurately in its floating-point limit in formula mode.  That's its ceiling, for you flying types.  It can be checked by hitting <z> for the exact param info and then <esc> to restore calculation.  So going that much deeper and still being able to specify the Julias for the more precise locations is a blessing.

Well, it makes me just a little happier, anyway...

Later!

Hi all, have been just keeping a scant eye on this thread and just realised one particular modification that I believe I've really only seen used by myself and Joe Maddry (though given that I only started in fractals in 1999 and never really had a proper look through the complete set of public formulas for Fractint I could be mistaken) and that is the use of the iteration count as part of the formula itself, I mean the following is tested by many:
z=z^iter +c
But I think few have investigated other ways of using the iteration count - in my case I came across the recursive algorithms for various polynomials at Wolfram Mathworld and incorporated them in my formulas for UF in as many ways as I had time for as my interested lasted (i.e. until I found something else to pursue) - you'll find these as the various "switch" formulas in mmfs.ufm at http://formulas.ultrafractal.com/ i.e. Switch Recursion, Switch Lucas, Switch Brahmagupta and Switch Morgan-Voyce. The Switch Gamma was just an offshoot of looking into such recursive relations as many involve factorials and the gamma function can be used for such evaluation as an alternative to simple recursion ;)
Joe Maddry's formulas are also in the UF formula database as jam.*
I definitely think formulas of this type are worth further investigation - both artistically and mathematically.

Thanx David, I (for one) will try it, with some variation.  Straight iter would bring powers to fantastically huge proportions quickly, so I'd consider something to limit it, like iter/z, etc.

But I had planned, oddly enough, to talk about integer powers and second parameters today.  For the multi-power index sets I'd been discussing here mostly, they are apparently crucial to them if they are to share the same TOPOLOGY as M.  That would be the simplest way to describe the narrowest definition of "M-like" there is, I think.

For example, f(z)->((...(((z^n)c)-d)^n)-d)^...  would require that both n and d would be integer distance from 0, with d no larger than 2, and pure real or pure imaginary, give results that are simply connected in one piece.  That condition yields identical topology to M and may be useful in the study of both kinds, by comparison.  That is where I think the significance would be found when considering issues like local connectedness.

Fractional d or exponent on z does not offer this in only 2 dimensions, which, when one considers the complex numbers as closed, is where it counts.  In addition, the sign on d is crucial as well, as previous posts suggest.  Only within those constraints do the multiple-power sets have the narrow M-like properties described.

Topological identity in this sense, I believe, would be a valuable key for complex analysts.  Not to drop names, but if only they were still with us, Julia, Fatou, Mandelbrot, Douady, etc., would have been very interested, and many talented living enthusiasts SHOULD be, and my faith is that they will eventually discover this as I did.  Unfortunately, my sense is deeply 6th, as my own specific skills are wanting.  My discovery was partly accidental, while I was in fact looking for it specifically.

That should explain my particular obsession with it.

I'll add thoughts on that to elaborate as they come...

As now, 6-15, I thought I should clarify that d=i works with functions like the above when n=3.  I have not found any (and there are few) integer d values that aren't non-escaping in M as c values.  Better to update this post, as this thread's gone elsewhere in page 10.

Later...

Here's an idea that I've just started playing with:

init:
  iter=0
  z=0
loop:
  iter=iter+1
  rot=cos(iter)+i*sin(iter)
  z=z^power+rot*pixel
endloop

Modifications can be made to include a slider from 0 (regular Mandelbrot calculation) to 1 (full rotation effect) or to alter the amount of rotation each iteration.

Of course I should have added that although the recursive relations are normally applied to simply give a higher degree version of a given standard formula - such as a Chebyshev or whatever - the key thing is that my formulas for UF allow the recursion to be applied across the iterations rather than to a given degree on each iteration - essentially like z^iter+c but such that "z^iter" is replaced with the rising degree poly form. Applied in this manner the results are distinctively different from "normal" formulas.

David-  I PROMISE I'll try it, but I can't say WHEN!  Things are a little hectic for Bunny & me right now- the headlines are catching up to everyone, so we're struggling a bit to maintain life as we know it.  Hope things are more comfortable 4 U.

And...a...later (if there is one).

David- if you're still glancing in, there's a question I was meaning to ask you for a long time:  I started this thread partly because I had noticed that it was easier to get a simply connected object over more dimensions, and harder to do the same in less dimensions, generally.  Since you are very accomplished in 3d, I was wondering if over the course of your experience plotting fractals, you have found the same.  Your thoughts?

Also, I downloaded the complete UF formula zip from the site you linked, but it looks like it can only be read thru the UF program.  True?

Thanx, later...

AFAIK anything that is disconnected in n dimensions can become connected in n+1 (or higher) dimensions - I don't know a stochastic proof for this but it seems "obvious" enough ;)

The formulas in the zip are just text files in disguise - i.e. load them into any standard text editor and you should be able to read them OK.
For a quick and easy comprehension of how most UF formulas work just look one or two in mmf.ufm first, once you have the general idea then following them shouldn't be a big problem ;)

Sorry for bumping into your talk.
Which software do you use for generating fractals fracmonk? You can download and unzip this whole formula pack, and in ChaosPro http://chaospro.de/ File->Import->Compiler Formulas and select all or only chosen ones inside the unzipped folder. 70-80% of them should work as expected. It looks that built in compiler has a bit different language from UF.

That's essentially correct, but doesn't cover any of the UF5 class-based formulas (*.ulb) and of the others some functions introduced in UF4 and UF5 are not supported but I think any for UF3 or UF2 will work perfectly in ChaosPro.

I was roughly aware of that, but after I've counted how many formulas were updated/uploaded to the database in last few days in *.ulb, then I see that there's missing much more than 30% :o

dwsel-  I'm a really old-fashioned guy, I guess: I use FractInt (and they all moved away from me on the bench...) and write formulae in Notepad (clean & simple).  Maybe other progs may offer more bells & whistles, but I try to keep it simple while I don't really have time to orient myself to their use.

David-  It's reassuring that you also view the greater chances of connectedness as more dimensions are available.  I felt it early on, and tended to observe it in practice later.  I'm comfortable enough to piece together a 3-d view from 2-d slices, but I'm sure some others may find it hard to visualize.  But I wonder if there is some object so hopelessly disconnected in 3d or less that it requires more than 6 or 8 dims to be connected in one piece, although we would not be able to visualize it in any way we're used to...and then, of course, there are String Theorists, (not to mention Theologists,) expert at finding questions that can't be proven one way or the other...I wouldn't know where to begin reading a proof, much less writing one!

'til next time,

Later...

Update- "theologists" is a word I coined for that subset of theologians who would condone killing over things they can't really prove, because (usually,) their one true God told them it was o.k.  My opinion is that things should never get so serious. Unfortunately, they do too often.

Otherwise, I've been trying to figure some rules of thumb for what power types to expect in combination for any given formula, but haven't had any time to do justice to it, and won't, I expect, for awhile.

Still otherwise, I've been looking at deforestation and its effects, and the sojurn of scarlet tanagers to our little island of woods after several seasons without them.  They make cardinals look drab in comparison, and cause my bunny and me to smile and be happier...

Little things, little things...later...

Not that I'm back exactly, but-

(This time of year, I'm usually building while not travelling to do building, most often with wood, less with stone, but it's all-consuming, & makes me a dull boy...)

Whenever I get a chance, however, I'm still exploring the many wierd new bugs with multiple multibrot-power-types that one can obtain by generalizing the method I've previously described in this thread.  My formula D32g yields an index set (pic 1 below) that contains M32, M8, & M2* power types, simply connected.  The function is f(z)->((((((z^2)c-1)^4)-1)^2)-1)^2**, for those who'd like to look for themselves.  The 2nd pic is of the whole julia set for a location ctrd. in pic 1, where there is a secondary mini of an M2.  The 3rd pic enlarges the center to show a J4 feature there, that ALWAYS appears about the origin in dendritic j-sets for this function, there at the origin, but nowhere else, nor in the index set.  Try to beat me to explaining why it's there...(it's easy, because I don't have a clue, nor any time to think about it)

But I thought I'd try to stay in touch...

Later!


_____________________
*  M2 is the designation for the "standard" M-set shape, as opposed to Multibrots of other powers.
** Oops!  Sorry, folks: I had originally misreported what was actually the formula for D32h (not shown): f(z)->(((((z^2)c-1)^8)-1)^2)-1, in which the index and Julias both contain 32, 4 & 2 power features, also not suggested by the function in any obvious way.  Also interesting to look at, however...

I met Irene inland- a bit less wind, as I figured, but the rain was biblical...very messy girl!

Intuitively I'd say anything disconnected in <n dimensions could be connected in n or more - I'm guessing there's a topological proof of this, but I've never studied topology.

David-  I've never studied topology (at least in any formal way) either, and at the risk of repeating an old comment to annoyance, I'm given to understand that Dr. M was never able to see the distinction between simple connectedness and local connectedness, either.  To clarify in my own case, however, I simply cannot understand the very CONCEPT of local connectedness itself, so my opinion on that is utterly useless.  The bunches of objects I've given that contain multiple power shapes in this thread are simply connected, but I still have not found the time to do more than list many of them taxonomically, with only the strong suggestion of a structural principle at work behind them, which has yet to be FORMALLY described.  Those who are expert at topology have not yet nailed down a determination of local connectedness for the simpler formula for M, as far as I know, (the big open question,) but I would think that if it were found to be so, it then might not be too hard to extend it to these as well.

Update-  I've been looking at multiple power shapes for odd powers again, with some work on combining both odd & even powers.  It seems that when it comes to powers, any combination can be made as long as only one of them is prime.  I'll try to show pix of other traits soon.

Later!

The recent storm(s) I mentioned in post 146 devastated the region in which I spend most of my time.  Deposition patterns from erosion have made future floods likely to be more severe than the rainfall that causes them would normally suggest.  I'm watching this dynamic closely, and have to get back.  New heavy rains are forecast for the next handful of days.  I'll elaborate on the last post when I can, however.

Try to stay "on top of things" yourselves...

Later!

Well, those interminable rains are finally over for awhile, and in fact, there have been 5 totally cloudless days in a row here, which is even stranger- here, that NEVER happens.  Good weather for digging out stream beds...

Then there's those people who want to drill for gas under us, and ruin our water, air, and landscape.  There's a big push for it, and NYSDEC (Department of Environmental Compromise) is inclined to fast-track it.

Cheers to O.W.S. crowd, making me feel like a few people actually listen!  Hope it's not just another deception, or eventual "object lesson".
I can recall plainclothes cops getting into demonstrations to incite riot and give the uniforms an excuse to bust everyone...ah, the good ol' days...when we weren't 99% spineless!

I'm very busy still, but I have machines making pix for me, the kind that take awhile.

Be good.  Be CAREFUL!

Later.

Too much time has passed, & I just wanted to say Hi.  Been working on index sets with Julia-like features a little more lately, and I will (eventually) get up some interesting pictures soon.

Elsewhere:  What's a better headline?  1. Winter Weather Withers Protest
                                               -or-2. Billionaire Mayor Crushes Dissent
I'm thinking 2...
Anyone remember the story about the goose that laid the golden egg?  So does it lay more eggs when you strangle it?  The answer is yes, for a very short while.  Then, no...

Bloomberg plans on cooking it for Thanksgiving, maybe.

Best laid plans...


Later!

May your holidays be joyful, and every other moment as well.

Lately I've been taking second looks at the family of functions that dominated my entries here, and lets finally call them Multipowerbrots, since the name Multibrot, which may better have been called Powerbrot, has long since been taken.

Anyway, most recently, I've been looking at a kind of circularity in them, that could come only from formula programming.

If you took  ...
                  s=z*z*c-d
                  t=s*s-d
                  z=t*t
                  |z| < 1000000
                  }

and did       ...
                  t=z*z*z*z*c-d
                  z=t*t-d
                  |z| < 1000000
                  }

instead,
then you would get the same results, as long as you started with the right initial z in each case, and they are integer distance values.  d could be either 1 or 2.

The d=2 case is interesting since it's the simplest case of quartic and squared M shapes possible for d=2, that being degree 8.  I have to look to see if the julias are also identical for any given set of c coords.  In any event, the julias will contain both j2 + j4 shapes if one picks coords in a mini.

Note: I use large bailouts for smoothness (and when they get away, they go fast anyway).

I can say more later, and show some pix then, too.

All my best, and thanx again to m'Bunny...

Don't think, this is worth for new thread, so put in some threads.

Adding +some number to colour algorithm allows to see, how exact numbers orbits to mandelbrot border. Imaginary numbers have curved trajectories, real have linear.

(http://www.ljplus.ru/img4/a/s/asdam/numberseeker.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/Cherrytree.jpg)

http://www.fractalforums.com/the-3d-mandelbulb/more-natural-mandelbulb-surfaces/msg39296/ (http://www.fractalforums.com/the-3d-mandelbulb/more-natural-mandelbulb-surfaces/msg39296/)

Would be good enought for Chaos pro. http://www.chaospro.de/formulas/display.php?fileid=224 (http://www.chaospro.de/formulas/display.php?fileid=224)

:) Nice to see you're investigating - but even the orbit trap formulas for Fractint let you do that in around 1995 or earlier - if you haven't done so already import all the Ultra Fractal formulas into ChaosPro and in 95% of the orbit-trap style formulas you'll find a parameter called "Trap Center" or something similar which accomplishes the same thing - essentially what your suggestion does is trap to a fixed point other than the origin.

Thanks. Well, chaos pro native formulas are quind of poor. I allredy have bunch of ultra fractal formulas on Chaos pro, but they not so smoothly implements in its interface :embarrass: Hadn't looked throught all, tested just some, and then went to exponential smoothing as Kali allways sugests;) Well, FE hadn't colour algorithms with variables, so at least Choaso pro users now could download easy to use algorithm.


Continuing with the main thread. Interesting feature is that you can write some random formula without any square function, linear z, some sin(z)-exp(z)+c whatever, just no cubes. There will be some set, but zooming in at some point still would show classical mandelbrot sets, it's even hard to get rid of them. So maybe mandelbrot set lake is something more fundamental than definition in wikipedia sugests.

Asdam- It's interesting, and I take it that it's an overlay, AND I can see the original formula coming thru in the third pic, but why cloud the point of it visually when we're talking theory here?

To be honest when investigating new ideas I'd recommend doing what I do - always search the UF formula collection for any keywords that are relevant and often a main formula and/or colouring (or even a transform) will turn up that's already using the idea - not necessarily as you intended but often involving essentially the same calculations - and of course you can do this even without UF or with an unregistered copy - for those who are more Fractint oriented than myself doing the same with the Fractint formula collection would probably also be useful.

When it rains, it pours!  I was in the middle of a lengthy report about my disastrous situation when the library computer I was on timed out and booted me off.  Starting over...

I was preparing pix related to post 153 this AM when I opened too many windows at once & froze up my ancient standalone.  Ctrl-alt-del did zip, so I had to reboot, except it refused under any circumstances.  Was earlier getting advice online, and have to consider options with a pro I know.  Worst case, I'll have to pop the main drive, put in another, reinstall XP, & hope to recover data from current main.  I don't think it affected external USB drive where most fractal-related biz takes place.  My backup is geographically remote and a month out of date.

Isn't that just wonderful?

Later!

Did what I said I would in last post, and with the replacement drive, it does the same.  Put a new battery in the motherboard, too, nothing, so I have to make new arrangements that'll take time.  But what I cannot show you now flashes complete in my mind:

(Continuing comment from post 153) The most fundamental necessity for the effects given by Multipowerbrot functions is that the relationship between z and c is multiplicative, as opposed to additive in standard M.

When d=2 in the post 153 funcs., you have f(z)->(((z^4)c-2)^2)-2, and you get larger filamentation than in M, despite the presence of the quartic shape.  In a quartic multibrot and higher powers, there is more "lake" and less filament (including larger and larger minis as powers increase).  This func. above gives Julias w. a 4-sym, while f(z)->((((z^2)c-2)^2)-2)^2 gives the same index set, but the Julias for it are 2-sym (and therefore the critical points are different).  So exploring them in greater detail is still an interesting enough universe on its own to keep me happy.  There are still new and interesting things to be found there.  You have plenty enough info to DIY.

Fare better!

Later.

Main reason reason were, that no colour algorithm had all the features I wanted to use animeting quaternion :D But that is way offtopic.
Not the overlay, dots appears where lies orbits with exact numbers. In first iteration there are just one dot, in second - two dots, in third - four rising in geometrical progression. Quind of interesting.

Didn't found this very simple variation in UF formula database. Throught I think, I alredy had seen this. Maybe as error.
This equation decreased size of set, but increased proportions of outside features. 0.12 produced unkempt mandelbrot, 0.25 creates almoust like bunch of grass, and -0,25 gives 'bloated' mandelbrot with minibrot rings on antenna.

EDITED: variation by Edgars Malinovskis.
Wow, this idea is spreading and evolving.

Nice variation!  Here are a Mandelbrot zoom using -2/0.1 as the factor instead of 0.12, and a Julia using 0.1/0.1.  Both use the color of the year for 2012, Tangerine Tango.

The second picture is marvelous, looks very natural. Hmm, shouldn't colour of 2012 be mayan, end of the  world and that stuff :fiery: Not connected with net I was searching with -0.3 and - 1.

(http://www.ljplus.ru/img4/a/s/asdam/1Mandelgrass_rings_all.jpg)
Looks as bloated Mandelbrot;)
(http://www.ljplus.ru/img4/a/s/asdam/2Mandelgrass_rings_outs.jpg)

This fractal have rings on antenna and minibrot rings on all stalks. Tried the most basic colouring, showing  just as it is.
(http://www.ljplus.ru/img4/a/s/asdam/3Mandelgrass_julia_minusone_r1p3_i0p5.jpg)
I think, I had alredy seen this feature, but I think I never had seen julia sets alike. Changing -0.3 with -1 and having julia seed x = 1.3 and y = 0.5 revealed pretty interesting julia set consisting of circles. Julia set goes nicely with -1, but then rings on mandelbrot are too dense.

This equation works well with another powers, for example cube. Looks like respective standart julias, but of circles and angles.
(http://www.ljplus.ru/img4/a/s/asdam/4Mandelgrass_julia_cube.jpg)

Another bloated Mandelbrot and almoust the same equation. From distance it looks the same exept have angle in its rear;) But non-analitical abs function does it's job quite well, it have triangles on antenna and square box minibrots on it's stalks looking like old TVs with antennas.
(http://www.ljplus.ru/img4/a/s/asdam/5MandelTV_outside.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/6MandelTV_julia_one_r1p09_i0p5.jpg)
Switching -0.3 with -1 and having julia seed x= 1.009, y=0.5 reveals julia set of angled spirals and almoust squares.

Works well in another powers, exept that squares are not so accurate as rings.
(http://www.ljplus.ru/img4/a/s/asdam/7MandelTV_julia_4th.jpg)

Downloaded the latest UF package and found this implemented in UF. Thanks Ikmitch, nice implementation, default cos function in formula creates very floral patterns. Was implemented just before the squares;)

Scratching mandelbrot fractal revealed these very long julias. 2nd and 3rd are as they are (r=-3.2, i=0.5), 1st (r=-1.95, i=1.45) is rotated to fit in the screen.
(http://www.ljplus.ru/img4/a/s/asdam/Linejulia2.jpg)
p.s.
Hmm, I allways tought mandelbrot an julia fractals as some sort of pair, I was disapointed realising that Julia actualy are some frenchmen Zhuly.
 

Hi.

First, if this is true, this is something awesome.
Second, that is not a mandelbrot-like iteration formulae. Where is the "c" variable ?
That formulae (and the other you gave in the http://www.fractalforums.com/index.php?action=gallery;sa=view;id=6995 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=6995)) does not work (i tested it!).

When i iterate over a polynom of z, then add c i never get smooth mandelbrot (except when i have z^n + c of course).
Your fractal seem to be awesome...
I want to explore theses Mandel-8/4/2-coexisting brots !
Thanks :dink: .

tit_toinou-  Terribly sorry to have only found out about a notation mistake I have made (& fixed) before!  It is f(z)->...((((((z^2)c)-1)^2)-1)^2)-1...  generally.  I'll fix it where you quoted it from too.  With some review, I noticed that post 38 generalizes it in another way, maybe a bit clearer to some, depending on notation style...                -And thanx for pointing it out.

I put together talis and tricorn. Could not find the name, but bumblebrot would be OK.
z= flip(sqr(z*z/(1+1i+z)+c))

Or easyer to see:
z= z*z/(1+1i+z)+c
z=z*z
z=flip(z)

Zoom of this reveals everything alredy seen spirals, ^4 mandelbrots and ^4 tricorns, but owerall shape is quite an interesting and exactly the same as in quaternion numbers. This don't remaind talis, but it have 2 kinds of nonsymmetric julias, smooth ones and branched.

Asdam-  I wonder if you examined your tricorn-like object far enough to determine whether it's connected.  I can already see that the Julias you showed aren't.  Pretty wild, in any case.

Later.

There are connected julias. At large it mostly resemble normal power 4 tricorn, and it looks all small features are connected. Here are another julia:

Very cool. Remember those are 2-d, that's generally harder to accomplish.

Later.

There alredy are  tonns of 2D stuff. Each Fractint formulas file contains 50 entries;)

But I think, there are gap as most of Fractints formulas are initialised by z=pixel. Thi is first iteration of mandelbrot starting z=0, but some formulas don't like beeing started with z=pixel.

Asdam-   So I was wondering-  don't you do your own formula files?  Then you can assign your own init. z, and try different ones.
Depending on the function, with some, zero won't work at all.  You need to find out at least some of the critical points.

I once coded some formulas in Fractal Explorer, and it allowed just z=pixel initial z. But looking at Fractins files most of stuff are initialised by z=pixel. I think, most of people are pretty laisy and just copy paste;)

*   *    *
Going back to Unit Vectors.
z= formula(z)
z=z+factor*z/function(z)

If function is cabs and the factor is some -0.3, Mandelbrot set have circles on antenna. Cabs function defined using power 8 instead ow 2 generates pillow like squares with round edges. So modulus function with power 8 are different than with power 2 :surprised: ??? ( (real(z)^8+imag(z)^8)^0.125 ) Not very different, but is different. The same thing appears in insides.

I'm making pix again, finally, w. various forms of Fractint on 2 machines.  Interesting story about that in Help and Support here, in "Still need help with DOSBox in Windows 7".

While I can foresee FEW opportunities to post pix, if I do nothing else, I will illustrate post 153 in this thread here.  I promise it will be SO cool...

Later.

Hello all,

as i tried to integrate Asdams formula into my tool i found another formula that produce intresting images:

z = z + ((c2*z)/z.sin()) + c;

This is an example of the Julia iteration method:

(http://yv3.bplaced.net/gallery/yFract/26451.jpg)

And this video that i made in 10 minutes describes how the change of c and c2 affects on the image:
http://www.youtube.com/watch?v=3H0zjJwGnEk&feature=youtu.be

I called this Formula "rotten apple" :)
Greetings

pix via bunny express

The 1st pic is of most of the index set obtained by BOTH formulae in post 153, where, in this case, d=2.  Results are identical.  It is centered on the same coords. as the 2nd pic, which is a 100,000,000 magification of the first.  If you were to zoom from the first to the 2nd, you would first go into a filament of an M4 mini on the antenna into an M2 there.  Then, just missing it in the vicinity of its "seahorse valley", you would continue to the left tip of the M4 in the 2nd pic.

The Julia sets for both formulae are shown in pix 3&4.  The 2 sym. version has a J2 at its center, and the J4 corresponding to the coords. at its non-zero critical points at 4 and -4.

The 4 sym. version has a J4 at its center, and others at 2, -2, 2i, and -2i, in similar correspondence to the index set coords.

Look closely, and you will see how the Julia sets could be "morphed" into each other's shapes, preserving the structure of each.  Both have infinite numbers of both kinds of minis in them, but the higher power minis dominate in both size and number in most views at any scale.  Zooming into their non-zero critical points reflects a journey similar to the one taken in the index set.

Stay out of trouble with your prof. if you are doing a paper on these Multipowerbrots, by kindly acknowledging their discoverer, Jeffrey Barthelmes, and this thread in Fractal Forums, in your footnotes.  Thanks.

 

Apple Ipod;)

Actualy this brot in the first page
(http://www.fractalforums.com/index.php?action=dlattach;topic=4881.0;attach=3955)
looks exactly as Fractal Explorer's (R.I.P.) Talis formula with the same torch lights on stalks. A pretty popular formula, number of times reimplemented in UF and once mandelbulb 3D.
init z=0; z=z*z/(complex+z) +c;
Or formula found in Fractal Science Kit "Tails" init z=1; z=z+1/z+c; Maybe  'Talis' is coined from "Tails", probably a sheep fat tails.

 

That one (end of last page) was from post 12, with my formula for it.  I almost didn't recognize my own shot!

The torch lights are actually inversions of infinity (I call them "infinity holes", and I believe they are centered on Misiurewicz [sp?] points, but am not sure) and the stalks are the usual dendrites, although they seem to have more regularity of shape than on standard M, oddly enough.

I just tried, how these super advanced 2D formulas would look in 3D.
Even power formula as expected produced just another quaternionic rotation surface.
z=(((((z^2*c)+1)^2)-1)^2)-1; (another proof of i=j aka y=z generating rotation thing)

But the result of odd power formula is pretty interesting one.
z= (((((z^3)*c+1i)^3)+1i)^3)+1i;
For an easyness and lack of quaternionic ^3 operator I rewritten it as chain of the formulas.

z=z*z*z*C+1i;
z=z*z*z+1i;
z=z*z*z+1i;

Vuala, here it is, a bitt strange horned quaternionic Brotosaurus.

Asdam-   I recognized the silhouette of your first pic immediately, since the connected 2d of it extends in probably all ways to the furthest distance from the origin on that plane, so none of the rest that is off that plane is in the way of seeing it.

I also program with the basic formula structure you showed.  You can see that it's most efficient, and I might hazard that that fact has to be related to whatever makes it appealing...

Asdam inspired me to go back to that formula, which goes like this in my file:

(initializing info)...
s=z*z*z*c-d
t=s*s*s-d
z=t*t*t-d
|z| < p3
}

The pix below show various details of the ci by di (2d) parameter plane slice subset of the 4d animal it actually is, where real c and d are 0, in this case.  In each closer view, the details are not surprising, given the degree 27 formula.  The last one betrays the fact that that particular 2d slice is NOT connected.  
I would suspect that the 4d hyperblob is connected over its 4 dimensions in one single piece, and most likely any 3 of those 4 in 3d, while the remaining one is set to 0.  By hyperblob, I mean the 4 axes that cannot be displayed at 90 degrees from each other all at once: a by bi by g by hi, where c=a+bi, and d=g+hi. (Not to be confused with any 4 dimensional single number of any kind, but just the combined parameters of a pair of complex numbers as the axes)

(Hope it's clear...!)

It was just out of curiosity-

Later...

I think in 2D now most in are interested in (low max iteration and exponent smoothing revealed) kaliset like pattern formulas.

First were simple formulas, then came long formulas, then all switched back to short universal formulas but with
sophisticated colouring techniques and layering. Now at least here in ff complex geometrical pattern formulas grabs the most atention. Entangled trees and somewhat arabic patterns, maybe it have something to do with middle east being at the centre of world politics;)

In UF mailinglists unit vector thing (thanks to Toby Marshall implementation with many functions) become quite an used, but mailinglist seems to be a bitt behind;)

I have few opportunities to post pix, but here's some of current interest, just done:

The first shows the whole "D27" index set, with the coords of the center of the second pic in it as well.
That spot is at the edge of the tip of an M27 bulb, and can be considered continental or non-dendritic.  The object has M27, M9, and M3 features in it.

The 3rd pic displays the whole julia, which has a J9 feature at its center at the origin.  It also has 3 large J27's near the ends of its major arms.

In a dendrite near its furthest extremity, the hybrid J3 shape in the middle of the last pic can be found.  Note that it is hard to pick it out of the field of shapes, since they sort of overlap as feature parts.

Later!

I was thinking that pattern sets being in current main 2D interest at FF, so this could be area of (2D) research.

Asdam,  "Pattern sets" meaning?

Later!

Like this http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/90/ (http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/90/)

I just implemented your formulas (as single formula) in Ultra Fractal database as Walbrot. http://formulas.ultrafractal.com/cgi/formuladb?view;file=em.ufm;type=.txt (http://formulas.ultrafractal.com/cgi/formuladb?view;file=em.ufm;type=.txt)

Sorry that didn't asked a permision, just couldn't wait to share it;) (If you are strongly against I wll delate.) But credits and links to this thread is included and you name would be seen in every time one uses the formula (placed info just below formula switch). But woun't be able to change the name of formula to something more pleasing sounding. :embarrass:

Added few extra modifications (unit vector) and power 3 soap like fractal (z=1/(z^3*c+realD)+z^3*c+realD), coz it generates soap buble julias with david stars.

Found nice julia spot, throught extreme complexity of geometry makes a bitt hard to find right colour method. All good mathematical names alredy are taken, so just remembered that mandel is almonds, but walnuts have much more complicated shape. At least there is no walbrots.

It looks that division placed first, then multiplication and just then addition calculates a bitt faster.
So z=z/cabs(z) * @vfactor+z is faster by about 1/20 than z=z+ @vfactor*z/cabs(z)  
The same goes for z=1/z^2*c+d+z^2*c+d
(http://www.ljplus.ru/img4/a/s/asdam/Walbrot_jul_EM.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/Walbrot_soap_EM.jpg)


z=((((z^2*c+realD)^2)-realD)^2)-realD and z= ((((z^3*c+imagD)^3)+imagD)^3)+imagD
with
z=z/cabs(z)  *  (-2)+z
Produced this. It have small tricorn fractal inside.
(http://www.ljplus.ru/img4/a/s/asdam/walbrot_even_uv.jpg)
And two small power 3 tricorn fractals (crosses)
(http://www.ljplus.ru/img4/a/s/asdam/walbrot_odd_uv.jpg)

Then there must be univerality for the tricorn, too, under the right circumstances.

Sorry, stuck for time,

Later!

Well, tricorn is derived from mandelbrot (there is no aviable description why it generates that shape), so it could appear (noone knows why). :tongue1:

Asdam,   I'm pretty sure it requires the conjugate version of z, or, at least, a "virtual conjugate", that would come about due to a programming detail that creates the effect by accident.

Let me take you back to nearly the beginning of this thread, where multiple power shapes first appeared in multiply connected form, that is, where the prisoner set is connected in one piece, but the escape set is not.  (See post 17)

Later, in post 23, the "San Marco" effect came about when the degree was really high, so that the iterations in the formula made it look like that particular Julia set (for c=-1), only it was a second constant, d=-1, that was taking its place, that was repeated more than the parameter being mapped as an index set.

In post 45, one picture, MC842, showed how multiply connected sets could be made by using a z=z+(1/z) instruction after so many z=z*z-d instructions and just before test.

In the first pic below, a long string of just such programming gives a degree 2048 index set with 7 (corrected 4-23, originally reported as 11, see post 208 for explanation) different power shapes in it.  It is multiply connected in one piece, and takes on the San Marco shape overall, with traceries of the mixtures of different powers in daisy chains. To merely identify some of them (for instance, M128 against M64), one would have to make multiple enlargements to tediously count how many lobes each has.  These preliminary pix zoom in a little for an idea of its details.

The 3rd pic shows M8, M4, and (a tiny) M2 in a succession from right to left, while the last shows the tiny M2 a bit larger.

More later, but not sure when...

I've noticed that when doing Julias for this family, especially edgy ones, false periodicity detection makes errors that fill in otherwise interestingly detailed areas.  The <G> command line fix is periodicity=no.  In these latest multiply connected INDEX sets, it is necessary for the first time to do it there, too, at some magnifications, so that errors don't give low quality pix.  That and the added division step realllllly ssssslowwwwwww things dowwwwwwnnnnnnn!

But I think the results are worth it.

Later!

Second picture have inetersing bell shapes. Maybe with right colouring.
Just remembered.

Well, this could be very hard and old question. It is listed as "unsolved problems in philosophy" together with questions like "what constitutes an art?" or "should I drink bear or vodka?" dating back to ancient Greece and maybe even older indian and jewish tradition.
http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_philosophy#Mathematical_objects (http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_philosophy#Mathematical_objects)

What are numbers, sets, groups, points, etc.? Are they real objects or are they simply relationships that necessarily exist in all structures? Although many disparate views exist regarding what a mathematical object is, the discussion may be roughly partitioned into two opposing schools of thought: platonism, which asserts that mathematical objects are real, and formalism, which asserts that mathematical objects are merely formal constructions. This dispute may be better understood when considering specific examples, such as the "continuum hypothesis". The continuum hypothesis has been proven independent of the ZF axioms of set theory, so according to that system, the proposition can neither be proven true nor proven false. A formalist would therefore say that the continuum hypothesis is neither true nor false, unless you further refine the context of the question. A platonist, however, would assert that there either does or does not exist a transfinite set with a cardinality less than the continuum but greater than any countable set. So, regardless of whether it has been proven unprovable, the platonist would argue that an answer nonetheless does exist.

Most people never encounter such questions, and blow them off when they do.  To consider them is a richer experience though often frustrating.  I'm very glad that you recalled this particular context in relation to these multipowerbrot sets.

Originally, the things I found were not randomly stumbled upon, since I was looking in the right place for them.  If there's an answer to whether they existed before that, waiting to be found, I'll never know.  I see very special things about them that I try to express, and wonder how good or bad I am at it.  I think that these sets are the most mathematically significant find in escape-time fractals since the discovery of the original M-set itself.  If I have my head in the sand when it comes to other kinds of fractals, (and I don't entirely) that is why.

The main thing is that they WERE found, for whoever is interested.

That said, you are confronted with a phenomenon that challenges you to deal with it.

What has really surprised me is how academics studying the local connectedness issues of "simple" M have not even acknowledged the existence of multipowerbrot sets yet.  I mentioned before that these sets can either help in that study, or enlarge the problem, but I've seen no evidence that they were even considered.  Do they know about them?  Do they FEAR them?  I've heard nothing of any of that.
But you won't find me personally chasing after them, trying to sell it. It shouldn't work that way, and I hope it hasn't come down to that yet, since it's about knowledge, not politics.

Maybe they think FF is not a legitimate publishing venue, and somehow, THAT's more important than the content...

Who knows?  Who cares?

What will be, will be.

Your thoughts?



 

Found something pretty strange, unexpected, and interesting about the real axis limits of the multiply connected object of post 193, (degree 2048) but I'll have to save it 'til I get some pix to illustrate.  I have them in the wrong size for display here, but will redo soon.  (and the flaky machine that made them is acting up...)

Have a good weekend!

Later.

The first 2 pix below are of the respective left and right real axis limits of the multiply connected (M2048) object I've been looking at lately.  They are at cusps of M2 minis in the centers of each of these pix, and are at 10 billion x magnification:

Left limit: (about)  >-.15450714937694...
Right limit: (about) < .65450714936977...

I thought that some of the digit places being identical was a pretty curious item.  The surrounding environment is pretty interesting, too, but there's a strict limit on the number of picture postings per day, I think...

The remaining pix are 10x larger than the previous ones, for more local detail.

Before I go and add anything more about the multiply connected set of high degree (M2048) I've been investigating lately, I thought I should clarify an important point about the simply connected Multipowerbrot sets, which are far easier to study:

That is, they are TOPOLOGICALLY IDENTICAL to the Mandelbrot set in 2d.  Anything that can be said of the M-set from a strictly topological viewpoint can also be said of Multipowerbrot sets, and for that matter, the simpler, older, and more well-known Multibrot sets that each have only a single power shape included in them.  That is the basis of what I said about whether the study of them helps or hurts the local connectedness issue in post 196.

Hope that helps...

Later!

Hi.
I also consider your finding very important.
But when you say things like "the set generated by this iteration formula is connected" or "theses set are topologically identical", do you have a proof ?
I mean, it LOOKS like it is true. Don't say it is.

I'm saying this because I've been told that theses kind of proof were for the best mathematicians.

However i'm trying to compute a nice image of your M842, with a Distance Estimator and a Sobel filter... But the derivative grows so fast  :o ! I can only compute two more iteration after diverging :-\ (with double).

You should definitely make a summary of all your findings.

t-t,     Like BBM, I have enough experience with simple, reliable formulae to be able to conjecture those things with extreme confidence.  True, proofs are not my domain, but they weren't his, either.  Yet, he could be as reasonably SURE.  I would put money on the idea that it can't be DISproven.  Now, having said the magic freaking word, before the vultures descend, I must remind one and all that we are talking about the SIMPLY connected versions of Multipowerbrots only, and in 2-d, on the c parameter plane.  That's nothing different from what I said before, but maybe more succinct.  Anyone who had a good look would be foolishly throwing their money away to me, and to me, THAT would be a very new thing.

And especially if I annoy you!

I would suggest not using distance estimation, and I don't even know what a Sobel filter might be expected to do.  The formula is incredibly simple for the 8-4-2 version, and is most easily done with the fractint formula parser.  Everything you need to piece it together is in this thread (somewhere!).  I probably said that I don't like a lot of generating progs. because many were written with very specialized 3-d apps. in mind.  KISS, and you'll get more.

My machines are working on M2048 pix currently, and that formula requires the use of NO timesavers.  One recent pic took 46 hrs. with fractint in DOSBox.  8-4-2 should take seconds in low mag.

Later.

We agree. Here is my image : (http://th02.deviantart.net/fs71/300W/i/2012/084/3/3/the_42_monkelbrot_by_titoinou-d4tvwur.jpg)
Applying this filter is like computing the discrete derivative. I used its norm to colour.
It is very efficient to show the border of the set.
This image is clearly conforting us in our belief that this set is connected.

t-t-   I see you turned it 90.

For those who'd like to do it as I do, in fractint:

M842(xaxis) {
  c=pixel, z=p1:;kcps=z0=-2,0,2 + more...
    s=z*z*c-1    ;M8,M4,M2 features
    t=s*s-1
    u=t*t-1
    z=u+(1/u)    ;for multiply connected version
    |z| < p3
}

Set p3 large for best results;

for the simply connected version, make line 5:  z=t*t-1  and leave out line 6, or set a semicolon before it, so it's ignored.
3-28-12: thought I should add that init z's that work would not then be as shown above, but -1, 0, and 1.

This is a cleaned-up version of my original formula, with one changed sign, that would reverse the orientation of the object.
It is untested, but *should* work...      ...!

Let me know how it goes, O.K.?  (Important)

Later.

If you were recieving education in something sphere, you could write master thesis on this;) That could count as publication.

I know-  sorry!  I've not been keeping up with my accustomed documentation.  I get a new notion, follow it, and then have to catch up with filing the pix so they aren't a mad jumble.  Still have to summarize, like you said, but there's other explorations in the objects I should do in more detail first, like locating significant values, as in post 198, or maybe to map occurrence of constants in the objects.  D842 has got a bunch, mentioned before.

I'll eventually get around to it, when I've got some time.  Helping Bunny right now...

Later.

"Playing away":  Again I've been in a spot beyond the reach of internet access most of the time, but I've returned to my remote backup, now the slowest of 3 machines at my disposal.  Still very busy with (survival in) the real world, BUT have that machine grinding away at higher-degree multiply-connected versions of Multipowerbrot formulae, in a way that I don't have to "service" it too often.  Most would say the pix are not that visually dazzling, as the higher the degree, the more numerous and larger minis clutter the visual field.  But I'm still very interested in how they structure themselves hierarchically as to powers.  Additionally, I think it important to discover the rules regarding each successive power doubling in multiples of 2.  For the simply connected ones, there's an alternation in the qualitative output (the overall structure of the connected index sets and associated Julias) for each doubling, so that resemblance only carries for every 2 doublings of the maximum degree that a formula is worth.  Degrees 4, 16, 64,... behave alike to each other, and degrees 8, 32, 128... behave alike to each other, in one formula structure.  One tiny change in the formula structure makes these 2 groups exchange their qualitative output.  THAT they do, I know.  WHY they do, I don't!

This phenomenon carries over to the multiply connected versions as the degree is doubled, but with different consequences in those.

I might have mentioned before that it behaves a bit like the periodic table of the elements, but it is purely an anatomy of numerical, not physical, behavior.  (Unless, as many have speculated, the physical world's most fundamental rules ARE in fact strictly mathematical.  Then, mathematics could no longer be considered only abstract, but very real...invisible as a magnetic field, but there, nonetheless.)

Idle musings?

Later.

Was up in the wee hours the night before last, and it occurred to me:

The biggest problem with the "real" world is the fake money.

Why is it so?

Later.

I just fixed an error in post 193 regarding behavior expected in M2048.  The formula structure outlined in post 203 leads to the following expections in the category of "different consequences" (from simply connected versions, as mentioned in post 206) found in the multiply connected ones, in the following scheme:

formula degree                      power shapes to be found
2                                           4,2
4                                           4,2
8                                           8,4,2
16                                         8,4,2
32                                         16,8,4,2
64                                         16,8,4,2
128                                       32,16,8,4,2
256                                       32,16,8,4,2
512                                       64,32,16,8,4,2
1024                                     64,32,16,8,4,2
2048                                     128,64,32,16,8,4,2
(etc.)                                     (etc.)

Earlier, I had bemoaned the notion of having to tediously count the lobes on the mini shapes that will appear on the plane with each formula permutation.  But if you don't, then you will miss the pattern revealed above, for instance, as I had done until recently.

Sorry about that, chief.

Later!

Now investigating a code error which happily resulted in a pretty wierd index set (degree 96, apparently), *connected* against ALL expectations, which will likely open up a whole new region of formula permutations (for each degree?) with their own peculiar rules.  Looking to bring the effect to the lowest degree possible (and see if it still "works") to know more about it before introducing any poorly-conceived theories about it!

(SEE NEXT POST about getting too far ahead of one's self...)

If only I could make ALL my mistakes privately, and find and correct them before letting them loose on anyone else!  (I try.  I DO, REALLY!  Honest!)

You must realize that it's not just about pretty pictures for me, though they come naturally with the territory.  I like the puzzle aspects as well.  I want to know what makes these things tick...

Notice in the last post that degree 2 yielded the M4 power shape in combination with the expected M2, seen very early on in this thread, because of the use of the inverse, so that it yielded a "quasi" degree 4 (in the distance between exponent 2 and -2).  I have since wondered whether the formula for determining the number of critical points (2d-2, where d is the highest exponent employed) is correct in all cases.

Can anyone help with that in particular?

Later...

So almost immediately, considering what I said in the last post, I find that I was likelier to be wrong than right.  (It figures...)  The M96 object I spoke of (1st pic below) had a certain misshapenness to the largest formation of non-escaping points, which experience has shown me always indicates that the object is *NOT* connected.  I had applied the same original code error to what amounted to a degree 12 multiply connected object (2nd pic), since experience also suggests that lower degree will reveal such problems more obviously.  There, the object had a clear discontinuity (last 2 pix), as when one uses a fractional exponent, which was in some ways pretty and interesting, but leaves one in a position of having an index set whose "critical" points can't tell you for sure about the nature of the julia sets it indexes.  The critical points would not then be very critical, would they?

I had put pix on a disk to show, then brought the wrong disk...next time, hopefully.  (5-2-12: This was what I thought yesterday- Erased a disk w. XP and burned the stuff I wanted to it, but the laptop I was using w. 7 actually found what I thought was erased, and ignored the existence of the new contents.  Weird, or what?)

Here's the thing:  Usually, one does not have to look far for evidence to show that an object is not connected.  In this case, there are features that signal that that is so, and yet, it might not be possible to obtain that evidence (ie., a separate "island" from the rest) beyond the reach of floating point precision, when that's your zoom limit.  And, of course, experimental evidence in no way constitutes proof, though I have never seen it to mislead if properly interpreted.

But I'm pretty satisfied that this little side road will turn out to be another of many dead ends encountered in this kind of experimental math.

There are always other things appearing to show more promise that might not be apparent if such dead ends were not scouted out.

I often wonder whether new approaches could be exhausted entirely, but then I think it's more like stage fright or writer's block, and I think that works the same way.  Just because you reach and it's not there, only means that you didn't reach into the right place yet...

Later.
  

Apparently most of my pix just barely exceed the max size of 256k, so I'm experimenting with the program and its "input procedure requirements" (my term), to get to show and say anything at all...

I wouldn't mind half as much if it let a posting go though and left out whatever it objected to, but all your carefully articulated notes are lost as well when you are forced to abandon the effort and start over to get anywhere with it at all...

The pix it seems to be willing to accept show a location just next to a spot where one would expect an M2 mini to be found, where surroundings are symmetrical, where instead, there is merely a hole.  The above restriction doesn't let me show the view that shows it best.  The next shows a slightly deeper view, but not the one that shows the M2 mini that actually can be found at the centers of both, upon deeper zooming.  Apparently, you'll just have to take my word for it that it's there.

More flexibility please!

One of the things I was trying to say in these attempts was that it is one thing to have this or that multiple of a power locally dominate some part of the plane, as well-negotiated subsets, which seems to happen with Multipowerbrots, and another thing entirely to have two conflicting systems interfering with one another in the same space, as in the discontinuities just mentioned in the last post.  The two have an entirely different visual character.

Later!  

p.s.
Legitimate publication place is arxiv.

Asdam-  Thanx.  I'll look into it, but you know, I'm one of those people who questions everything, like what makes one venue any more "legitimate" than another.  I'm not hacking out a career in this, but looking around and sharing what I've found, so I don't need to care too much about formality.  What I'm looking for is others who find these things somewhere near as fascinating as I do, and want to help flesh out and further develop the concepts involved.  Maybe it would trigger someone else's wildly different take on it, or that some pretty cool applications might occur to others as a result of exposure to it.

I've returned to concentrating on successive squaring resulting in power shapes associated with multiples of 2, and lately, as applied to adding multiplicative inverses of expressions to get multiply connected effects.  Below are an assortment of lower-degree transformation variations, each resulting in another curious index bug.

Today I intended to begin showing a nice zoom series from the last pic in the last post, but the laptop I borrowed for this, with its trusty Windows 7 OS refuses to read the cd that was made with XP.  This has happened before.  In fact, last time, I erased a disk that was written with 7 to put stuff on it from an XP machine, and the laptop acted as if the disk was unchanged from its original condition, reading the erased data as if it wasn't erased.  When I put the disk back in the xp machine, it read it as intended.  I bet it'll turn out to be the same deal this time.  You'd think I'd have learned...

Sorry, folks, maybe next time.  But let's call the program MS Typical.

I'll bet that I'd have to transfer such data to my 7 desktop, then reburn the disk with that machine to get the laptop to recognize it.  I don't know why anyone puts up with MS, other than the fact that everyone else does, and we're all just SHEEP.  They just SUCK.

I feel so much better now...

Later. 

The situation I described in the last post was true.  I put the stuff on the disk into the Win 7 desktop, erased the disk, and burned it back from there.  It should read right on the laptop now.  A curious thing happened when I tried to shut it down after making yesterday's post, however.  It literally never got done logging off.  I had to put it in sleep mode, and brought it home, and opened it, and it was still trying to log off.  Slept it again, and took out the battery.  Started it up on wire in safe mode, and looked at bizarre messages (like one "critical", as opposed to "error", or "warning") in event viewer before doing a system restore. That got rid of 2 MS updates, and nothing else, and then it seems now to be working O.K.

Hmmmmm...

So the pix below begin from the last pic in the last post, M16a.  It has a ring shape more pronounced due to higher degree and a slight tweak in the usual formula (for formula structure, see post 203), leaving the last subtraction of d (=1, in this case) out before the addition of the multiplicative inverse, which gives the multiply connected topology.

The journey begins near the very large M4 mini, in one of its valleys.  Notice there the dendrite system that daisy-chains smaller minis together, following the chasm all the way in.

Further next time (if there IS a next time...)

Later!

 

I've been following this thread for a while, it's been interesting, especially with regard to different power Mandelbrots appearing in the same fractal. I've noticed that the standard Mandelbrot shape keeps turning up sometimes misshapen sometimes not, lately I've been finding multibrots along with the Mandelbrots. Here is an example using a formula I've called Double 3F Onion M, onion refers to the fact that the function calls are nested so the brackets resemble the layers of an onion, 3F for 3 functions in each onion, double for two onions and M to signify that the onions are multiplied together (I have an other where D signifies division).

z = alpha*f1(f2(f3(beta*z + gamma)))*f4(f5(f6(delta*z + epsilon))) + c

alpha to epsilon are all complex parameters and are all set to 1, c is the location in the complex plane and the initial value of z is set to c
f1 = sqr
f2 = tan
f3 = cube
f4 = sqr
f5 = sin
f6 = tanh

(http://fc00.deviantart.net/fs70/i/2012/129/f/a/three_islands_by_element90-d4z1krt.jpg)

Obviously, the set is not connected, at the edge of the large island at the compass points multibrots are found, this one is at the eastern edge edge.

(http://fc06.deviantart.net/fs70/i/2012/129/4/3/at_the_edge_by_element90-d4z2okt.jpg)

@fracmonk:
I think you have done a lot. Now, it is our responsibility to use your whole bunch of formulas so that they (and you) get noticed.
I think a big video in a M842 connected set would be SO awesome............

element90-  You're realllly deep into assigning functions, I see.  I'm curious whether what you found was accidental or on purpose.  Sometimes you find yourself in a situation that can be hard to describe to the uninitiated.  Either case may have implications about (let's call it) "natural occurrence".

t-t-  Thanks, you're very kind.  I too have a fondness for the degree 8 inclusive, since it marks the extent that many of the most popular constants are operative, but ANY doubling can have interesting things in it, and I just happen to be looking at one of the M16's at the moment.  You can see that they have behaviors in common that I'd love to see generalized coherently.  It's the "coherently" part that's tough...

The zoom series I began last time continues below.  We go between 2 M4 lobes into another "daisy chain" dendrite, and into an edgy spot on a M8 mini...

('til next time)

Later.

Generally speaking I've no idea what I'll get from any new formula. I'll take an idea regarding a formula and extend it and add extra parameters just to see what will happen. I'm currently developing version 3 of my software which will change the way it deals with formulae, a formula may have a number of parameters that have values that are fixed and a value that is the location in the complex plane (I call this value c), for version 3 a formula will have a number of parameters any of which can be substituted with c, so the basic Mandelbrot formula will be used for both the Mandelbrot and its Julia. This will reduce the number of formulae defined but will increase the number of possibilities, taking Double 3F Onion M formula as an example the number of variations due to parameter substitution will be 64.

element90-  Keep us posted, o.k.?

fracmonk

Will do. I announce releases here in the "announcements & news" section. If I find anything fruitful I might even start a new thread.

One of the main features or Saturn and Titan is the ability to orbit plot all the fractal types except Lyapunov. The orbits can be plotted as escape orbits only, captive orbit only and all orbits with the option to omit a number of points at the start of an orbit. So far I've discovered that Julias produce poor orbit plot images and fractals using the Mandelbrot algorithm (both divergent or convergent) produce interest images, I'm mainly concentrating on captive orbit plots and all orbit plots as the escape plots (Buddhabrots) haven't been that successful. There is an exception to this general rule however and that is that Pickover Popcorn (I extended the formula with an extra parameter, the seed for the Julia or location in the complex plane for Mandebrot) produces good orbit plots for Julias but poor plots for the Mandelbrot form, both produce forms nice gnarls when treated as ordinary escape time fractals (although often the bailout is set to none).

I've drifted off topic some what.

I'll leave you with a variation that produces a seriously altered "Mandelbrot". I have posted this before elsewhere on this forum but you may not have seen it. Since the the image is held on a different site I'm not wasting the forums valuable disc space.

(http://fc07.deviantart.net/fs71/i/2012/107/4/f/altered_by_element90-d4weax6.jpg)

Parameters

Formula: Combination 4

z = transform(z)
z = (alpha*(z^beta) + c)^gamma + (delta*(z^epsilon) + c)^zeta

alpha = 0.25 + 0i
beta = 2 + 0i
gamma = 1 + 0i
delta = 1 + 0i
epsilon = 2 + 0i
zeta = -2 + 0i

Programs: saturn and titan
Number of A transforms: 0
Number of B transforms: 0
Transform sequence: A
Number of Complex Plane transforms: 0
Initial values of z: 0 + 0i
Image centre: -2.208 + 0.128i
Image width: 15.008
Rotation about image centre: 0 degrees
Maximum iterations: 600
Bailout: norm(z) > 160
Colour selection: outer for bailout otherwise inner
Outer colouring: iteration
Inner colouring: fixed colour

The set is clearly disconnected, on the needle we get this:

(http://fc02.deviantart.net/fs71/i/2012/124/b/6/mandelbrots_and_rings_by_element90-d4yhqf2.jpg)

element90-  Your pix are along the lines of what I spoke about in post 210, but maybe more esoterically (dis)organized.  Excuse my very subjective, purely personally perceptive take on the visual field there.

Clearly, your choices are not entirely random, and you've developed what looks like a fun toy.  What I said before relates to users understanding how it works enough to get pleasing results from it.  I vaguely recall doing pix a bit like the second one in fractint, again using multiplicative inverses.

Keep going!


My own offering for this time reaches the destination coordinates and shows the Julia set for them.  One more shows the center of the Julia magnified.  Notice the smaller J4 minis in the periphery surrounding the center J8.

I plan to show a few more views of this Julia to show correspondence between it and the index set, but I don't know when I'll manage it, exactly.

Life is often, well, um, chaotic...

Later.

The nonzero critical point z0=2 is at the centers of all the pix below, and the next bunch to come, most likely.  They start with mag 1 and increase by 10x each, just as the index set had.  It's not hard to see the usual similarities in local detail as magnification increases in both.  Notice the abundance here of small j4's, and also the general shape of the daisy-chains of minis in general.

Later.

The zoom series into the Julia is completed below.  Since the coordinates in the index set were within an M8 feature, this winds up in a J8 feature at the critical point z=2.
Next time, a look at the hybrid J2, if I can find it...if you recall, when the coords are in a higher-power feature, then all Julia minis of lower power than the coords one are hybrid, that is, never as found in a normal multibrot, but heavier at their extremities.  A J2 then would be generally barbell-like, unlike those found in Julias for coords in standard M.

To be honest, I think the Julias for standard M are very pleasingly proportioned in comparison.  I don't know whether I perceive them that way due to familiarity (that I have come to expect them the way they are) or if it's something completely unconscious that governs that sense.

Comments on that?

Later.

...so let's start this post over.  I got an error message saying that it exceeded 256K.  I don't know if that meant 1 pic, or all.  So if you got less than intended, don't complain to me.

The first 2 were explained in the last post, and the next two are views of J4 minis, for which the same observations about nonstandard features apply as well.

(After trying this and that, I found it would not take the last pic, a larger view of the center of the 3rd one.  Such is life.  I have decided to try not to fight chaos anymore, if I can help it.  It's a natural force.  Altruism is not altruism, by the way, if one feels forced to do it, whether by coersion, compulsion, or an externally imposed sense of duty.  It must be a free choice.)

There's some really neat stuff to be found in these objects beyond 16-digit floating point.  I feel like I'm always only scratching the surface.  Wouldn't it be great if someone felt charitable enough to allow us to go beyond that limit for "formula" in Fractint?

Later.

Here's the pic that was rejected yesterday.

But there's MORE!

Finally, I tried something I used to do routinely in formulae.  Yes, squaring or cubing the constant c also doubles or triples the symmetries in these sets as well.  Great way to get snowflakes, for instance, if you like snowflakes.  If you don't then never mind...

Anyway, the second pic shows a degree 2 with c cubed, and the last 2 are degree 4 variations.

I've been exploring near the origin in the very last one.  Most prisoner points on the real axis are in 4th power shapes, while all (?-looks that way) on the imag. axis are M2.

Later!

As suggested in last post, here's some pix from the last index set posted there.

The 1st one has the coordinates of the index set dead ctr.  The second is a rather lacy whole Julia for that spot.  In this one, all the minis are found at narrow spots in the filigree.  The 3rd greatly magnifies z=2, a critical point, where a J4 is found (corresponding to the feature local to the index set coords).  The last shows the "barbell" J2. (This would have been the case, but again, the server/program rejected the pic, which was well within stated criteria.)

These phenomena show up again and again, and are characteristic; the entire behavior, I'm starting to grasp, is that of how lower powers are locally mimicked within higher power behaviors.  It's easier to see than to describe...

Have a good weekend and then some.

Later.

Quite a long time ago, I became curious about the theory that all the minis and dendritic structures in the M-set were composed of miniatures of M.  With the minis, this is obvious, but the theory goes on to say that assuming one could zoom in as deeply as one desires, any dendritic structure would be shown to be composed of minis also, just really tiny ones.

If this is so, then what about structures including multiple powers?

Any thoughts on that?

Later.

Hey, I've been away from the Internet since last post here, practically, and I bet I'm happier than you!
I don't imagine anyone missed me all that much...if you are interested at all in this thread, please let me know, so that way, I can either pay more attention myself, or forget about it...

Playing with the formula structure for multiply connected forms of the Multipowerbrot type,
I came by an elegant spinoff.  I cut it out of a now enormous formula file that I use, and
it contains my sort-of-shorthanded notes, which help me keep the many permutations, how to
use them, and what they do straight...

M4La(xyaxis) {;deg 4,M2 only MC
  c=pixel, z=p1, d=p2:
    s=z*z-d     ;d=1:z0=-2,2:2sym
    t=(s*s-d)*c ;d=2:z0=-2,0,2:8sym
    z=t+(1/t)
    |z| < p3
}

Notice its brevity.  It yields the 1st 2 pix below.  The last 2 pix are subsequent doublings
of degree, to 8 and 16, respectively, obtained by repeating the squaring and subtraction of
d as in line 3.  Alas, they have only the classic M2 shape in them, and so, they are not in
the Multipowerbrot family.  For degree 2, not shown, the calculating portion would be simply:

    t=(z*z-d)*c
    z=t+(1/t)

For each degree, the number of equally largest M2 figures is double that degree.

Also, I had mentioned before that I always use the same palette without rotation.  It creates
a standard for comparison for everything I show here.  For theoretical consideration, I think
it creates less confusion.  I hope that's not tiring!

Later.   

MM4Lad2.GIF is pretty interesting.

Asdam-   Yeah, and what I was trying to point out is what you get from the formula structure, no matter what the degree doubling.  Nice, symmetrical rings...

Looking at Im d in some of the formula structures, where it works.  You can get these odd blobs, with Julia-like shapes, assymmetrical, linked with little M2's, but also apparently connected (when you're lucky!)

Maybe I can get some pix together later.

And

Later.

Now all folks went to colour method based formulas;)

Asdam-    Your last message implies what exactly?  Is it that a consequence is horrendously (and unnecessarily) complicated coding that's no fun at all?  Just guessing...

Here's the fractint formula for the degree 4 version of what I spoke of in my last post.  If d=i or -i and your init z=2 or -2, you will get an index set object like pic 1 below.

M4L {;deg 4
  c=pixel, z=p1, d=p2:
    s=z*z-d   ;d=-i,i:z0=-2,2,more?
    t=s*s*c-d ;M2+J-like features,MC
    z=t+(1/t) ;d=1:z0=-2,-1,0,1,2 NC
    |z| < p3  ;highly distorted
}

By repeating the 3rd line (1st line of calculation), that squares and subtracts d, the degree is doubled, resulting in the 2nd and 3rd pix, degree 8 and 16, respectively.  Notice how an off-origin julia-like symmetry begins to emerge in higher degree.

Leaving that line out altogether gives a degree 2 version, which actually then can have y-axis symmetry, as shown in the last pic.

This formula structure variation will yield another totally different shape that is most likely *not* connected (NC), with very distorted sausage-like lobes and broken minis, when d=1 is used.  See formula note above.  The effect becomes deceptively more regular in higher degrees.  I include that note for those who like such Frankenstein's monsters that require minimal effort.  (Generate & post yourself if you like...)





In the U.S., Independence Day is celebrated tomorrow.  Enjoy!

Translation from modern Orwellian:

DO NOT USE THOSE ILLEGAL FIREWORKS ON DEPENDENCE DAY.  Turn them in, along with your neighbors and family members, with pictures taken with your cell phone.  Use an online substitute fireworks show instead.  Don't drink and drive.  Have another barbequed soyburger.  They're healthier for our profit margins.  Take a pill (prescription only, it's O.K. then).
Obey all rules of the kinder, friendlier international totalitarian police state. 
Become whatever your TV is showing you.  Never ask why we show or tell you anything.  If you don't think there are already too many laws, imagine new ones, and obey those, just in case. 
Only with your outstanding cooperation can we watch you so closely.  Buy our latest iSuck product.
Things are getting better all the time.  We won't eventually murder you.  We promise.
 :)Have a nice day. :)


Later.

_________________________________________________________________________________________________________________


Diagnosis Criteria : Compositionality : regarding Orwellian eye to brain to subject text civilization satire perfect : Sound of your verbal articulated voices are more than good, excellent choice of wordings, clear and so in tuned with the prevailing farceties : tok = Theory of knowledge mood achieved excellent : verbal descriptions constructs : Un reserved : Un-restrained achieve goal of telling it as it is! : Dynamic style coming from the hippy dual perspectives juxtaposed realistically : professional seconds into [ text at  police state ] Excellent pause as we enjoy reading what you have to say poignantly to they max : the truth hurts, your word are biting truthful : Thanks do more!

I meant, UF colour methods have hard time with some of these formulas;) Multiwave coloring for Mandelbrot by Pauldelbrot works well, but some orbit trap based methods not so well.

Asdam-  I suspected something like that-  -it's one reason why I stick to fractint.  I can communicate the basics as easily as I can think up a formula and test it.  No muss, no fuss.  Gets ideas out quick & easy.

weavers-  My first thought was: "Uh huh..."  -a critique on delivery?  How about where YOU stand?  You know, on the subject matter...



and hey, how come no one ever tries MY questions on for size?  No seriously, check the history...

Later.

I had a nice, moderately-sized post, with a formula, some small pix, and good explanation of it all, went to post it, and was told the upload box was full.  I left out the pix then, and tried to post it, and then was told I already did.  If I did, then it should be there.  Do YOU see it?  Didn't think so.  Maybe next time...

I've been looking at AI and the techno singularity lately.  There's nothing to worry about, apparently...

Later.

Just kidding-  human deception, corruption, arrogance, and incompetence in
the name of control, and the pathological competitiveness it breeds and
excuses, will create a technological singularity with a character that
will make the Unabomber look like a saint, in retrospect...there's a bunch
to worry about!

What follows here is what I tried to put up last time, with a little
updating:


Here's an adjustable (by d) function I've been looking at since last week
that exhibits a *very strong tendency* toward connectedness in the c
parameter plane.  Only the M2 type (standard M shape) is yielded by it,
everywhere in it, in the examples below.  Adding the multiplicative inverse
in this (any many other) function(s) makes the connectedness (when
available, as here) multiple.  I've wanted to further explore that
(previously mentioned) idea of *cohesiveness* in certain functions not
found in others.  I have this feeling that that distinction will one day
become very important...

M4F(xyaxis) {;deg 4 for Re d only!
  c=pixel, z=p1, d=p2:;M2 features,MC
    s=z+(1/z)   ;d=1:z0=-1,1:2sym
    t=s*s-d     ;d=2:z0=-1,1,-i,i:4sym
    z=t*t*c     ;d=-2:z0=-1,1:2sym
    |z| < p3    ;d=-1:z0=-1,1:2sym
}               ;d=0:z0=-1,1:2sym

M4Fi(origin) {;deg 4 for Im + some cmplx d
  c=pixel, z=p1, d=p2:;only; M2 features,MC
    s=z+(1/z)   ;d=-i,i:z0=-1,1
    t=s*s-d     ;d=-2i,2i:z0=-1,1
    z=t*t*c     ;d=-3i,3i:z0=-1,1
    |z| < p3    ;d=1+i,2+2i,etc:z0=-1,1
}

;The only difference between these is the timesaver for symmetry type.
;The second version will work for all d listed in the comments in both. 
;Shown below as examples are maps of sets for Re d for now, as labeled. 
;(A negative d value subtracted in the code is actually positive then.) 
;Others next time.

;For Julia sets, the following *should* work in all cases, but to be sure,
;remove the (origin) spec:

M4FJ(origin) {;deg 4
  c=p1, z=pixel, d=p2:;M2 features,MC
    s=z+(1/z)   ;d=1:z0=1:2sym
    t=s*s-d     ;d=2:z0=-1,1,-i,i:4sym
    z=t*t*c     ;d=-2:z0=-1,1:2sym
    |z| < p3    ;d=-1:z0=-1,1:2sym
}               ;d=0:z0=-1,1:2sym (etc)

Notice that all examples offered so far involve whole numbers, and no
values exceed 2.  With d values >2, and fractional d values, index sets
like those we've seen before here in this thread, that have fractured and
distorted M-like fragments and localized julia-like features appear. 
Often, as in other functions shown here before, they appear as if they
might be connected, but one would have to look too deep trying to prove
instances that are obviously not connected.  In other instances, lack of
connectedness on the specified 2d plane is obvious.

In my experience, it is easier to find connectedness in an object shown in
more dimensions rather than less.  There is no question that a 4d index
set of this function for the c by d complex parameters would be connected,
but probably not finite.  I have not yet looked far enough into that
property.  Fractional d values that are real (and pure Im as well) should
yield a connected 3d object that is also probably not finite, from what I
have seen.  In that, after stacking parallel 2d slices at tiny incremental
distances from each other, observation could reasonably suggest continuity.
In other words, various combinations of c and d complex parameters could be
connected in 3d, but would require that many dimensions to be so.



Judging by recent comments, I would renew my call for a version of fractint,
or another workalike program with its basic map design concepts, that can
display true 3d in the formula parser without resorting to trig. 
Ridiculously fast processor speeds now take away the (what? 20-year-old)
argument that it would take too long to calculate.  Currently, it can
*calculate* in as many dimensions as you can keep straight mathematically,
but can only display 2 of these dimensions at once.  An assignable 3d pixel,
along with a simple scheme to color-code proximity in a depth dimension,
would really be a great thing...

Ambitious showoff programmers Wanted.  Inquire Within.

Later. 

I don't know if it's just me or the equipment I've been using, but I have fidelity issues with the way the pix appear here.  Rather than a faithful pixel-by-pixel display, they look like they are being "interpreted", as by the (really crummy) schemes that MS uses in the picture display programs they incorporate in their recent OS versions.  Some spots look smeary or out-of-focus, not sharp.  I truly miss MS Photo Editor, which once came with XP.

Is there a reason for that?  A fix?

Comments from others?

Later.

Here are more pix from that adjustable function.  The first is for d=-2,
significantly different from d=-1 in the first report.

The next 2 are for pure imaginary values for d: d=i, d=2i.

I find the last most interesting, since it has a 4sym, but turned 45 degrees.
That's d=2+2i, and worth comparison with d=2 in the last report.  Its
traceries have a certain grace to them, and the entire figure seems to be
made exclusively of the M2 form. 

It can be argued that for each d value, you are really dealing with a
different function.  I was concerned here with the similarity of the
structure of the formula from one to the next, and how there can be both
profound similarities and profound differences simultaneously when comparing
the figures that result.

What are the numbers trying to tell us?

Bear in mind that we are dealing with on-off, ones-and-zeroes, dichotometric
Boolean math at a more fundamental level here.  That most of the functions in
this thread, where the most interesting things seem to happen, are dependent
on zero, one, two, and powers of two seems poignant to me.

Dream big, dream small, but dream!
 

Later.

Spent more time driving than sleeping in last 3 days.  Saw Furthur @ Bethel Woods.  Almost Dead.

Later.

hmmm... not much time to wonder about tapestries behind bands, but... Dead Heads on the Bus?
whew!!

Apophyster-

Sadly, the stage background had little to do with fractals.  Sorry, it's off-topic, them voices tell me so...

Don't worry 'bout me...

My apology that I go OT.  You and so many others are far advanced in math beyond me.
So I gawk at all of your experiments, yet oh... I'm such a numb skull.  ::)

Apophyster-  [Nah, nah,] you misunderstood!  It was ME going off topic, and making a little fun of myself at the same time.  The stuff I do here is simpler than you think, [when you look at it right].  Anything I can make clearer somehow, just let me know.  Does that skull have one eye, two, or three?

(OK, enough references to Dead culture and lyrics...I get carried away sometimes...)

Later.

Two and a bolt.  Been a while since I checked the bolt for new eyes.
There are continual new representations..  :tongue1:
Personally, IMHO you may feel free to carry sometimes on, anytime on then.  I'm on Sugar Magnolia just now. 
Or just ending.
Since '67 iym dead to the bone, through the breast, and all 'round m' heart!
 :) Friendly Fredley

A-

Then you understand, and got it all.  See?  No numb skull after all.

Funny, I considered a defense of GD, if one was needed, but left it out of here days ago.  
They have origins in CIA LSD experiments, but look at this:
You have to make your way in a violent, desperate, and often merciless society, and you come to realize deep down in your heart that you're not gonna change it much in any meaningful way.  Information control limits influence.  What do you do?
As poets, they do music.  Another kind of poet does math.

You keep your fingers crossed that it won't wind up hurting anyone.

Or you make movies that make the rest of us more careful about what we wish for.

Later.

As far as the CIA category is concerned, it helped get Larry Ellison and Oracle going.
So however the spy guys interjected themselves into the band and other facilities and people in the BA and elsewhere, may not be so unsettling as what they're doing in the IT field.

I read very little about that type of thing though.  People may have dug up more info since I've read aything about the subject.

If there's another time around I hope the band and all the asscoiated troupes will be there to collectively turn on the love light again.

A-

Exactly.  Here came Oracle, there went your freedom.  Everything needs a critical eye.  Even the GD.  I have no major complaints, though.  They've gotten me through a lot, even if I came out of it a little crispy.

As far as turning on the love light, that rests with each of us, and it's in what we do.

ALL of it together, not just this or that part-time cause.  It's a lifetime of research to figure out how to avoid working for the bad guys and also avoid winding up...well...dead!



Sorry, no pix at the present time.  Working on variations on the formula that gave squaring-effects-without-squaring, seen very early in this thread, the first bunch of pix, in fact.  Designated it M1La, the variation that gives me all the current puzzlement.  I should add that they are remarkably slow, or they reveal nothing.

Later.

The formula variation for squaring effects without squaring I
mentioned last time:

M1La(xaxis) {;deg 1, Re d M2 features
  c=pixel, z=p1, d=p2:;z0=-2,2 seem
    t=(z-d)*c   ;to be critical
    z=t+(1/t)   ;d=2:z0=-2 notable
    |z| < p3
}

The very first pic I posted in page 1 of this thread can be
duplicated with this variation of its original formula by using
init z=2 with d=0, and gives a double M-set inside the unit
circle, arguably a lambda form, mapped there.

The first pic is yielded by d=2 when init z=-2.  I have not seen
a SINGLE M-set shape packed inside the unit circle like this
before.  In both cases, the main bulb edge maps to the unit
circle, the first one twice, and this one once.



YOU DO NOT SEE THE PIX BELOW, since they were rejected because of size, all having lesser storage requirements than the one you do see.  Rules are rules...nothing over 256K(?) allowed, but they weren't, and the site said they were.  Who am I to argue?  I'm only human...

...and I have better things to do than to drive miles to a computer to have this awesome dude bot treat me this way!  I'm thinking about having nothing to do with it any more.  (That may make you HAPPY!)

Different values for d cause an emerging M-set shape outside the
unit circle.  This one (pic 2) for z=2, d=3 is pretty elegant.

Another example is seen in the next pic, z=2,d=2.5, in which
there is complete separation, but there are still residual
dendrites that are not a part of it outside the unit circle.

If you study the formula, there are at least 2 combinations of
init z and d values that will result in division by zero: z=2,
d=2, and z=-2,d=-2.  The closer you get to d=2, the further away
the exterior M-set gets away from the unit circle.  The last
pic, z=2, d=2.05, shows the emergent M well distant from the
unit circle using a d value arbitrarily close to 2, which makes
one rethink pic 1 as a whole set, which may have its "nemesis"
at infinity.

(...but there's More!)

Later.

Why don't you share Fractint formula file.

Asdam-  Nice to see you here again.  It's right there in my last post, at top.  Don't know how you could miss it.  It's for fractint formula.



This is my Olympic Semiathlon event where half is done on one day,
the other half now, and I care half as much and try half as hard.
Since it is not an official Olympic category yet, it is outside
the jurisdiction of the IOC.  The IOC should know I am not trying
in any way to exploit or damage their "brand", which they might
want to burn into EVERYONE's backside, apparently, without
exception.  I am not criticizing them, which is also probably
illegal by now.  In fact, I want nothing to do with them, and
let's see how that works out.

Here are (with any luck) the pix that didn't stick last time, 2,
3, and 4.  For explanation, see my last post notes (after the
complaint).

And again, there's more, but it will have to wait until next time.

Later.

I was thinking about formula file in attachment. Most of Fractint formula files are pretty simple text files with lots of formulas.

This thread is inspiring, there follows a picture using

(https://copy.com/F3qIcaSiJBi3)

For version 3.0 (eventually ...) the formula will be altered to

z = alpha(beta*z^gamma + delta*z^epsilon)

when any of the complex parameters alpha to epsilon can be substituted with c i.e. the location in the complex plane.

(http://fc08.deviantart.net/fs70/i/2012/211/f/3/chained_mandelbrot_4_by_element90-d594tgy.jpg)

This is from the blunt end of one of the Mandelbrots in the fractal generated using that formula. The "chains" and "beads" attached to the more normal Mandelbrot structures are of course missing from the standard Mandelbrot in a similar relative location.

Keep up the good work it is appreciated.  

Just so you know, I was able to view the images!
Lurker the Fred E

Apophyster- Yeah, when the bot finally let me put them up...

Element90- Not familiar with your generator, [but that's a bit misleading, since cubing exponents would never produce that pic.]  Also, exponents have to be integer, to be clearer.

element90-  I'm sorry, I was dead wrong about that [in brackets just above]. 
I translated it to fractint formula as soon as I got back:

e90cubinv {;sample test
  z=p1, c=pixel:;z0=-1,1
    t=z*z*z
    z=(t+(1/t))*c
    |z| < p3
}

-and from it, I got a symmetrical ring pic not unlike those
found in post 229, all M2, which I'll post later, if you like, having
already reached today's limit.  It has a 6-sym.  I totally
CRINGE when I do stuff like that.
 
Sorry!

I felt so bad about that that I made a special trip back to
amend this!  Please forgive my intolerable error.



Asdam- I purposely give only a few, so that people DIY using the principles given.  It's the only way they'll move to understand what's going on with this stuff.


Part 2:

1+sqrt(2) (more or less) is the center of the 1st pic, where the
external M2 and a residual antenna extending from the unit
circle last touch before separating when the d value assignment
exceeds 1.828427124746189... on the way to 2.  I stared at this
number for awhile before I saw the pattern in its digits; it's
2*sqrt(2)-1 (more or less...)

This number was refined in precision by looking for a cross in
the second pic (greatly enlarged) while keeping either the
intersection resulting in the cross (d too small), or a gap (d
too large) in the center of the pic, which also has 1+sqrt(2)
at its center (more or less) as a repelling point.  In this
version, the external M is still attached to the unit circle (d
too small, but not by much).  It doesn't lend itself to
exactitude, since both d and the center of these pix are based
on the same trancendental number.

Notice in the unit circle (1st pic left) the similarity of the
contents there with the 1st pic in post 250(?).  You can't miss
it, it's the only one there, unfortunately.

When the d value enlarges, as in the third pic, there was promise
of mapping the cardioid to the unit circle, but probably can't
happen until d reaches infinity (so...No...).  In fact, the set
would then BECOME the unit circle, but there are easier ways to
get one, like the Julia set for f(z)=z^2+0.

Doesn't the second bulb there (last pic) look a little odd, though?
Check where the branching is, nearer the ends of the dendrites than
in standard M.  The behavior is at once similar and different
compared with standard M, once again.

Later.

While I'm here, let me lay this on you: (preprepared)

The reason why I had returned to this area of my study was inspired
by a theory about information falling into black holes winding up
accumulating on the surface (or event horizon) of them in 2d form. 
Then, there are entwined particle theories, and symmetry-breaking,
to which the "nemesis" effect might relate.  But you must understand
that I have a slightly smaller budget than CERN devoted to physics
fairy stories.  What can you really find out about a bearing ball
by throwing thousands of basketballs at it, only to see them turn
into smaller balls, assuming your elaborate detection equipment is
not lying to you, and telling you what you want to hear, so you can
add a few more canards to an already inelegant theory?  Keep it
alive when it really wants to die, but you've invested too much
reputation and funding into it?  (And dispose of billions of dollars'
and euros' worth of industrial capacity that could have otherwise
improved the lot of millions of people living today?  A better method
for destroying the world?) 
O.K., I'm a little cranky right now...

So I might as well say here that this is just a small sample of the
curiosities available in multiplicative inverses and their likeness
to squaring when observed as fractals in the complex realm.  There's
more to be found, and not enough time in my life to do it all.

The same could be said for Multipowerbrots, which are algebraically
related.  It is like finding new universes; except it is merely a part
of this one, a fragment of its DNA, if you will, except more
picturesque than chemical DNA.  That is, if you go with the theory
that mathematics is the DNA of the design of the universe, then it is
like that. 
Junk DNA of the universe?

The reason why I decided to begin sharing all this stuff when I did
was the effects of age.  I had begun worrying how long I would have
the ability to do it.  It takes intermediate skill, which is a
no-man's land in math, where you are either a beginner or halfway to
Gauss, commanding the arcane notation and nomenclature that preserves
a priesthood from newcomers, who seem to get lost there, as I do. 
Sadly, there doesn't appear to be any good bridge between the two. 
And then, when you do get there, would you write a 15-page proof for
2+2, or try to sell a math that doesn't quite follow existing rules?
At least I know this is not that.

Honestly, I expected more interest, but we're not in the '90's any
more.  While there are more people acquainted with fractals today
than ever before, the interest is more casual than at that time and
lacks the enthusiasm for depth of understanding that existed before. 

Maybe the scientific community overreached in places when it came to
the application of chaos, but in many cases, it found paydirt.  I am
very likely overreaching here myself, but without throwing questions
out there, there is zero probability of answers, as opposed to small
but nonzero chances.  All this stuff could easily have died with me
otherwise, and may anyway.  I was only lucky enough to stumble into
some pretty interesting things.  Why did they appear so, and what do
they mean?  I see no direct line between this stuff and some future
application, and therefore, almighty money, but, fool that I am, I
believe that some things are beautiful in themselves, and worth
appreciating for that alone.

If you do not seek, you will not find.

Later.

Regarding post #58: imo, well said.

It wouldb be more ease to look by yourself if formula file would had been present;)

Asdam-  My formula files are a work in progress; there are too many unresolved questions in them, and untested conditions.  In 3 basic volumes, there are simply connected, multiply connected, and totally experimental kinds.  It would cause too much confusion.  There is adequate info here to see the patterns among them that make them work.  (so stop whining!)

Just kidding.  You can do it.




Thought for the day-   When a value is really large, its multiplicative inverse may go unnoticed...

Later.

In my earlier post I used the the following formula:

z = c(0.5*z^3 + 0.5*z^(-3))

where c is the location in the complex plane and the initial value of z is also the location in the complex plane.

The implementation of the formula allows for non-integer and non-integer complex powers just to add extra flexibility as is the use of separate power parameters instead of using one and using two terms with the same power where one is the inverse of the other.

The result is a ring of 6 Mandelbrots linked together in a similar way to an image already posted by you. The use of the location in the complex plane is unsatisfactory as determining the factors to adjust z^n and z^-n to get well formed Mandelbrots is tricky. This formula only appears to produce Mandelbrot islands and I'm inclined to believe that multibrots won't appear at all.

It is far easier to use the critical values as the start value adjusting the factors to get a a critical value of 1.

So using z = c(z^3 + (z^(-3)) and z0 the resulting picture is as follows:

(https://copy.com/QHrz7gv8PtCT)

Comparing this image with that for the previous formula using the initial value of the location in the complex plane the only difference I've noticed is the size.

I then wondered what would happen if I used 2 and -4 as the powers again I got 6 Mandelbrots in a ring but the connections between them was different.

z = c(2z^2 + z^(-4))

(https://copy.com/V3vrnkeF2xcF)

So what of z = c(5z + z^(-5)) would there be a ring of 6 Mandelbrot? Well, No:

(https://copy.com/xRJHlgehEq6Pi)

The limit used for this one is norm(z) > 16000000 instead of the usual norm(z) > 16.

The images of powers 2 & -4 and -2 & 4 are only slightly different, so what of z = c(5z^(-1) + z^5)? The difference is anything but slight:

(https://copy.com/tEY0ior13TKT)

So far, ignoring signs the powers have all added up to 6 resulting in 6 Mandelbrots, 6 buds or 6 lobes in terms of the largest features. So what happens if the powers have the same sign and allow add up to 6, will the result also have a six of the most prominent features? The answer is no.

For z = c(z^5 - 5z) the answer is 4 major lobes:

(https://copy.com/cQtpJJgEq3G1)

For z = c(2z^2 - z^4) the answer is 2 Mandelbrots:

(https://copy.com/4omx2zQD6sjH)

Now for powers that are both negative, for z = c(2z^(-2) - z^(-4)) the answer is two distorted Mandelbrots:

(https://copy.com/9JjdzbzgTGzz)

And finally z = c(5z^(-1) - z^(-5)), the answer is a group of four.

(https://copy.com/tZDQlizAcDne)

The final picture does contain properly formed Mandelbrot islands which can be found easily by zooming in.

After much experimentation there appears to be a rule that the number of the most prominent features increases by 1 as the sum of the powers (ignoring signs) is increased by one. For the initial example of the ring of 6 Mandelbrots, the minimum power, whether positive or negative, is two, so the smallest ring of Mandelbrots is 4. I'm not sure about the number of variations in the construction of the ring, there can only be one version of the 4 Mandelbrot ring, there doesn't seem to be any difference between the two possible variations of the of the 5 Mandelbrot ring but there are at least two different versions of the 6 Mandelbrot ring.

element90-   ABSOLUTELY SUPERB STUFF!  I'm especially intrigued by the second-to last one.  Will give them a serious look.
You should know, however, that they're not printing right, at least for me where I am.

Later.

Here are more:

z = c(-3z^(-3) + z^(-9)), z0 = 1, c = location in the complex plane

(https://copy.com/HxJOteU0kGrT)

Now for non-integer powers.

z = c(z^2.5 + z^(-2.5)), z0 = 1, c = location in the complex plane

(https://copy.com/3HrCt9uz2TRy)

And the first attempt a throwing imaginary numbers into the mix.

z = c(z^(2i) + z^(-2)), z0 = 0.623832938258 - 0.258400063681i, c = location in the complex plane

Fortunately my calculator has complex numbers so working out the critical value was straight forward.

(https://copy.com/dEQ4DzzRHZSb)

The picture produced using an imaginary power is similar to those produced using the compasses formula with imaginary powers. Like compasses the image is sensitive to the bailout value, for the above image I used norm(z) > 1.6e8.

I've started to look at using imaginary powers for both terms and they are even more sensitive to the bailout value and zooming out is required before anything can be seen. The width in the complex plane of the picture above is around 0.84 for a fractal with with powers of 2i and -2i and a bailout of norm(z) > 1.6e15 the width required is about 800000. Experimenting I found that size of the image was affected by the size of the bailout value, for a bailout of norm(z) > 1.6e9 the width required is about 2000.

I'll show pictures using imaginary powers for both terms in my next post and in the meantime I'll move onto complex powers.

I had to take a close look at element90's function that makes
those interesting wide loops in the dendrites.  I had run
across similar shapes in my experiments, but never got them
to close like that.  Another thing I hadn't realized was that
functions containing only negative exponents (or, restated,
having z only in denominator expressions) could escape.  It
never occurred to me!  Is it still considered degree 4?  My
Fractint version looks like this:

e90circs(xyaxis) {;z0=+,-sqrt(2)
  z=sqrt(2), c=pixel:;no
    s=z*z           ;periodicity
    t=s*s           ;testing!
    z=(1/s-(1/t))*c
    |z| < p3
}

I tend to favor functions that can be programmed so
efficiently.  This because of faith in Ockham's razor, which
usually suggests that simpler explanations are likelier to be
true, but I suspect that natural phenomena tend to follow
simpler fractal rules as well, perhaps as a related
consequence.  It is why that kind interests me more than more
complicated functions.  Consider the logistic equation if you
need an example.  (Off topic:  As I recall, I think it was
Bob Weir at a Dead concert some 25 years ago or more, while
tuning up, saying: "...  Is it not so?  Is it not written in
the sky?" -and it STILL makes me laugh!)

The critical point was a fortunate early guess on my part, and
so, not too terribly hard to find.  Some detailed insight into what
methodology element90 uses to find usable critical points
would probably enlighten us all.  I know I'm not that good at
finding them myself. 

The first pic below shows the whole set, a "large" one, for
one that fits within computation limits for Fractint without
errors and cutoffs, AFAICT.  Ones that do at all are usually
much smaller.

The second is a closeup of a couple bulbs on the main bulb. 
Large and small loops join at key points on the familiar
branches.  They are infinite in number, but to fractal buffs,
that's nothing new...

One thing I have never seen before is dendrites emanating from
the Seahorse valley cusp.  The third pic shows a closeup of it. 
Notice how the loops rejoin the same bulb to the left, while
they bridge between the bulbs to the right.  There's a lot
going on in this object.  In the last pic, zooming in further
on that dendrite main ring in a different location, a mini
there is connected in the same way that the 2 largest M2 shapes
are, at the main bulb's pinch point.  This is apparently common
there.


More about that...later...

To finding the critical points, I just solved the differential of the fractal function set to zero.

i.e. taking the general form

z(n+1) = f(z(n))

I solve

f'(z) = 0

So with

f(z) = c(alpha*z^beta + gamma*z^delta)
f'(z) = c(beta*alpha^(beta-1) + delta*gamma*z^(delta-1))

Using

alpha = 1
beta = 2
gamma = 1
delta = -3

c(2*1*z + (-3)*(1)z^-4) = 0
c*2*z = c*3*z^-4

c cancels out leaving

2z = 3z^-4

cross multiplying gives

z^5 = 1.5

so z = 1.0844717712 to the accuracy available to my calculator (an HP-42S that I've had for some time ...).

It is easier to select the values for alpha and gamma to give a power of z equal to 1 and hence a critical point of 1, you don't have to of course, you can have it equal a negative number, an imaginary number or indeed a complex number and then determine whatever the critical value happens to be, for example z^5 = -1 would yield a critical point of 0.80916994375 + 0.587785252292i according to the HP-42S. The resulting picture will be identical in structure only the size will change.

Since my calculator can to do complex numbers I can "easily" work out the critical points when using imaginary and complex powers. I've only successfully used this technique when the solution to f'(z) = 0 is straight forward.

Fracmonk, it seems that you're having fun with this variation, I haven't explored the variant the variant that much, yet... No pictures today.

I've been side tracked, so no pictures using complex powers.

First of all, I've tried inverting the complex plane which leads to the teardrop shaped inverse Mandelbrot. For the example I'll show first the fractal in the unaltered complex plane:

Using z = c(alpha*z^beta + gamma*z^delta) where zed on the left hand side of = is z(n+1) and the zeds on the right hand side are z(n)

alpha = -3
beta = -2
gamma = 2
delta = -3

giving a critical point of 1 so z0 is 1.

(https://copy.com/g2EYHDmBcE4w)

Saturn has a transforms tab where the complex plane can be transformed so that c = 1/c or c = c^(-1) which modifies the formula so that it becomes:

z = (alpha*z^beta + gamma*z^delta)/c

since c cancels out in the calculation of the critical value 1/c also cancels out so using the same parameters z0 remains at 1.

The inverse version looks like this:

(https://copy.com/MzkwJdQxrOVZ)

In common with most Saturn fractals there a z = transform(z) line applied before fractal formula if a transform or transforms have been defined (also using the transforms tab).

For

alpha = 1
beta = 6
gamma = 1
delta = -6

a 12 Mandelbrot ring would be produced, however using the simplest transform "translation" with a value of 2 the formula becomes:

z = c*(alpha*(z+2)^beta + gamma*(z+2)^delta)

so

f(z) = c*(alpha*(z+2)^beta + gamma*(z+2)^delta)
f'(z) = c*(beta*alpha*(z+2)^(beta-1) + delta*gamma*(z+2)^(delta-1)

so substituting values we get

f'(z) = c*(6*1*(z+2)^5 - 6*1*(z+2)^(-7))

to find the critical value

c*(6*(z+2)^5 - 6*(z+2)^(-7)) = 0

which is

6*c*(z+2)^5 = 6*c*(z+2)^(-7)

or more concisely

(z+2)^12 = 1

since the 12th root of 1 is 1

z+2 = 1

so the critical value, z and hence z0 is -1.

The 12 Mandelbrot ring has fragmented, is disconnected and there are only 6 of the largest Mandelbrots.

(https://copy.com/ykPT3KgXKtfz)
(https://copy.com/eIbi7dbsrLuk)
(https://copy.com/pyjvbD2bwOW6)
(https://copy.com/RlyLwxjoSrfj)

The last two are really interesting! More should be done with those...

element90-

I think I've been hijacked!  Very interesting ride...

With post 265, this may be redundant, but I preprepared
this, and it may show how our thinking somewhat overlaps.  
Bear in mind that I haven't been able to have a look here
since I posted last.  This is how cave men do it:

(And I wish I could take your posts back with me printed,
so I could study them, but I can't from here...it'll have to
wait, but you seem to have things well in hand...)

I'd been busy over the weekend, finding myself
mournfully sans Bunny, following my own advice of DIY,
using trial-and-error, fractint testing, a calculator,
and a lot of good ol' human pattern recognition to
"crack" this problem of finding critical points, and
remember, you crawl before you walk in this life.  

I came up with this:

For functions (from element90) with this structure:

f(z)->c((1/(z^a))-(1/(z^b))),

Where a and b are real integers and a>b, and there are
no coefficients, a critical point (for init z) can be
found using the following formula:

              __________
     (a-b)/ a-b
z0=  \  /  ___ + 1
         \/    b

(Copy+Paste the following into Notepad just as it
is if the version above doesn't space right in your
display.  I anticipated that sort of problem here, but
now I notice that the same applies to formulae I've
given.  They'll look right in old fashioned typewriter
spacing:)

          __________
    (a-b)/ a-b
z0=  \  /  ___ + 1
      \/    b

Or, the verbal explanation:

You use the (a-b)th root; if it's 1, there's no radical;
if it's 2, then it's sqrt; if it's 3, then 3rd root, etc.
Take the (a-b)th root of ((a-b)/b)+1 within the radical
to get z0.  Most yield infinite digit strings and require
more precision for better results.

For example, for a=2, b=1, the answer is 2.  All a with
b=1 result in an infinite index set, like pic 1 below (the
origin is in the light circle in the lower right here).  
It is NOT a contortion of Mandelbrot's set, AFAICT.  Check
the progression of dendrite branchings from bulb to bulb
of comparable size.  Each successive large bulb adds 2,
INCREASING with distance from the cusp.

Since element90 appears to like sixes, consider a=8, b=2.  
When you take the 6th root of 4, (same as 3rd root of 2,
BTW), you get 1.2599211... as the critical point for
pic 2 below.  
If b>=2, the index set is of finite size.

While one could use:

e90circsuniv {;z0=? no periodicity
  z=p1, c=pixel:;   testing!
    z=((1/(z^p2))-(1/(z^p3)))*c
    |z| < 1000000
}

-univerally for functions of this structure, it would be
terribly slow.  For the example above, it would be better
to use:

e90circs82(xyaxis) {;z0=6throot(4)=
  z=p1, c=pixel:;    cuberoot(2)=
    r=z*z           ;1.2599211...          
    s=r*r           ;no
    t=s*s           ;periodicity
    z=(1/t-(1/r))*c ;testing!
    |z| < p3
}

The programming for your generator may have different
requirements, of course, but these fractint examples
should be easy enough to follow.  (Why I stick with it.)

The lowest combination of integer exponents resulting in
a finite sized index set is a=3, b=2, giving pic 3.  Pic
4 shows a completely dendritic Julia set for c=20.25,
the precise distance in which the main loop dendrite
crosses the real axis (Pic 3 right).  Compare the famous
f(z)->z^2+i.  
Good stuff for number theorists.


Later.

Below is the beginning of an endless progression of simply
connected functions inspired by the formula structure
introduced here by element90.  In this case, the exponents
are then positive ones.  The structure goes as follows:

f(z)->((((z^a)-(z^b))c)-1)^2

in which integer a-b=1, and critical points are z0=b/a, so
that the critical points of the below pix, in order, are
1/2, 2/3, 3/4, and 4/5.  It turns out that a-b=1 appears to
be the only case in which simply connected index sets, in
one piece, as those shown, can be formed.  Counting the
stems from the bulbs, the number of branchings on the
largest dendrite to left in each is 2a-2.  They get more
"furry" as the exponents increase.  (So do up, say, a=98,
b=97, with z0=97/98, if you like.  Go ahead, make my
day...)  Sadly, Fractint has a hard time with these, and
errors increase as exponents increase, and the sets get
larger too.  Actually, some set exponent combination could
act as a benchmark test for the accuracy of both hardware
and software calculation schemes that makers claim can
handle such.  It's a killer.  It seems most computers
these days are geared for tying everyone up with twit'er
and fakebook, etc., and not traditional computing,
ironically, (since they still call them computers) fast as
they may be.  

As an example, the formula for the last one shown here is:

DM54(xaxis) {;deg 10
  c=pixel, z=p1, d=p2:;quasi M2
    q=z*z       ;d=1:z0=4/5=.8
    r=q*q       ;only!
    s=r*z       ;nonstandard
    t=(s-r)*c-d ;dendritic
    z=t*t       ;structure
    |z| < p3
}
  
Note:
If a-b>1 AND b=1, then the sets will look similar to the
a-b=1 case, but will be surrounded by infinite numbers of
satellite islands of varing size, unless I missed something
and have that all wrong (which is always more than
possible!).  For the curious, for b=1, I used sqrt(3)=
.5773503... for a=3, and cuberoot(4)=.6299605... for a=4, to
get coherent shapes, but as a "critical point", in those
cases, it would only indicate the existence of one or more
non-escaping points at or about that location in the island-
bearing Julia sets, not indicating more than partial
connectedness, reflected in the nature of the resulting
index sets themselves.  You've seen that effect before here
in this thread, in post 41, and very recently, 266.
  

Off topic:  I LOST the Olympics!  And I wasn't even trying!
My arms are too short for world-class swimming, and too long
for weightlifting, and worst of all, I gave up letting
coaches abuse me (not sexually, but the usual classical
abuse) long ago.  I found I was just not specialized enough
for the robot junkyard.  And of course, now I'm...old...
...and maybe in the way...
(For Fred)


Later.

Not to hijack the this very constructive thread heheh, but, no one's an obstruction (in the way). The rest of the herd is just in too much of a hurry!

More on topic, I've been picking around in Fractal Explorer trying to use some of the basic formulas from this discussion (as best as I can understand them).
I don't get the critical point stuff really.  (Fractal math flunkie.)  But I've been able to produce the ringed sets of mandelbrots.
Kinda darn cool really.  Lots of fun to zoom on. 
I'll try to post some images if I can figure out how to do that here.
If not I'll try to put some up on DA.  For many of the images I zoomed to the limit of FE, about e-15.
I'm not able to get close to any of the beauties I see around FF.  But in FE I see myriads of little mini brot spots all over the place.  I've never witnessed that kind of thing before. 
I will include parameters when I get some of these uploaded.
Fred

fracmonk - I discovered that for one particular form of 'my' formula (that you've picked and run with) is sensitive to its bailout value. The form is where one power is set to zero and the other power is negative.

So for

z = c(z + 1/z)

the following are produced by changing the bailout value:

norm(z) > 2

(https://copy.com/Fiaw9z2jpVG2)

norm(z) > 4

(https://copy.com/MjgIU8Hgi3xd)

norm(z) > 8

(https://copy.com/WMttQLHJrBWO)

norm(z) > 16

(https://copy.com/BhShJ1Lk4P3V)

The number of the main buds or bulbs is determined by the sum of the powers, ignoring signs. When there are only two buds they merge into each over as seen above. As the number of buds increases the bailout value can reduced such that the buds are not cut off.

z = c(9z + 1/(z^9))

the result has ten buds and a bailout of norm(z) > 2 doesn't truncate any buds or Mandelbrots.

(https://copy.com/v0n8HhFCkaLG)

I've been playing with the variation of the formula you mentioned in your last post. The formula isn't available in Saturn and Titan and there is no mechanism to add custom formulae, so I'm using Gnofract4D to try out this variation with extra parameters.

z = (c(alpha*z^beta + gamma*z^delta) - epsilon)^zeta

What I've found is that if I adjust alpha and beta so that the critical value is 1 I get completely different pictures to you. If alpha and delta are set to 1 I get the same results as you. With the earlier formula I could use any values for alpha and gamma work out the appropriate critical value and the resulting picture was the same in all but size and location. If I try the same thing with the formula above I get different pictures for each critical value. A good check to see whether the critical value is correct is the presence of well formed Mandelbrots and each picture does indeed contain well formed Mandelbrots.

No pictures using the above formula yet, Gnofract4D has some bugs such that it doesn't correctly generate high resolution images, it has other annoyances like automatically changing the number of iterations. I'll post some pictures once I they can can be correctly or nearly correctly generated. I've just installed the newest version to see whether that's any better.

The following pictures are produced using Gnofract4d.

z = (c(2z - z^2) + 1)^2

z0 = 1 and c is the location in the complex plane.

(https://copy.com/USrn8xluoV2Q)

z = (c(z - z^2) + 1)^2

z0 = 0.5

(https://copy.com/R6cLCgC8kO6O)

z = (c(z - z^2) + 1)^3

z0 = 0.5

(https://copy.com/2bKE0MTL9dQy)

z = (c(z - z^2) + 1)^4

z0 = 0.5

(https://copy.com/8P7HXsWxOmB0)

z = (c(z^3 - z^4) - 1)^5

z0 = 0.75

(https://copy.com/SPN7GBTiVlOK)

This shows the limitation of Gnofract4D there are a least three areas where calculation wasn't completed, at the centre of the fan and in the spirals above and below the main body. For some of the images Gnofract4d also refused to increase the number of iterations above 256.

z = (c(2z - z^2) - 1)^3

z0 = 1

(https://copy.com/sinqxUjBGCkj)

element90-  
Go really large on your bailout value if you have any control over it, and you'll get better results.  Look more closely at the edge of the unit circle in the last pic in post 271. you see the buds get notched there too.  They're just smaller bulbs.  I get the same problems myself sometimes, no matter what.

You last post suggests all this stuff is getting knit together, and entering its baroque period.  I'm not sure, but is your prog. picking out critical points for you?  And I didn't know that some of this stuff has has been worked into type lists.  I guess that's good, but the puzzles of working out the formulae and critical points yourself are more rewarding.  Do you agree?

As promised, I wanted to talk about the loop dendrite
emerging from the seahorse valley regions of objects
like those in post 264, actually, that particular
object in this case.  

Pic 1 has a julia-shaped formation common to those on
the main dendrite pretty deep in that valley, that
one would expect to find there.  It's a bit distorted,
so that an endless spiral appears to be at its center.  
This pic has its center one of the larger minis on
either side, probably its actual structural center,
but it's hard to tell, since the components are
somewhat repetitious, as one would expect.  
Pic 2 zooms into the center of the 1st pic, to a
location close to the complex equivalent of the
Feigenbaum point of that mini there.            >>>but you won't see it, since the posting bot
                                                                 >>>rejected it for size.  C'mon, were doing FRACTALS here!
                                                                 >>> Get a REAL server, with some CAPACITY!
The 3rd pic
shows the whole Julia for that location, which was
the main thing I wanted to show here.  It is also a
loop shape.  The last zooms into the critical point in
it, but not too much.  See note about pic 2.

Have a decent weekend.

Later!

  

Apophyster-  

You have to go to the very basics of what the M-set is to understand critical points.
Originally, Dr. M was looking at Julia sets for f(z)->z^2 + c, and noticed that if they were connected at the origin, they were connected everywhere.  He came up with a way to index them, by plotting out one picture with that critical point, zero, as the starting z for every c value, mapped as pixels in the picture.  The pattern of it turned out to be pretty remarkable.  The M-set.

Here, we're doing the same thing.  Once you know the critical point for a function, then you can make up one picture, an index set, that summarizes that behavioral aspect of all its Julia sets, if only one constant is involved.  More complex constants would then require more dimensions to map, but here, we're only dealing with 2 at a time.  At least, we're supposed to...

Later.

 

element90-  I think we might both have been looking in here at the same time, but now I think you've left.  It would have been an interesting dialog, maybe.  Regarding the start of your post 271, that variant of "your" formula is in Beauty of Fractals, you know, and rather ancient.  It was a good starting point for me at the beginning of this thread.  I might have mentioned that then.  I would imagine that things both you and I have done have grown from that, not the other way around, no?

Later.

I work out the critical value in the same way as before using pencil and paper, the introduction of the extra constant and powers had me going for a bit as the differentiation gets more complicated. However,

g(f(z)) = g'(f(z))f'(z)

Now for g(f(z)) = 0 can be met by f'(z) = 0 so I only have to deal with inner function and I can forget about the powers. For some formula I often get an expression to solve that has a c in it, that is the location in the complex plane which stumped for a while  until I realised that 1 can be provided that the first iteration results in c.

For formulae of the form

h(z) = f(z)/g(z)

h'(z) = (f'(z)g(z) - f(z)g'(z))/(g(z)*g(z))

to solve h'(z) = 0

it's more difficult, g(z) can not be zero, so f(z) and f'(z) must equal 0 for at least one of the formulae I use this is impossible, so I've come to the conclusion that there is no critical value for some formulae. This thread has been very useful to me as I hadn't really understood how critical values were determined and I can't remember where I came across the rule that the critical value can be found by solving f'(z) = 0, working out critical values for this thread has helped enormously. I think I'm right in say that if the there a "well formed" Mandelbrots in the resulting image the critical value is most likely to be correct.

If I gave the impression that this stuff is knitting together please disregard such a suggestion as I think the surface is just be scratched.

The reason for using smaller bailout values is because using very large values for the examples shown the result is an incoherent mess and in my opinion even with clipped bulbs the results can be pleasing to eye.

fracmonk - I don't know what your statement "I didn't know that some of this stuff has has been worked into type lists" means.

fracmonk - I've stopped uploading any image files to fractal forums I just insert links to images elsewhere on the net, you can upload them to any free "cloud" storage on the net and then you can around the problem of losing you images from your posts.

Here's a dendrite spiral coloured using a more exotic colouring method that just colour by number of iterations:

(http://fc06.deviantart.net/fs70/i/2012/226/e/c/spirals_on_a_thread_by_element90-d5b0vgr.jpg)

These things crop up all along the thin threads and if you delve deeply enough into them you'll find unsurprisingly a Mandelbrot.

The formula I called Cczcpaczcp i.e.

z = c(alpha*z^beta + gamma*z^delta)

is the subject of a multi-part guide on my blog, it's currently up to 5 parts and I think there'll be at least an other 3 parts before I'm finished.

Fracmonk - re post 275, I put quotes around 'my' formula only to indicate the implementation in my software I pretty sure that anything I've come up with has already been thought of before. I've got a copy of the Beauty of Fractals and I've implemented the Magnet Model 1 and Model 2 from there, I don't remember seeing a variant the formula above, I'll have to have a look as I haven't looked at it for some time.

stunning last image! you created an art piece ;)

element90-

My own copy of BoF is hundreds of miles away right now, but the object I was referring to had the formula f(z)->z(c+(1/c)) I think, so I was wrong about that.  I think the results are exactly the same, though.  I have my machine grinding away at a hi-iteration, no periodicity, hi-bailout version with the formula you gave.  The difference in expressions between the one above and the one you gave is the z/c in the above case, and c/z in yours, I think.

It was I who said this stuff seemed to be knitting together, but you're right about merely scratching the surface.  I had a preprepared thought, shown below, that I had put together over the weekend which would in part point out spots where our thinking is in parallel.

By "type lists", I meant the formula collections for various generators, that these things are being absorbed into them.  Also, I believe much of what we've been playing with here has NOT been done before, AFAIK.

Remember, I wrote the following with no knowledge of anything beyond post 275:


A (somewhat self-) critical note:

I doubt that more than a small handful of people care
all that much about what we do here in this thread.  
I think I've already made the case for its origins,
and told the story of how my own work evolved.  I saw
things beautiful and profound, and wanted to share my
amazement at them.  The biggest problem I have seen is
that newly discovered objects appear with such rapidity
that there is no time to study or explain them
adequately.  I don't think it's enough to say: "Oh look,
here's another new shiny thing!" and not explain it
clearly before moving on to something else, also
inadequately explained.  I'm sure I've done that myself
on enough occasions.  We simply happen into new forms
that quickly.  But there are patterns to these things
which ought to be understood.  I think it's important
to bring these things to light.  "New Theories and
Research" meant to me that we should be talking more
about HOW and WHY these things work, and not wasting
everyone's time trying to prove that we're freaking
geniuses or anything.  Leave the showing off part to the
Mensa crowd who have only that much to prove.  I prefer
it when people make themselves understood, and are not
going out of their way to appear mysteriously awesome.  
It has to be be simple enough to understand, or it's
useless.  I TRY to make it so, and apologize if ever I
fail in that way.  Many so-called smart people fail
miserably that way.  Worse, some will go out of their
way NOT to be understood.  Then it's doubtful, at least
to me, that their intentions are wholesome.  Have you
noticed that too?

(Kinda wandered off-topic...)

If I haven't made it clear before:
One of the things I've been trying to do is to avoid
coeffients to deal with these things in their most
irreducible forms, and explain patterns I've found in
them as families in the simplest terms available.  The
fact that their formulae are all related either in form
or algebraically is a profound thing not to be
neglected.  So that's why I avoid using coefficients
myself, though I see nothing at all wrong with them.  

A while back, I communicated the idea that some
functions are connected in complex space while others
are not, and how it would be nice to find some quality
that would make one able to recognize the difference
in advance.  I may be closer to that now than at that
time...or some fact will turn up that makes it a still
more complicated problem than I originally thought.  I
still can't tell about that.  And remember, no one
seems to care anyway...  

That's just one worthy Holy Grail, and that's the kind
of thing that comes to my mind when I think of the idea
of what 'research' means to me.  There are as many
Grails as there are seekers for them, and more we
probably SHOULD pursue that still have yet to reveal
themselves.  I think there are enough of them to pass
around when company comes over.

"Here!  Have a Grail!"

"Thanks!  Don't mind if I do."

-or-

"Just had one.  I'm full.  But thanks anyway."


Also, I've seen enough blank rectangles, each with a
little red square with an "X" in the middle of it.  I
think this is called link rot.  It is why I post my
necessarily miniscule pix directly to FF, so you can
see what I'm talking about as long as FF is there, when
some other linked sources perish inevitably.  (You have
likely seen my complaints regarding the limitations.  I
do not trust "The Cloud" in any way, shape, or form,
and you shouldn't either.  See posts 248,249.  You have
to be out of your mind to trust your data to unknown
others, but then, look around, plenty crazy people...)  
Anyway, I THINK that the metadata for each pic goes
with it, so there's more about it available in it if
you take it to fractint together with its formula.  
There, you SHOULD then be able to play with these pix
in any way you like.

As an example, here's a formula and a pic that goes with
it below.  You SHOULD be able to call up this pic in
fractint with the accompanying formula inserted into a
.frm file and zoom into it, etc.

DM98(xaxis) {;deg 18
  c=pixel, z=p1, d=p2:;quasi M2
    p=z*z       ;d=1:z0=8/9
    q=p*p       ;only!
    r=q*q       ;no per. ckg.
    s=r*z       ;nonstandard
    t=(s-r)*c-d ;dendritic
    z=t*t       ;structure
    |z| < p3
}

The pic is one of few in a series involving a zoom into a
"furry" spiral with small enough storage requirements to
be accepted by the FF posting bot.  You should be able to
find the zoom destination.  By switching the z and c
assignments, you will have the formula for the Julia set
then, too.

I looked at real axis limits for patterns in the values
between 21, 32, 43, and 54 exponent combinations.   If
there's anything there I can recognize, I'll let you
know, unless you can't wait and have to look
yourselves...

Many thanks are due to element90 for bringing this
formula structure into the mix.  



Later.

element90-

I must apologize that my personal circumstances are
not leaving me enough time to do your contributions
any justice at all right now, but they are very
powerful in my estimation.  Without my having to
more than "scratch the surface" of them, they have
helped focus some of my own observations.  I will
try to explain how.

If you took the simplest formula, for Multibrots,
f(z)->(z^n)+c, you will find that n must be integer
or you will get discontinuities in the object, AFAIK. 
I have noticed that many Multipowerbrots that have
taken up the bulk of my own contributions to this
thread require that any second constant used must
also be integer, for the same or similar reasons. 
This leads inevitably to, and here's the big idea, a
quantum effect. 

While we know that in the case of exponents, the
discontinuity already infects any parallel 2d slice
for any fractional n in the c plane, the second
constant d in the Multipowerbrots I've shown here
SHOULD have continuity over (a MINIMUM of) 3
dimensions, in some, not all cases, as a solid
object.  So that quantum effect would then be
expressed only in exponents.  I'm not clear on WHY
this is so about exponents, but it is so.

That's where my thinking has been going lately.  Your
work brings out new forms consistent with this notion.

Here's where I go into COMPLETE AND UTTER speculation:

If this kind of math does in fact say anything at all
about numbers governing the design of our universe,
we may be mapping actual elementary particles, and
there may not be only one, when everything is said
and done.  While there may, in all eventuality, be
many such possible forms, the patterns that govern
these forms should be painfully finite.  If the
objects aren't the particles themselves, then they
might dictate the laws that make the forms.  The
latter is more likely.

As in string theory, luckily for physicists living on
government funding, it's all too small to test!

How's THAT for mad scientists overreaching with
physics fairy stories?

Obviously, it could use plenty of analysis, merely as
mathematics and geometry that is beautiful in itself. 
It's only a big maybe right now, although the
experimental results we get here are indicative of
SOMETHING.  But what?  The habit of these numbers
staying in one piece under the right circumstances
cannot be ignored.

Given time, some conclusion will be drawn about that
(besides "fracmonk is NUTS!").

But those personal circumstances are always a
hindrance.  Everything happens in its own time, I
guess...

Have a great weekend, unless, as G. Carlin once noted,
you insist on having a crappy one.

Later.

No comments at all?

To be maybe a little clearer:

Variables and constants can be considered continuous in as many dimensions as their number require.  The same can be applied to coefficients, which usually only govern scale and proportion.  The inherent behavior of fractional exponents, like quantum behavior, is not continuous when mapped onto the complex plane.  Is there a relationship there?

Later!

So, here's the thing:   If you think it's totally preposterous, say so, and say why.  That's what we're here for, no?

Later.

element90-

I checked out your post 272 a little better, but my online time at the library is severely limited, and it still won't print right so I could study it right later.  But I think your approach to cp's is practical, and I still think that while some formulae won't calculate right under any circumstances, some others might under ridiculous conditions.

Very busy with other things, I let the machine grind away with the hope of making a clean version of the barebones formula from post 271.  set to 100000 bailout, 20 million iterations, no periodicity checking, it took (only) 140 hrs. 21 mins. ...but it did come out right.  You know, I hate it when machines and their programs of any kind think they know better when it comes to what you want to do.  Just leave the driving to us, they say, and when they didn't think of everything, you might just make the evening news...

Life is just a bowl of...

Later!

Following up on element90's reply #265, it looked like a good opportunity to try feeding some values into the many parameters used in his generalized formula.  I tried a couple of different methods and found some interesting views when the base terms alpha & gamma range between -1 and +1 and the power term beta ranges from 1 to 5, with the power term delta = -1*beta.  I then set up a path following a Lissajous curve that travels within these boundaries and made a rough animation over a long path.  I excerpted an interesting portion and made a proper animation.  Nothing to look at if you're interested strictly in the integer values, but there's a lot here that's interesting to me at least.

Ryan

http://vimeo.com/moogaloop.swf?clip_id=48652524&amp;server=vimeo.com&amp;fullscreen=1

Ryan D-

That must have taken awhile to put together, no?  I'm especially curious if a:
one of your dimensions was exponential (gamma?), and continuous thru fractional values,

and b:

If the the discontinuities I mentioned are reflecting that, in your view.  I seem to see them clearly as stills.

Really action-packed illustration of the power of that formula structure that element90 brought us
anyway, I think.  I really appreciate it, for one...

Later.

3000 frames, 2560x1440 resolution, render time was a bit over 20 hours.  I started with a low resolution animation (probably 256x192, that's what I usually use) that covered a longer trip along the Lissajous curve, found a patch that looked interesting and made a slow traverse along that portion of the path.

Of course, the time actually spent working on it is far less than 20 hours.  I've written a simple program to generate the batch file needed for the animation, and that's just a matter of some programming details to get the path where I want it.  Pick the number of frames, hit "Enter" and it spits out a massive batch file to make successive calls to Fractint.  It's not very romantic, it produces stuff that looks like this (3000 lines worth of it) -


I don't keep the individual frames any longer than is necessary to render the final video, but while they're still on the hard drive, it's a good opportunity to check out some fine details.


The batch file snip above shows the parameter structure I used, as follows:
alpha real / complex
beta real / complex
gamma real / complex
delta real / complex
z0 real / complex

The rest of the batch file is the same, all parameters use real values only, and I have always used z0=1.  (There is some investigating to be done with this, using element90's method of calculating critical points and substituting that in to z0.)  This applies to element90's formula, z = c(alpha*z^beta + gamma*z^delta).

As mentioned, I'm using a 3D Lissajous curve to determine the parameters.  That is just a trace of 3 concurrent sine waves, one on each axis (or parameter, in my case).  As a sine wave, there are no discontinuities in the parameters, in fact the general structure is entirely repetitive.

http://demonstrations.wolfram.com/3DLissajousFigures/

Agreed, however, there are some definite discontinuities in the video.  The first one around 10 seconds in occurs when gamma crosses from the negative into the positive.


There are other surprising views, such as that around 1:00 in where 3 minibrots turn into 4 not by a split of 1 into 2 but rather by a merger of 2 into 1 while at the same time, 2 others appear out of nothing.  As I implied earlier, this probably isn't much for those into mathematical rigour but I like it nonetheless.

Ryan

Ryan D-

I gave myself one of those cringy feelings upon realizing that the exponents were beta and delta...sorry!
For that belief that I must crawl before I can walk, I've been avoiding situations where planes get bent, much less folded, spindled or mutilated...I need to keep things humiliatingly simple for myself!

In parallel, I was looking at the "time" dimension (the progression of the frames) as one assignment you used, but I'm still not clear on which it was, unless you decided to reach a point in the progression, make a previous variable fixed, and begin changing another.  You can never get done going down all available avenues in a multidimensional animal such as this.

It "looks" as if z0=1 remains critical most of the time, based on the M-shapes retaining their form.  If so, then the coefficients may not be continuous, either.  Are YOU sure of it?  Again, just curious, and I do not have a complete understanding of the journey, mind you...

Both in thinking of the how to, and the implementation, not to mention ridiculously long render times, I have spent longer on simpler things!  Labors of love these are.

Before getting into anything so complicated, I'm trying to nail down some rules or expected behaviors for much simpler things, and as usual, wish I had more time to devote to it.  I feel like Salieri to your Mozart, if you saw "Amadeus", but I harbor no jealousy as such.  Kudos on an interesting journey, and again, it's not clear to me the path it took.  Which reminds me of a favorite bumper sticker: Where are we going, and why am I in this handbasket?

Have you seen it?  Not that it applies here, but it came to mind anyway.  I'm a sucker for a good laugh, off-topic as it may be every time...

Later.

The parameters in my path simply follow sine waves.  Alpha and gamma are unmodified sine functions ranging from -1 to +1, beta is a sine function scaled and offset to range from +1 to +5, delta is the negative of beta.  Yes, the sine function is continuous.  That's intuitive to me, and as with many things, the proof causes me more confusion than just accepting what I already "know".

http://www.math10.com/en/algebra/functions/continuity-sine-cosine-function/continuity-sin-cos-function.html

Ryan

"I see!"  said the blind man as he picked up his hammer and saw.

The continuity I was looking for was in the object, in however many dimensions it has.  In just 2, it doesn't happen in every parallel plane, and probably never will, no matter what the function.  (Someone will prove me wrong about that, but please make it nontrivial, O.K.?)

Later.

Anyone up for a really good puzzle?

I have been very busy with OTHER THINGS (that...just won't let you be...), but I still find a
little time to knock this stuff around a bit, just barely.

I've been looking lately at functions that have no single numeric value as a critical point,
but ones with critical points that are algebraic expressions.  f(z)->(z+c)(z+(1/c)) is a
(relatively) simple example, as is one earlier in this thread in (my) posts in 88-97. 
z0=-(c+(1/c))/2 in THIS case, and already I was hard-pressed to figure that out, since I'm
not all that skilled at it.  I had originally looked at its Julia sets to determine the real limits
of the double M figure to the left in the first pic below, without realizing THEN that the 2
separate M figures to the right existed.  For each prisoner point c in this index set, there
seems to be a single connected Julia set, although the index set is not in one piece itself. 
The Julia sets resemble those of standard M without exception, although they are not
centered at the origin.  I would posit that the fact that the components of the index set
are obviously countable (4 M perimeters in 3 separate figures) is what makes this possible,
as opposed to being surrounded by infinite numbers of satellite islands.  That would then
mean that index sets need not be in one piece for this to be so, but the number of its
whole components on the parameter plane then still must be finite.  Then too, the argument
can be made that c and 1/c in each case of c is still a (split) constant, but a constant
single number when added up, all the same.  Here is an efficient fractint formula for the
function:

Mcinvc(xaxis) {;deg 2 index set
  c=pixel, d=1/c, z=-((c+d)/2):
    r=z+c       ;3 M2 figs. with       
    s=z+d       ;4 perimeters
    z=r*s       ;total
    |z| < p3
}

McinvcJ {;deg 2 Julia sets
  c=p1, d=1/c, z=pixel:
    r=z+c       ;3 M2 figs. with       
    s=z+d       ;4 perimeters
    z=r*s       ;total
    |z| < p3
}
 
Accordingly, I have found another function whose Julia sets are intriguing, and suggest the
existence of some similarly algebraically-determined critical points by virtue of the fact
that I have found one-piece Julia sets from it, f(z)->((2/(z^2)-z)c)-d.  This one may or may
not have an index set in more than one piece, I'm guessing it's in one piece, but I'm stumped
as to what expression involving c and d should be assigned as init z0 to obtain the index set.
Giving credit where it's due, this was inspired by the functional structures that element90
brought here recently.  Everything effects everything else. 
The Julia set formula is:

DMDC221J {;cp not found
  c=p1, d=p2, z=pixel:;quasi M2?
    s=z*z           ;d=1:z0=?
    t=((2/s)-z)*c-d
    z=t*t           
    |z| < p3
}

Pic 2 is of a one-piece Julia set using this formula, that appears to correspond to what would
be the approximate real right continental limit of the index set for d=1, c=.33881.  The others
show details of that pic in which it is easy to see that there are an infinite number of
bridges having zero width crossing the real axis.  At least one of them is critical...yeah, but
which one?

Anyway, can anyone please show me what I'm missing here (besides the index set)?

Later!



 

The function f(z)->((z^2)/c)-(c/(z^2)) appears to create an infinite set (detail pic 1), but many of (if not ALL) its
(specifiable) Julia sets are finite sized.  The problem with the index set is not in zooming IN, but zooming OUT, when the limitations of a given generator determine how much of the set you can zoom OUT to.  The 2nd pic is
of fractint's limit that way, and the 3rd pic is an inversion, showing how the same index set looks on
the 1/c plane.  Limits again show themselves when zooming in to the origin of the inversion, as in the
last pic, where limits to the bailout value blank that local region.  You won't in any event get to see
all of this set, but there's still a lot of it you can.  Try it:

Zdivcinv(xyaxis) {;deg 2
  c=pixel, d=1/c, z=p1:;z0=2
    r=z*z     ;infinite set?
    s=r*d     ;mostly foamy
    t=c/r
    z=s-t
    |z| < p3
}

ZdivcinvJ {;deg 2
  c=p1, d=1/c, z=pixel:
    r=z*z       
    s=r*d
    t=c/r
    z=s-t
    |z| < p3
}

I picked a spot that made an interesting Julia set, and did a zoom to it, and then of the critical point
z=2 in the Julia, in many formats using FFW, trying to determine correct x mag factors to yield
correctly proportioned pix, including monstrous, time-consuming 2048 sq. pix (xmag=1.3333...) that would
never ever fit here.  When I don't have time to give the work attention, the machine still grinds
away...like inflation grinds away at buying power when they make so much fake money.  That's off-topic,
of course, but always in my thoughts as time marches on and entropy unfolds as it does.

I might add that distortions in the proportions of M shapes are often a matter of what the formula will
correctly yield, as in this case and the set shown in post 261, differing from MALFORMATIONS of the M
shapes, as can be found in the last 3 pix in the set in post 210, indicating an incorrect (in that case UNFOUND) critical value.

Later.

Note:
Please don't mind me if I get snarky sometimes.  I've been long in the dark heart of the Land of Disenchantment with the Arrow of Time poking out my lower back, and yet, it's work or wither, and winter's coming...

Later.

A question - why does your M-set equivalent above equate z0 to a fixed value (p1) rather than the pixel value?  Obviously the J-set must use the pixel value, otherwise you'll just get a single colour over the entire screen since every pixel would start at the same value and follow the same escape path.  But if you're trying to create an M-set that serves as an index to the J-sets, it seems to me you'd need to have an equivalent initial value for z.

I implement it as follows - I use the ismand variable to allow toggling with the space bar, and through force of habit I always force a non-zero default bailout value.  As a direct toggle, the M-set now has z0=pixel, the same as the J-set.  I've also generalized it by adding p2, allowing for exploration of varying power terms of z.  Your hard-coded value is p2=2.  P2=sqrt(2)=1.4142134 is interesting.  (Virtually all the formulas I fiddle with will be generalized in some way so I can vary parameters to create animations.)  Symmetry was removed because this formula now looks at both the symmetrical M-set and the non-symmetrical J-sets - if I were to create an M-set animation, the formula would be rewritten to remove the ismand toggle and the appropriate explicit symmetry would be restored.

rdfmonkZdivcinv {;original hardcoded with r=z*z, z0=p1(suggested=2)
  if (ismand)
    c = pixel
  else
    c = p1
  endif
  d=1/c, z=pixel:
    r=z^p2     ;infinite set?
    s=r*d     ;mostly foamy
    t=c/r
    z=s-t
    |z| < 4 + p3
}

Following are a couple of M-set images for p2=sqrt(2), one with a considerably zoomed-out view (magnification about 0.05) and one of a zoomed-in mini-blob.  There are also two J-set images, one located at the Elephant Valley equivalent to the zoomed-in blob and one along the chain that disappears off the edge of the zoomed-out image.  Fun stuff, I need to take some time and go through a bunch of the previous formulas to see if there are worthwhile animation candidates.

Ryan

Ryan D-

I used z0=2 in the index set because 2 seems to be critical in the connected Julias. 
From the discontinuites and separate islands in your pix, I can tell you're not starting with the critical point for whatever other values you may be assigning.  I see these kinds of qualities in my own initial tests often enough, and then know I don't have the right c.p.  Not that you can't have interesting fun otherwise, but I'm kinda obsessed with connected sets, and why the critical points are critical, and sometimes I THINK I know all the patterns that produce the results I get, and find there's usually something new, some exception that keeps me looking...

It would be great, for instance, if you had a reliable formula for adjusting the init z correctly for every change of, say, the exponent of z, that may in fact be pixel-dependent in some cases of fractional exponent.  I don't know about that myself.

I'm just trying to digest the "simple"...it's complicated enough for me...and besides, I think of integers as the most basic constants, anyway, followed closely by rationals.  By that thinking, 2 is no less magical than its (trancendental?) square root, but every number has its unique identity, plain or ambitious...

Later.

Hey!

We were lucky enough to be in a decent spot in NYC for the storm.  We had little rain and very high winds, but most everyone we know did pretty poorly in the tidal surge.  Most we know are w.o. power & phone, one burned out @ Breezy Point, others flooded catastrophically in Long Beach.  Been busy...

Machines doing long pix...

Later.

Also off-topic for here, but still fractal...

The second storm would have been almost inconsiderable, if not for the damage from the the first, not unlike rubbing salt in a wound.  No wound, salt's not a big deal.  Saw a fuel tanker stuck in a line for gas today...

Snowcover, not well tracked, reflects sunlight and heat away from earth, so cold gets colder.  On the other hand, lack of reflection makes warm warmer.  Logistical equation strikes again?  (That is, due to sensitivity to initial conditions, by the law of averages, on average, there's a wide departure from the average.  Let's call it the Law of Average Irony.  Any subscribers?)

Later.

Here i have buddhabrot render of one of your early formulas: http://www.fractalforums.com/fractal-programs/problems-with-implementing-budhabrot-in-uf/msg55319/ (http://www.fractalforums.com/fractal-programs/problems-with-implementing-budhabrot-in-uf/msg55319/)

Alef- Not me, anyway.  I don't recall ever having had anything to do with that.

Sorry, I've been either too busy or not well or both.(Like Now)
Otherwise, I'd be paying more attn.

Started working on an illustrated summary of Multipowerbrot funcs and how they work.  I will draft it in Word, but eventually, maybe with Bunny's help, put it in html form.  Many pix in it will probably be same as those seen here, but either way, it won't fit in FF's size constraints, except the text portion, which I will be happy to post once finished.  Some of that should, for most, be easy to follow, making reference to things most fractalists know.  Others, like zoom tours, will need the pix.  Meantime, the thread here remains a a good resource for those who would like to study them in depth.

Pretty sure that ALL points in a Fatou dust (for f(z)=z^2+c) escape eventually, but has anyone done a proof for it?  They say if z0=0 escapes, it's not connected, but then, there's nothing to connect, right?

If anyone can shine a light on that, I'd greatly appreciate it.

Later.

I have for some time been playing (mostly unsuccessfully) with simple M-like formulae in Fractint, and if I get anything mildly interesting, most of the time they have disconnected julia like areas.

Today, however, I created the following very simple alteration of the mandelbrot in the formula parser:


Now, for anyone who doesn't know the formula parser, anything before the : is performed for each pixel once, then from that to the last but one line is iterated for every pixel, then the test is the line at the bottom.  I have tried a few different combinations of the "z = z * z ..." lines and this one creates the following image which is finally connected:

(http://www.needanother.co.uk/uploads/smf01443.gif)

Zooming in to the valley on the right at 0 axis, produces the following zooms:

(http://www.needanother.co.uk/uploads/smf01444.gif)

(http://www.needanother.co.uk/uploads/smf01445.gif)

(http://www.needanother.co.uk/uploads/smf01446.gif)

(http://www.needanother.co.uk/uploads/smf01447.gif)

(http://www.needanother.co.uk/uploads/smf01448.gif)

(http://www.needanother.co.uk/uploads/smf01449.gif)

(http://www.needanother.co.uk/uploads/smf01450.gif)

(http://www.needanother.co.uk/uploads/smf01451.gif)

(http://www.needanother.co.uk/uploads/smf01452.gif)

(http://www.needanother.co.uk/uploads/smf01453.gif)

(http://www.needanother.co.uk/uploads/smf01454.gif)

I very quickly reached the number precision limit of the version of FractInt I am using. The next image was very blocky so not included.

So far quite interesting for me, and I now need to ask kindly of members of the forum who have Fractal eXtreme (64-bit) and the necessary know how to write me a plugin with this formula included... (I hate grovelling but, when needs must...)

Hey Simon!

The limit for the fractint formula parser is 16 digits, giving you about 12 accurate with large pix.  I cry and whine, but no one listens...

So that's an M4 Multibrot, yes?

Later!

If I was feeling really manic I'd pop into a dosbox session and run the true dos version, which I believe supports arbitrary precision, but it just takes far too long to generate images for my liking.

The windows version that was created is a lot faster but has its own issues.

I have sent a pm to the only person I know with the suitable skills to create the plugins, however the author of Fractal eXtreme, Bruce Dawson is also a member of the forum I seem to recall, so maybe I should ask him.

Hi Simon,

I've tried your formula but it looks nothing like you example.
My actual code (for floating point) looks like this...

I have a spreadsheet that helps me expand the complex multiplications into individual real and imaginary equations.
So for Z=Z*Z*C
expands to
zr=(zr*zr - zi*zi)*cr - (zi*zr + zr*zi)i*ci
zi=(zi*zr + zr*zi)i*cr + (zr*zr - zi*zi)*ci

And I've just spotted a mistake on line 6 "zi=2*zi*zr+ JuliaI;" but I've already clobbered the zr variable in the previous line.
Doh!
Back soon (Hopefully)

Jeremy Thomson.

That image looks kind of interesting, too. Might be worth a look at that in Fractal eXtreme also!

Simon,

   Sorry!  I misidentified your object as M4, but it's M4 on an M6 body, and I only realized that when I took a second look at the first pic.  I think arbitrary precision only works with Mandel type in FractInt.  In formula, I wouldn't care if each 10x mag. took twice as long as the last, as long as I could go beyond 16 digits when I wanted.  I dream...

Also, I'm not familiar with what ismand does, and haven't used conditionals for years now, as it turns out.  When I did, I might have knocked out the largest circles of lambda, for instance, to save time in calculation when machines were that slow...but I'm obsessed with utmost simplicity in formulae leading to utmost complication in outcome, these days.

Later.

If you include the ismand check then FractInt gives you a prompt at the time you select the formula where you can set whether you run in mandelbrot or julia mode, and while you view the fractal if you press space it will then give you a crosshair which you move to your desired point then press space to show a julia of that spot.

I hope that helps.

Simon

So, tell me if I'm rambling but based on what FractInt does with complex numbers, and trying to convert this into code for Fractal eXtreme, I have been trying to get my head around the way these things are done.

Here's my take on converting the complex maths into non-complex code:


I don't know if I'm on the right track with this, as it's all a bit over my head, but I was good at maths at school so shouldn't have too much trouble with it if I put my mind to it.

Happy coding...

Simon,

To be honest, I've lately been overreliant on the auto complex calc that FractInt does, and it has been pretty reliable because of the straightforwardness of the formulae I use lately.  The last time I resorted to "manual" was when trying out a 4d scheme that required it, and it was a disaster, not so much math-wise but results-wise.  I'll read your code if it will print (it's in a rolling box here) and see if I can interpret it right, O.K.?

Later.

Please do, but I do feel we're slightly hijacking this thread so can we create one in the Fractal eXtreme plugins forum instead.

Simon

Simon,

No problem, but make plain how you got that shape, O. K.?

Later.

I need help implementing one of the formulas in this thread...

It's on page 2. Fracmonk describes the formula as: The oddity of M4 shapes in an M2 general structure (1st pic below) is obtained with the formula f(z)->((z^4)c)+1. And the results he gets are:
(http://www.fractalforums.com/index.php?PHPSESSID=4dfad40c3d7917a187702c4dea5a16a8&action=dlattach;topic=4881.0;attach=4032;image)

I tried to implement it myself in glsl, but the results are:
(http://img16.imageshack.us/img16/1074/failurexb.png)

Clearly, my result is not the same. Here's my code (all the important bits, at least):

Pay special attention to this line:
newPoint = complexMult( initPosition, complexSquare(complexSquare(newPoint))) + dvec2(1.0,0.0);

If someone could help me find where I went wrong, that would be awesome.

You maybe need to begin with the Complex 0 (or 1 it is the same after one iteration). Here you begin with c just like if you would do with the regular Mandelbrot because c=f(0) (in order to simplify one iteration) !

You are correct. Thank you very much sir! :D

I've made a slight change to the standard Mandelbrot formula and came up with another new M-like fractal which I feel has some interesting features:

Here's the formula in FractInt:


The important bit is between the endif: and |z| which are the two assignments to z.  All I do is first multiply z by abs(z).

Here's the initial image...
(http://www.needanother.co.uk/uploads/smf00321.gif)

...and some zooms...
(http://www.needanother.co.uk/uploads/smf00283.gif)

(http://www.needanother.co.uk/uploads/smf00284.gif)

(http://www.needanother.co.uk/uploads/smf00285.gif)

(http://www.needanother.co.uk/uploads/smf00286.gif)

(http://www.needanother.co.uk/uploads/smf00288.gif)

(http://www.needanother.co.uk/uploads/smf00291.gif)

(http://www.needanother.co.uk/uploads/smf00292.gif)

(http://www.needanother.co.uk/uploads/smf00296.gif)

(http://www.needanother.co.uk/uploads/smf00301.gif)

(http://www.needanother.co.uk/uploads/smf00302.gif)

(http://www.needanother.co.uk/uploads/smf00305.gif)

(http://www.needanother.co.uk/uploads/smf00306.gif)

(http://www.needanother.co.uk/uploads/smf00308.gif)

(http://www.needanother.co.uk/uploads/smf00309.gif)

(http://www.needanother.co.uk/uploads/smf00312.gif)

(http://www.needanother.co.uk/uploads/smf00314.gif)

(http://www.needanother.co.uk/uploads/smf00316.gif)

(http://www.needanother.co.uk/uploads/smf00317.gif)

(http://www.needanother.co.uk/uploads/smf00318.gif)

(http://www.needanother.co.uk/uploads/smf00319.gif)

(http://www.needanother.co.uk/uploads/smf00320.gif)

...and some julias...
(http://www.needanother.co.uk/uploads/smf00294.gif)

(http://www.needanother.co.uk/uploads/smf00295.gif)

(http://www.needanother.co.uk/uploads/smf00304.gif)

(http://www.needanother.co.uk/uploads/smf00311.gif)

What do you think?

Looks like another candidate for conversion to a Fractal eXtreme plugin.

I've tried power 3, 4, etc. and they look quite good too.

Looks really good

Thanks.

I've started writing a Fractal eXtreme plugin of the fractal I've written.  So far, it zooms up to 45 times, now got to complete it by writing the trickier bit...  Will continue this thread in the Fractal eXtreme plugin section of the forum (here (http://www.fractalforums.com/plugins/)).

Back in reply #272 of this thread I shared some pictures using Gnofract4d. I'm adding the formulae to version 4.0.0 of Saturn and Titan. Now that I've had more time to play with them I noticed some odd things about one of the formulae.

The formula

z = (c(z^3 - z^4) + 1)^5

where initial value of z is 0.75 and c is the location in complex plane.

Oddly, along the real axis the pictures exhibit the blocky nature of fractals that have been produced with insufficient precision at low levels of zoom, i.e at about -8.94 + 0i and a width of 0.03 I get this:

(https://copy.com/DhczZjjdq9yH)

Closer to zero and the width that shows this problem gets significantly smaller at -8 + 0i it only gets blocky at a width of 1e-7.

Elsewhere in the fractal I get this:

(https://copy.com/EcMretMoSqDD)

Which Gnofract4d makes a mess of:

(https://copy.com/EPPE34wquHtA)

Zooming in further with Saturn I get:

(https://copy.com/NSQ7uBHzkyIy)

The whole nature of the spiral unexpectedly changes its character.

Does anybody know what is going on? I haven't yet been able to see what an increase in precision will do to the unexpectedly blocky parts of the fractal as Saturn currently only increases precision automatically, there is no manual override (this feature will be added soon).

It sounds like the operations are formed in a way leading which lead to catastrophic cancellation (http://en.wikipedia.org/wiki/Loss_of_significance).  Example: x²-y² is a numerically bad computation.  Sound familiar?

Roquen: that looks like likely explanation.

Increasing precision solves the problem but the amount it is increased by is down to seeing what the image looks like for each increment. Saturn now allows precision to be set manually and allows increments of 16 bits.

Well that's good news.  However increasing precision simply pushes back the problem.  Using GLSL as an example of a couple simple tricks:


and likewise pullout powers when possible: z³-z⁴ = z²(z-z²)

The problem occurs with a formula that has complex parameters for complex powers so in C++ std::complex<T> std::pow(std::complex<T>, std::complex<T>) is used, so I have no means of removing the "catastrophic cancellation". All formulae in Saturn are built in, there is no facility for editing formula like there is in other programs such as UF.

In case anyone's curious, I'm still kickin', but I've been better.  Have been looking in without logging in.  Did some new stuff, related to the old stuff, nothing presentable yet.  I need the time and energy...hope you're all feeling vastly better than I am!

Later.

Glad to see you are still hanging around.  Sounds like you are not doing all that well, so I hope you feel better soon, at least better than I have been.     :)
 

Paul,

Thanx for your kindness.  Compassion seems scarce and unfashionable.  It's very good to hear from you again after all this time.  What's been on my mind lately is that from the most complex human organizations right down to the simplest microbes, we're just under attack all the time.  We just have to fight all the time to survive as long as we can while we can, and I guess that's the game.  My experience lately observes a kind of pathological competitiveness that forces us to do things in undesirable ways.  For me, chief among these what I see as a nightmare of electronic dependency that many embrace as the best thing since sliced bread, but I see it ending badly.  Involvement for me in that is minimal, and many might say that it's an age thing, what they may not get is that with age comes experience, which is undervalued.  I find things, as here, and must use an electronic venue to communicate them, but I'd rather there was another way.  I need *simpler* badly!  That you're still in the fight is extremely gratifying to me.  Be well.

Later.

Personally, I am just trying to hang on long enough to see the end of modern civilization (and most likely, mankind), which at the current rate of destruction should not be that far away.


Dependency on electricity is not a good thing, but at this point in history, it is where we are currently at.  The biggest problem is HOW we create most of that electricity.  We have nuclear plants melting down poisoning everything around them for decades, fossil fuels burning and polluting, dammed up water altering landscapes and restricting aquatic life, etc....
 

Funny to complain about this in fully computer forums.
Sliced bread is great thing. Well, I think we can survive another few nuclear accidents, there even are people living in Chernobil area. Sea level rise could be worst I think.

But doing population census in neigbourhood I noticed marked cultural shift among generations. Many senior russians as an education listed church community schools. But then it's paranoia and nationalism among youth. Less open, less willing or able to communicate, hiding some secrets. It's even with writing a  postcard vs. new facebook likes and SMS.
The same thing all around European countries. And no progress exept in smartphone market. + wild unnemployment rates amoung periphery european youth and cousy supermarket in place of factory which once produced this thing:
(http://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Minox_CLX_BACK.jpg/800px-Minox_CLX_BACK.jpg)

I was just trying to convey my extreme unease with being forced to use these vicious devices to communicate or be incompetent without them.  By vicious, I mean that they are designed in part to betray us.

I have no particular objections to electricity use; it would be better if such expanded awareness and knowledge, but most is not used that way.  Means of generation, as Paul suggested, often leaves much to be desired, as well.

fracmonk, here I rendered orbit plot of your even power multipowerbrot from somewhere of the star of this thread. z= sqr( ( sqr( z*z*c +1 )) -1 ) -1 Thread had become hudge;)

It needs just eyes.
(http://www.gregofhawaii.com/wp-content/uploads/2012/11/cheburashka1.jpg)

Alef,

Pretty complicated orbit, if it's an orbit.  I'm absolutely no judge, since I don't use whatever your code works on.  It happens all the time.  I just use fractint, and it's enough for me.  It's sort of like how, no matter what their language, everyone in Europe spoke Latin as a common language once upon a time, so they could understand each other regardless.  In fractals, especially this kind, fractint provides a common language.

Later.

It's been a long time, almost a year, since I last posted on FF, and months since I last logged in.  I've had a couple serious illnesses that tended to compromise my energy and still do, and life's challenges otherwise don't quit, you know.  Honestly, work on this topic has not been on the front burner.  I've let one machine do really long pix with FractInt, and MAY find time to post things that pick up where I left off.  Surprisingly, I've had NO messages since before my previous post, so at least I didn't feel that terribly needed.  I'll try to check in more in future.

Later.

Be sure to check out the kalles fractaler imppementation of pertubation theory, improving render time of deep zoom mandelbrot by a factor of 100 or more

Thanks, I will.  Strange how only very recently I returned to deep zooming of standard M.  I was getting a little tired of the many black blobs in the pix my own formulae usually produce.

Many theoretical physicists insist that time doesn't really exist.  Benjamin Franklin famously remarked that time is money, and everyone's so concerned with both!
Be well.

Later.

So I just checked it out (kalles fractaler), and got the picture quickly.  I watched a 4+ minute youtube video by Karl Runmo that went very deep very fast.  Very impressive!  Most striking was that mine was only the 159th view of it.

But I have still an age-old complaint that only standard M is ever taken beyond double precision.  I still dream...

Later.

Burning ship and z3 mandelbrot is available in kalles fraktaler as well, check the iteration window

After looking some more, I see that's true.  I just logged back on to correct myself, but you you beat me to it.  I have been checking the products of the program, not so much the program itself.  It'll do (so-called) Multibrots, for instance, standard M in any degree.

But what I meant was *any* formula that uses the basic operations, not nearly as ambitious as "formula" in fractint, but at least that much.  KF is a killer, though!  I only wish I had the time to learn how to use it.

Later!

I believe there is still an infinite amount of discovery that we still have to accomplish. :)

I have mostly been playing with stardustforever's fractals and making variants of them. One fractal I discovered was the normal Heart on the top on the Celtic Heart on the bottom! I have also been getting interested in simon_snake's fractal, the Simonbrot.