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OK so this is actually 4D, but it's of course the 3D view I'm talking about....

Here's how to get he power 8 from 2 complexes z and w:

    float m = |z|+|w| ;square of 4D mag
    w = 2.0*z*w
    z = 2.0*sqr(z)-m
    w = 2.0*z*w
    z = 2.0*sqr(z)-(m=sqr(m))
    w = 2.0*z*w
    z = 2.0*sqr(z)-(m=sqr(m))

Not quite perfect - but view from all angles ;)

Edit - fixed it as follows:

    float m = |z|+|w| ;square of 4D mag
if m>0
    w = 2.0*z*w
    z = 2.0*sqr(z)
    float n = cabs(z)
    z = z*(n-m)/n
    m = |z| + |w|
    w = 2.0*z*w
    z = 2.0*sqr(z)
    n = cabs(z)
    z = z*(n-m)/n
    m = |z|+|w|
    w = 2.0*z*w
    z = 2.0*sqr(z)
    n = cabs(z)
    z = z*(n-m)/n
endif

It may well be this is just an alternative way of the standard Mandelbulb - but it doesn't look quite the same - here's a very quick test render (at *30,0000 when you see the whole thing at *1, solid at 1e-6 DE)

(http://nocache-nocookies.digitalgott.com/gallery/16/141_18_07_14_6_46_32.jpeg)

It's definitely *not* the same as any of the current ones (at least not that I've seen) - here's a full view of the z^2 version....


(http://nocache-nocookies.digitalgott.com/gallery/16/141_18_07_14_6_58_56.jpeg)

Althougn the first version I posted is "flatenned" I kinda like its spikiness so here's a render I did before "fixing" it by reducing the magnitude of z instead of just subtracting from it.

(http://nocache-nocookies.digitalgott.com/gallery/16/141_18_07_14_7_07_02.jpeg)

Nice pictures Dave, but I do not understand what your code is doing.
Can you explain?

how sweet, a new candidate :D

right, please elaborate your idea, combining complex number has always been an attempt to create a 3d number system, what was your idea ?!

the z^2 variant is definately a lot bigger in the z-direction
and the z^8 variant has some whipped cream along the bulbs, but quite similar!

That spikey version looks like a great candidate... except for the whipped cream of course...
Do you have a full view of the power 8 version.

Great work David... keep them coming!  :beer:

The idea came from thinking that if you have more dimensions than 2 for the vectors then the *angles* should be more than 1 dimensional - this struck a chord since the problem's always been that the 3 (or more) values concerned need changing simultaneously and single dimensional angles can't do that, even if you have more than one you always have the problem of order of application.
So I played with various ideas for complex angles - initially trying atan2(z)+flip(atan2(real(z)+flip(real(w))))) but that was interesting but not even close - in fact like just about nothing I've seen before ;)
I tried that and several similar options because I was attempting to strictly stick to a pure 3D or triplex type system but after about 3 hours of trying I gave up and switched to 4D.

The method is straightforward - it's just that we use the angles such that the (complex) sine is z/magnitude and the (complex) cosine is w/magnitude (where z and w are complex giving 4 dimensions).

Then I just used the normal equations for multiple angles i.e. sin(2a)=2sin(a)cos(a) and cos(2a)=2*cos(a)^2-1 - hence for the original squared version:

  float m = |z|+|w|   ; magnitude squared
  if m>0                   ; we use a division so make sure we don't have zero
      w = 2.0*z*w      ; w = (new mag)*sin
      z = 2.0*sqr(z)    ; z = (new mag)*2*sqr(cos) : note no -1 as we need to apply along the z vector not just to the real part
      float n = cabs(z) ; magnitude of 2.0*sqr(z)
      z = z*(n-m)/n    ; equivalent to (new mag)*(2*sqr(cos)-1) if it was just real
  endif

After the above just add constants as normal.

Also multiplication and I think division are pretty straightforward and I think it's a field - but maybe someone else can work that out ;)

One possible alternative that I can think of that in a way makes slightly more sense is if the magnitude is defined as either cabs(z)+flip(cabs(w)) or maybe cabs(real(z)+flip(real(w)))+flip(cabs(imag(z)+flip(imag(w)))).

The requested render using ((z^2)^2)^2 - am working on straightforward z^8 ;)

(http://nocache-nocookies.digitalgott.com/gallery/16/141_18_07_14_1_12_04.jpeg)

One point to mention is that the object is truly 4D but we're just viewing in 3D - and I think there are some disconnected bits on the z^2+c version plus of course some "whipped cream" however I think this may be explained by the fact that the DE used is doing the calculations in 4D therefore it could and will detect points of close approach to "solid" outside the 3D space *and* because it's a 3D slice of a 4D object the disconnected bits may well be connected via the 4th dimension which of course is cut as far as the view is concerned.

BTW the weird thing is I came up with this one after playing for hours trying to find the "correct" version of the proof for Fermat's last theorem i.e. the simple one that led to him just jotting the theorem down but not providing the proof....have got someway towards that too, obviously I'll be mentioning it somewhere if I find the solution ;)

The other weird thing is that for some reason I initially chose to use "2*cos(a)^2 - 1.0" for cos(2a) instead of cos(a)^2-sin(a)^2 and if you switch to using that instead.......

And if you switch to 1-sin(a)^2, well here's the zoom into a "bulb" from z^8+c using that method...

(http://nocache-nocookies.digitalgott.com/gallery/16/141_18_07_14_2_19_53.jpeg)

Nice. There are some streching on its back, but expected 3D mandelbrot features are not only on z=0 sides but alsou on y=0 top and bottom. And some strechings could be a feature of something I don't know, maybe low iteration number or DE;)

In picture 2 fractal looks as zoom on each stalk would look different. As on  my version holy of grail (http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/), which however hadn't nice stalks on the top. And there are thicker stalks on x>0, and much thinner objects on smaller bulb where x goes to -1 (?). Maybe try different zooms in different regions of z^2 as I did in that thread.

Interesting zoom in 3rd picture, most of holy grail versions haven't anything what would resemble 2D stalks. Throught these things on stalks looks as z^8 lakes and not as z^2 minibrots. Maybe cutting them at z=0 would reveal mandelbrot lakes or maybe it's the iteration number and DE. As when you zoom on stalk with low iteration number and see something round, then increase iteration number and mandelbrot appears.

Alsou woun't be bad to see a julia sets.

Anyway, nice insect:) The most grailish of them all.

Nice! I think that's the best power 2 bulb yet. The power 8 isn't as impressive though, because the whipped cream is easily visible, unlike the standard bulb where the whipped cream gets tucked into the crevices. I don't understand the math behind it yet, but I think you may be onto something.

If you want to test your theory that the whipped cream is an artifact of the DE algorithm, then the best way to test it is to do a high quality brute-force render. I suspect that the whipped cream is just part of the shape, and I don't think it's possible to get rid of it without some dramatic rethinking behind the way we construct 3D mandelbrots (of course I have no idea what that entails, haha :embarrass:).

Ah - no the disconnected bits (which definitely exist on the z^2) are the bits which I think are an artefact of the DE - the DE detecting points close but in fact outside the 3D slice of 4D space - I've had the same issue before when rendering true 4D fractals with the 4th term being non-zero.

The whipped cream is again I think possibly an issue with the image being a 3D slice of 4D space - the flat non-fractal bits could be missing fractalness because we simply can't view the rest that's outside the 3D slice - we're viewing the 2D (ignoring fractalness) surface of a 3D slice of a 4D object - this is a bit like viewing the 1D (ignoring fractalness) outline of a 2D slice of a 3D object.
Edit: Even simpler consider taking straight line segments through the standard complex Mandelbrot - assuming the line at least crosses some inside and some outside the amount of fractalness of the points on the line will vary wildly - for instance if the cutting line is the real axis then there's going to be no fractal detail at all with respect to the boundary points of the Set.

I love it when a mistake works - when rendering that last pic for using 1-sin(w)^2 for the double-angle cosine I messed up and corrupted w.

Anyway this is the result - "The Griffin (an ambulatory plant)"

(http://nocache-nocookies.digitalgott.com/gallery/16/141_18_07_14_11_25_52.jpeg)

I don't know about anyone else but I was wondering about the 3 different values calculated for cos(2a) using either 2*cos^2-1 or 1-2*sin^2 or cos^2-sin^2 even though complex numbers are supposed to work exactly like their real counterparts - the answer is simply because I used the magnitude of the hypotenuse rather than the complex value of sqrt(opposite^2+adjacent^2) which of course is a complex number itself.
I also realised the reason we have 3 possible values for cos(2a) (and in fact sin(2a) as well) is because we are in 4D and therefore have 3 independent angles (the 4th being fixed by the other 3).

The whipped cream is again I think possibly an issue with the image being a 3D slice of 4D space.

It probably depends on a fractallness 4 dimensions. Quaternions are fractals in just 2 dimensions. Maybe try a bitt zoom around small bulbs and antennas in pow2. It seems quite a promising. There looks as could be a different structures not so visible from the outside.

ALEF have right.
The graphic is not like 3D MANDELBROT set because is too stretched. As well Mandelbrot set 2D is pure fractal and not have neither stretched convergent point (like HENON attractor). So this function is yet a failure attempt to render a 3D MANDEL style, as other many mistakes.

IMO it's a fail for triplex, but I challenge anyone to prove it's a fail as a 4D version since it's impossible to see if the whipped cream exists when viewing the entire object (since we can't - unless someone's worked out a way).

I'm now working on multiplication, division and hopefully log and exp.

The whipped cream (as far as I can tell) disappears if you zoom in far enough increasing the "solid" level appropriately - for instance the whole area in the one below looks like a pipe in an area of stretched pipes until you zoom in this far (*125650):

(http://nocache-nocookies.digitalgott.com/gallery/16/141_20_07_14_10_00_56.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16429 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16429)

Please read the post with that one again - that was a bug in the formula that I just rather liked - not a 3D or 4D Mandy !!

Here's a simplified square (and power 8):

I checked the unit vector multiplication table and it cane out:


And then I realised how to fix the zero problem - use both the c^2-1 and 1-s^2 versions of the cosine as follows:



Whipped cream == gone (at least on the z^8)

Pictures!!!!  O0

Pictures - OK, OK - just one small point - although the (j,k)*(j,k) unit vector multiplication is all zeroes in fact this isn't the whole story - because I'm using the real value for the hypoteneuse instead of the complex value then with the exception of when the values on axes (i.j) is (0,0) the magnitude is still maintained correctly.
Also I have one more version to try - at the moment I'm essentially using either cos^2-1 or 1-sin^2 depending which avoids the zero issue but it could equally use cos^2 - sin^2 which gives the normal bicomplex on its own if I remember correctly - so instead of splitting the first two depending on the magnitudes of z and w we can split the calculation 3 ways -> when magz>total mag*2/3, magw>total mag*2/3 and the case when the magnitudes of magz and magw are similar.
Now to post another pic....

This is from the z^8+c using the 4D formula where the hypotenuse of a complex sided triangle (i.e. 4D) is taken as the real magnitude rather than the actual complex value, this allows three choices of calculation for the cosine (and the sine too I guess !). In this image the cosine used is essentially either cos^2-1 or 1-sin^2 depending on which is most likely to avoid asymptotes or whipped cream effects (basically this depends on the relative magnitudes of the initial z and w).

I found some "Super Villains" in the plain 4D Mandelbrot for z^8+c using this method (view x,y,z of x,y,z,w). This is at magnification *252098 with solid at 1e-9.

(http://nocache-nocookies.digitalgott.com/gallery/16/141_21_07_14_1_24_08.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16432 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16432)

Another pic - "Evolution", this is another z^8 Mandy but this time with cos(2a) as 1-sin(a)^2 instead of cos(a)^2-1, just to repeat the two are different because the hypotenuse is taken as real (the real magnitude of a 4 dimensional number made up of two complex numbers) with the opposite and adjacent both being complex.

(http://nocache-nocookies.digitalgott.com/gallery/16/141_22_07_14_5_27_45.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16433 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16433)

I did a few pictures of the first code and now I have a time explain what I meant.
I twas thinking everything what is not a fractal in lower number of dimensions would not be a fractal in higher number of dimensions. And fractal in higher number of dimensions will be a fractal in lower number of dimensions. At least for for quaternions, all the 2D slices of quaternion are fractals, exept YZ and YW axis are circles. I don't think something could appear in higher number of dimensions unless its very strached and that is how it goes in lower numbers of dimension.
However unused 4th dimension allows it to look throught the slices of 4th dimension or even to switch dimensions.

Alsou I think if it's a 3D mandelbrot then pow2 version should show becouse 8th power allways generates nice overall shape just by being perfectly symmetric. But maybe a pow 4 3D zooms would look the best becouse pow 2 looks like ribbons and stalks and pow 8 have too much small round things.

Anyway, no DE rendering, 20 iterations. Pow8 version is nice crispy. Maybe could do better zooms but that would take more time and I forgotten all the Chaos Pro setting which must be changed to get something out of it. And dont wanted render -> antialiasing what renders its two times as big so for times as slow.

All the parameter files included. This would be downloadable on the Chaos Pro only after the day, so the formula too.


Pow 2. Not so visible but all the smaller bulbs stalks and the seahorse area present.

(http://www.ljplus.ru/img4/a/s/asdam/1_Testbrot_pow2_old.jpg)

First smaller bulb:
(http://www.ljplus.ru/img4/a/s/asdam/2_pow2_bulb.jpg)

Somewhere on the x<0 bulb:
(http://www.ljplus.ru/img4/a/s/asdam/3_Testbrot_pow2_zoom2.jpg)

Pow 4. Maybe the bests zooms? never tried throught.

(http://www.ljplus.ru/img4/a/s/asdam/4_Testbrot_pow4.jpg)

Pow 8

(http://www.ljplus.ru/img4/a/s/asdam/5_Testbrot_1.jpg)

Julia -0.3, 0.5, 0.7
(http://www.ljplus.ru/img4/a/s/asdam/6_Testbrot_julia.jpg)

Somewhere below and to the right.
(http://www.ljplus.ru/img4/a/s/asdam/7_Testbrot_zoom2.jpg)

Seems perfect exept area of some ^#%&^.
(http://www.ljplus.ru/img4/a/s/asdam/8_Testbrot_pow2_cutY0.jpg)

This is strange. ZY x=0. However if you compare 3D cuyouts this seems to be the wort place even for pow8.
(http://www.ljplus.ru/img4/a/s/asdam/9_Testbrot_2D_X0.jpg)

Anyway, I think if it's now what we were seeking then its close to it.

All the parameters and Chaos pro formula files of the post above. I hope they 'll appear. EDIT: now they uploaded.

Thanks Alef - you basically confirmed my own findings - I'm still struggling a little with division.....

Did the same but now with max 150 iteartions. It's the same first  formula without "Whipped cream == gone (at least on the z^8)" post becouse this thread failed to be saved on my flash thus I could test it;) Maybe the streched horns are where division problems occours, overall its pretty nice and crispy.

Pow2
(http://www.ljplus.ru/img4/a/s/asdam/Testbrot_pow2_ultra.jpg)

Pow 4
(http://www.ljplus.ru/img4/a/s/asdam/Testbrot_pow4_ultra.jpg)

Zoom on the satellite on upper left sub bulb of the leftmost bulb of the power 4 thing. Chaos pro cutted off some piece on the right.
(http://www.ljplus.ru/img4/a/s/asdam/Testbrot_pow4_ultra_Zoom.jpg)

Parameter files included.

I had couple of old parameter files from this thread (http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/)
They all worked fine even in the most deep zooms just that the result was more fractallish. Throught I should say that i didn't spend much time to tweek them. Here thicknes of the fractal stalk depends on bailout value and when it have too much things around nothing is seen.

Not very elegant render, but it shows that it is very grailish. I hope someone could render better (old PC anyway).
(http://www.ljplus.ru/img4/a/s/asdam/3D_grail_3.jpg)

Bunch of parameters, of this and some other zooms on the stalks.

The versions above using 2*cos^2-1 and 1-2*sin^2 led to this unit vector multiplication:



So I decide to use this but play with 16 possible values of the 2*2 bottom right of the above:


This led to two forms both with perfect x/y and x/z Mandelbrot outlines and featuring bulbs and antennae if anything better than other attempts so far - just not spread around the entire fractal, rather sticking to the 90 degree positions around the x axis.

These are both expressed by the following vector multiplication table:



More later - I now have to beat 2 litres of semi-frozen home-made chocolate, banana, vanilla and brandy ice-cream then go visit my brother and family who are staying nearby and then come back and beat the ice-cream again....

Some pictures of the two types for the unit vector multiplication table given in my last post:

1st version:

z^2 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_23_58_162401796.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16465 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16465)

z^2 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_23_58_164652409.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16466 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16466)

z^2 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_23_59_16466266.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16467 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16467)

z^4 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_23_59_164671844.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16468 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16468)

z^4 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_23_59_16468644.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16469 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16469)

z^4 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_27_33_164691389.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16470 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16470)

z^8 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_27_33_16470702.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16471 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16471)

z^8 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_27_33_164711900.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16472 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16472)

z^8 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_27_34_164722339.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16473 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16473)


2nd version:

z^2 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_27_34_164731440.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16465 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16465)

z^2 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_29_58_16474147.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16475 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16475)

z^2 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_29_59_16475870.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16476 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16476)

z^4 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_29_59_16476328.png :o)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16468 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16468)

z^4 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_23_59_16468644.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16478 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16478)

z^4 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_29_59_16478575.png)

lhttp://www.fractalforums.com/index.php?action=gallery;sa=view;id=16479 (http://lhttp://www.fractalforums.com/index.php?action=gallery;sa=view;id=16479)

z^8 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_31_22_164791122.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16480 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16480)

z^8 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_31_22_164801334.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16481 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16481)

z^8 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_31_07_14_10_31_23_164811430.png)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16482 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16482)

Did this version. Is it Unit Vector matrix?

Now all of the streched horns and hairs are gone and perpendicular cut is solid. In former strange areas there is line like structures but these look fractal. Looks quite a convincing, throught there are question about angles. I got similar with formula where z was like abs reflected x. But with DE it dont looks as having lines and angles??? Re-rendered some of fractals.

pow2
(http://www.ljplus.ru/img4/a/s/asdam/1_Testbrot_pow2.jpg)

previous parameter of the above, now much cleaner throught still a slightly strange.
(http://www.ljplus.ru/img4/a/s/asdam/2_3D_grail_3.jpg)

Old coordinates from another formula worked fine, throught it's a bitt messy.
(http://www.ljplus.ru/img4/a/s/asdam/3_Old_antennas_800.jpg)

Zoom on its stalk, nicer.
(http://www.ljplus.ru/img4/a/s/asdam/4_Old_antennas_zoom1.jpg)

Old parameter of very deep zoom on some stalk.
(http://www.ljplus.ru/img4/a/s/asdam/5_3D_grail_10_minibrot_verydeep.jpg)

pow4. Looks unclean becouse of small details but there are some noticable angles.
(http://www.ljplus.ru/img4/a/s/asdam/6_Testbrot_pow4.jpg)

pow8
(http://www.ljplus.ru/img4/a/s/asdam/7_Testbrot_pow8.jpg)

Alsou tried some slices from 4th dimension. No renders, but looks like any non zero 4th dimension starting value curves the fractal, at least julia sets. Maybe it gives some hint about its (somewhat spherical) shape in 4 dimensions. Didn't tried to swap the dimensions, say instead of XYZ calculating XYW or XZW (the most outside) but probably it would be fractal from all dimensional viewpoints.
Improved formula file as it will be auto downloadable only after the day.

Not shure what it http://www.fractalforums.com/index.php?action=gallery;su=user;cat=95;u=141 (http://www.fractalforums.com/index.php?action=gallery;su=user;cat=95;u=141) is, but
I think sincos version looks the best, almoust like pencil drawn version.
(http://nocache-nocookies.digitalgott.com/gallery/16/thumb_141_01_08_14_4_17_37_16503631.jpeg)
 and the sin version looks strange. Alsou Unit vector matrix version z^2 have some smooth holes I hadn't (If that's what I rendered). Maybe a DE artefact :hmh:

Actually I never tried the version with all zeroes in the bottom-right 2*2 of the unit vector multiplication - that table just came from when I checked what the cosine version did *just using unit vectors*, I didn't check what happened with mixed values e.g. (0.5,1,2.7,-2) or whatever.
I don;t think the cos and sin versions fit into nice vector multiplication at all ;)

Anyway here's the post thread post for the new images for the cos, sin and sin+cos combined versions:

Cos version:

z^2 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_09_25_16482127.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16484 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16484)

z^2 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_09_26_1648470.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16485 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16485)

z^2 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_09_26_164851538.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16486 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16486)

z^4 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_09_26_164861411.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=164878 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=164878)

z^4 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_09_26_164871693.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16488 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16488)

z^4 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_11_28_164881011.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16489 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16489)

z^8 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_11_28_16489944.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16490 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16490)

z^8 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_11_28_16490799.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16491 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16491)

z^8 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_11_28_16491635.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16492 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16492)


Sine version:

z^2 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_11_29_16492278.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16493 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16493)

z^2 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_13_22_164931957.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16494 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16494)

z^2 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_13_23_1649497.jpeg)

[urlhttp://www.fractalforums.com/index.php?action=gallery;sa=view;id=16495[/url]

z^4 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_13_24_16495318.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16496 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16496)

z^4 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_13_25_16496479.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16497 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16497)

z^4 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_13_26_164971464.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16498 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16498)

z^8 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_15_56_16498988.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16499 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16499)

z^8 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_15_57_164992376.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16500 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16500)

z^8 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_15_57_165001197.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16501 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16501)



Sine+Cos combined version:

z^2 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_15_57_165011987.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16502 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16502)

z^2 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_15_57_165021839.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16503 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16503)

z^2 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_17_37_16503631.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16504 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16504)

z^4 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_17_37_165041732.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16505 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16505)

z^4 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_17_37_165052040.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16506 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16506)

z^4 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_13_26_164971464.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16507 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16507)

z^8 x-z sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_17_38_165071594.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16509 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16509)

z^8 x-y sliced

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_18_21_165081309.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16509 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16509)

z^8 full view

(http://nocache-nocookies.digitalgott.com/gallery/16/141_01_08_14_4_15_57_165001197.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16501 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16501)


The only thing is that the combined version has a calculation issue that bothers me - there's no smooth change from using the cos <-> sin so I'm going to investigate a "cookie-cut" version.

Good work David, and great support once again Alef!

Although I do not subscribe to the word "true" in the 3d or 4d context, I think these are the best I have seen so far.
Like Benesi it is the control of the surface derivation processes that deal with the hairy , whipped cream artefacts of early days.

It is analogous to a tight pressure on the surface dynamics keeping the wilder " coronl mass ejections" tight and bubblicious!  However filamentation or strands has to be part of expected outcomes for these trochoidal rotations. It is just a difficult question of controlling how much as elegantly as possible.

Theoretically the 4D one could allow us to get rid of them altogether - I suspect in these 4D ones the remaining problem bits would in fact disappear if we could actually view in 4D rather than just some 3D slice (or projection since we'd have to project 4D->3D->2D anyway).
I've been considering doing a 4D->3D->2D projection such that the ray-stepping is 4D and the render to 2D is made up of many 3D->2D renders each with the viewpoint start coord at a different 4th dimension coordinate such that the result is made by each pixel being coloured at least partly by the number of the passes at different 4th dimension coords where surface is found - however it would take some time to render such for it to be meaningful....

Great work David, very interesting to follow.

I like the shapes of the Cos version (even z^2) very much...

I wish Luca would do his magic with these to include them in Mandelbulb3D...

I'd recommend waiting to see if someone can come up with a method for at least fractional real powers and possibly 2D/3D/4D powers as well as other higher functions ;)

I suspect I can get fractional real powers but probably not the rest, I'm not really well versed in techniques for deriving such...and don't have Mathematica or Maple or similar to help.

What software are we using for these?

This is a dual number over complex.  So will be expressible in Clifford as well.



Replace the rule of dual number (e²=0)


Replace e -> j and ie -> k


Ugg..did all of this in mathematica and the order got moved around.

It's a bit tricky to get Mathematica to do non-commutative multiplication.
It states that "**" is supposed to be just that, but that operator doesn't seem to be particularly well-defined.

I define wrapper types.  The strange thing here is that the mathematica expression was in the order I expected...it just spewed out a different ordering when I wrapped it in TeXForm.  Never noticed that happening before....but then again I don't use it that much.

yeah, weird. you'd think, Mathematica wouldn't switch order between an expression and its conversion to TeX.

So did you already edit it to correct the ordering?
In other words, is this correct?

I have about zero free time for the next couple of day.  A quick look at it appears correct.  Note that dual-complex product does commute so no special work is required..I was assuming you were thinking about GA.  I just used replacements:  to convert the rules.  If you want to toy around with basic ops then you could define the rule:

that only works in simple cases, it could be beefed up to be even integer powers instead...or somebody probably has a dual number package (or see other equivalent systems below)

One thing to note about dual-complex forms is (that I'm aware of) they are naturally 2D systems embedded in 4D.  See here:

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/other/dualComplex/

Has links to other interpretations like SE(2) and Clifford/GA

you mean they are 2D embedded in 4D in that they are describing translation?

I actually was talking about GA before, and that TeX I gave below simply was what you gave before, rearranged to be more sensible ;)

I still gotta learn a lot about all the pattern matching abilities in Mathematica. It seems to be really powerful but thus far I always have to look up stuff and it's hard for me to find applications from the samples they give in the documentation, even if that is actually rather well done.

As I understand writing a formula for M3D is kind of hacking. Could be not so hard, just need to have a hex editor but then it alsou needs assembly language. Xe Xe maybe in some hacking forums you can find right guys. Or if you had even created some PC game trainer.

p.s.
Maybe an odd powers of 3,5,7 and sincos formula as it looks as imagined 3D mandelbrot. In 2D complex power mandelbrot is discontinious, so I think adding two more number components in power could make it just more discontinious. Throught it would be interesting in 2D to observe Z=Z^(x,y,z,w) +C even if it would not be very smooth.

Has anyone tried to hybridize the 2d-Mandelbrot into 3d ?

Using the standard z = z^2+c
On even iterations use c = x+yi
On odd iterations use c = z+yi

Or any onther combination of x,y and z. The one above will have the 2d M-set on the x=z plane.

Just a thought, would try it out myself if I wasn't travelling.

Keep up the good work!

This is related to bicomplex (http://en.wikipedia.org/wiki/Bicomplex_number).  Note the sign difference on j².  So is one reflection if k²=-1 and two if +1.

Also same as:  http://mathworld.wolfram.com/HypercomplexNumber.html when k²=1, otherwise one reflection (note mathworld mentions a povray function julia_fractal) and some authors call this version bicomplex as well:
http://www.3dfractals.com/docs/Article01_bicomplex.pdf
http://arxiv.org/pdf/1304.0781v1.pdf
http://www.scielo.cl/pdf/cubo/v14n2/art04.pdf

But then again, can also be expressed by a quaternion or split-quaternion expression as well.

Is this of any use...don't know.

Dunno about Alef but I'm just testing then in my Wip3D formula for Ultrafractal - I've added an extra "4D" option with a "func" call to make it easy to try any new formulas - but haven't released it publicly yet partly because I want to add convergent bailout first.

Actually that unit vector multiplication table is commutative - *but* mathematica probably dislikes the zeroes ;)

OK back to the sin and cos combined version - I worked out a seemingly perfect reciprocal and decided to try doing some 3D/4D Newton renders - though as yet just using my old brite-force 3D/4D render formula coz my wip3D one still isn't updated to handle convergence.

For the combined sin/cos version the (unoptimised) reciprocal is as follows using complex values (note that |z| returns the *square* of the magnitude as per normal Fractint and UF behaviour):

  func inv(complex &z, complex &w)
    float t = 1.0/(|z|+|w|)
    complex we = t*w
    complex ze = -t*z
    float ms = |ze|*|z|
    float mt = |we|*|w|
    complex nz
    if mt>ms
      nz = 1.0/(we*w)
    else
      nz = 1.0/(ze*z)
    endif
    z = ze*nz
    w = we*nz
  endfunc

If you examine the above you'll see that when w is (0,0) on entry the reciprocal becomes 1/z for z and (0,0) for w as it should be (and vice-versa).

And in the next post will be 2 views of each of the standard degree 3 and 5 Newton's for the roots of (+x,0,0,0).

 :hmh: deleted that last post and the 4 pics coz I found a bug - fortunately it means the solid non-planer surfaces have gone - problem is the Newton though now correct is a little more complicated than I anticipated - it'll take a while to get decent renders.
For those interested I messed up when copying and then forgetting to change some variable names !!

Interesting idea.

Few days ago I had about the same thought on alternating the formulas in 2D for the Mandelbrot set and Burning Ship fractal. Does anyone know if there is software that already does this alternating-hybrid thing?

There are alternating formulas in most software listed here in the forums I think - the first were probably done in Fractint like most things fractal ;)

The next step is escape-time IFS where the "transforms" can be any escape-time formulas.....plus added controls for which "transforms" are allowed where etc.

There are also formulas that implement rising polynomials over iterations - such as expanding the sequence of Lucas Polynomials on each iteration - see:

http://mathworld.wolfram.com/LucasPolynomialSequence.html (http://mathworld.wolfram.com/LucasPolynomialSequence.html)

The recurrence relation can simply be used as an iterative method.

OK I've now done some videos showing the structure of the Newton for the cube roots of 1 using the 4D sincos combined method.

Here are the links:

Newton XYz starts with the clip window (limited z slice) showing the complex equivalent view then slides along the z axis....

https://www.youtube.com/watch?v=ywmbN1zegeA (https://www.youtube.com/watch?v=ywmbN1zegeA)

Newton xYZ starts with the limited x slice at -8 and slides along the x axis to +8...

https://www.youtube.com/watch?v=1GwSuuIXNKU (https://www.youtube.com/watch?v=1GwSuuIXNKU)

Newton XyZ starts with the limited y slice at -8 and slides along the y axis to +8....

https://www.youtube.com/watch?v=cVQPspjurxU (https://www.youtube.com/watch?v=cVQPspjurxU)

Didn't read thorougly as I prefare to read hard texts when sitting in cousy chair and drinking a tea but bicomplex numbers just produce square like mandelbrot.
(http://www.bugman123.com/Hypercomplex/Tetrabrot.jpg)

Everything I posted is made by Chaos Pro -> update Formulas -> MalinovskyFract -> DavidsGrail. Very easy to write 3D formulas throught slightly outdated renderer, and not so easy to get nice images as Mandelbulb3D. And Ultra Fractal you alsou have to write your own 3D raytracer (most of the authors are individualists so no plugins;)  ).


I think bottom right zeros in multiplication matrix should not be inherently wrong. There are dual numbers where epsilon represented unit with property e*e=0.
http://en.wikipedia.org/wiki/Dual_number (http://en.wikipedia.org/wiki/Dual_number)
More strange is that both k*k and j*j is 0. But 4th dimension is absent thus j is like k and J*J = k*k =0. Thus in 3rd and 4th dimensions this fractal must look identical (like quaternion in i, j, k). So this could be extruded (linear) or revolved between 3dr and 4th dimensions. Thus it practicaly is 3 dimensional and not 4 dimensional, exept julia set when z,w aren't 0.

Maybe there could be multiplication matrix with 3 (or 1) zeroes only for k and j*j=r.
But then if k allways =0 there are just r, i, j parts and r*r, i*i, j*j are perfectly different r, -r ,0. Here r*r + i*i + j*j =0 as r*r + i*i =0 of complex, not shure is this important.

Twinbees 3D mandelbrot site have this: On October 13th, 2006, Marco Vernaglione put out the question and challenge to the world with this memorable document. Looks very much like sincos version.
(http://mandelbulbs.s3.amazonaws.com/new/q85/Imaginary-3-dimensional-Mandelbrot-set.jpg)

http://www.renderosity.com/mod/gallery/index.php?image_id=1308487&member (http://www.renderosity.com/mod/gallery/index.php?image_id=1308487&member)

Notice simmilarity.
(http://nocache-nocookies.digitalgott.com/gallery/16/thumb_141_01_08_14_4_17_37_16503631.jpeg)

Rendered 4 zero matrix mandelbrot from unusual perspective.

There are some strange angles and strands on the top but alsou there are bunch of mini mandelbrots (universality?) near the object. Hadn't seen these small repeated minibulbs on the mandelbulb fractal.

(http://nocache-nocookies.digitalgott.com/gallery/16/5956_20_08_14_6_39_55.jpeg)

Very beautiful and VERY INTERESTING render Alef!!!

And you are right, looks very much like Marcos drawing, I remember it from Daniels site!

Thanks KRAFTWERK  :D

wow, this all looks better than the mandelbulb (in my opinion).

Yep, because the original mandelbulb is ugly anyway. Never really understood why that power was chosen for the bulb.

I think pow8 was chosen becouse of its simmetry but probably alsou becouse square mandelbrot have strange ungrailish things on its top and pow8 hides them :embarrass:

Indeed, 4 zeros vector multiplication table of cos formula is the same as a dual complex numbers. Throught what happens could differ, probably so far only the author understands what is going on :dink:

These strange angled stalks on top have something to do with bailout value, on minimal bailout they aren't visible. Probably a slow escape.

Looked at 3D slices of 4D, or in plain language rendered it with normaly absent 4th component of pixel beeing w=-0.11 and w=0.11. This curved fractal to the side and all the smaller bulbs are gone. I think w and z could be identical, there are no smaller bulbs on z axis and probably in 4th dimension its somewhere linear or revolved :hmh:

pixel w=-0.11
(http://www.ljplus.ru/img4/a/s/asdam/Wiev_4thdim_0point1.jpg)

pixel w=0.11
(http://www.ljplus.ru/img4/a/s/asdam/Wiev_4thdim_opposite4tdim.jpg)

p.s.
Sincos looks like spacecraft. I was watching news about another political ^&^#% and they showed spacecraft who looked just like sincos bulb.

More detail of the sin/cos combi version for z^2+c - solid at 1e-8

Top of the whole thing showing many minis above:

(http://nocache-nocookies.digitalgott.com/gallery/16/141_24_08_14_4_33_18.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16571 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16571)

The Minibrot M1 from the above:

(http://nocache-nocookies.digitalgott.com/gallery/16/141_24_08_14_4_34_17.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16572 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16572)

The Minibrot M2 from the above:

(http://nocache-nocookies.digitalgott.com/gallery/16/141_24_08_14_4_35_14.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16573 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16573)


That's the good bit - the bad bit is that all 3 forms, cos, sin and cos/sin combined fail as mathematical fields because although unlike quaternions they are commutative and unlike bi-complex they have a full division algebra, they are neither distributive nor associative :(

Who cares?  O0 Those minibulbs looks scaringly Grailish...  :beer:

congrats, indeed a very good candidate ;)

I mentioned the field thing because I believe if a true R4+ field is found then we'll have the perfect Grail ;)

Also an interest point I thought I'd add is that neither the plain cos nor sin versions of this form have minibrots at the "top", they only appear in the combined version.

This sounds as almoust an universality, most of "grails" didn't had such nice self repeating and mini mandelbrots. Could the author publish sincos formula for pow2 for those who have no idea how to turn ^-1 into ^2? Unless of corse this is going to arxive or there are another reason for not sharing it.

Looking at pow4 zooms gave the idea that without DE it have too much details. But a walk throught 4th dimension was more promising.
Julia sets should have very elaborated 4D geometry as when you render it with non zero 4th dimension pixel value you sometimes can see something appearing, features changing and spirals rotating. In Mandelbrot set strange areas with disconnected lines and angles are just in the same regions where with changing 4th dimension pixel value solids appears (in animation at right corner). So it could be that these distorted lines are remnants of some 4th dimensional objects. So I was wrong, it's fully 4 dimensional.
Hadn't tested the fractal with switched dimensions throught.

Cos Julia set (-0.5; -0.3; 0.15; -0.25)  rotating and moving throught 4th dimension with pixel value from -0.3 to 0.

(http://fc05.deviantart.net/fs71/f/2014/238/8/e/mandeldragon_vid_by_edo555-d7wqsig.gif)

smaller
(http://fc09.deviantart.net/fs71/f/2014/238/6/0/julia_set_of_3d_mandelbrot_by_edo555-d7wqsig.gif)

And a cos mandelbrot set going thought 4th dimension from -1.24 to +1.24.

(http://fc08.deviantart.net/fs70/f/2014/238/1/0/mandanim_forgif_by_edo555-d7wqt8k.gif)

smaller, 534 kb
(http://fc09.deviantart.net/fs70/f/2014/238/b/4/3d_mandelbrot_by_edo555-d7wqt8k.gif)

I had forgotten what it means. At least there are division, so it's more practical than bi-complex.

The images you posted here are interesting, but unfortunately they don't look much like the picture from Marco Vernaglione.

I have an idea about this, but I am not able to realize it myself (at least not for a long time).
Because the roots of the expanded Mandelbrot formula are indeed located where the bulbs are expected.

First iteration, x, has root in 0. Thats the main bulb, so it can be rotated around the real axis.
Second iteration, x^2+x, has a new root in
X2=-1
And around this points there is the second big bulb. So, just rotate them around the real axis.

Second iteration, (x^2+x)^2+x, has (new) roots in
X2=(-0.1225611668766536+i0.7448617666197443)
X3=(-0.1225611668766536-i0.7448617666197443)
X4=-1.7548776662466927
So, you could put the 2D bulbs there, rotate around the axis of the angle from (0,i0) and put additional bulbs rotated 90 degrees.

Third iteration, ((x^2+x)^2+x)^2+x, has (new) roots in
X2=-1.310702641336833
X3=(0.28227139076691393+i0.5300606175785253)
X4=(0.28227139076691393-i0.5300606175785253)
X6=(-0.156520166833755-i1.032247108922832)
X7=(-0.156520166833755+i1.032247108922832)
X8=-1.9407998065294843
So, you could put new spherical bulbs there, and put additional bulbs rotated 45 degrees.

etc etc
Would that be possible?
Would it give a good result?

Like this?
 :D

Edit:
Made a fun movie of it as well:
https://www.youtube.com/watch?v=vMfbp2xmAQk

Yes I think that's a way to visualise what we're aiming for - but we want it from a straightforward z^2+c formula rather than a manipulation that produces what we're after ;)
We're getting close enough now for me to be convinced the required form does exist, probably as a 4D system rather than a 3D one though ;)

The flip side is that having a preconceived notion of what it should look like might be steering you in the wrong direction.

indeed. For one thing, why would it have a 4-fold symmetry?

Though basically all we need would be to geometrically separate each bulb of the mandelbrot, rotate it around its local axis, and instance it a couple (symmetric) times around its mother-bulb. If we somehow can describe this process geometrically and efficiently, we would get exactly that movie except with arbitrary detail.

I was hoping that the relation between the center of the circles in the 2D Mandelbrot and solutions of the expanded z=z²+c polynoms could give any hint to create the shape preconceived by Marco Vernaglione.
But maybe I am only spoiling the thread, sorry.

A function that would result in Marco Vernaglione's shape would only be interesting if the iterations are too low to determine if points are in the m-set or not, which results in spirals etc. If the iteration is enough it would only result in prefect spheres. So Marco Vernaglione's shape is probably not even interesting.

It is fascinating though, that the polynoms quickly get ridiculous high degrees and that the circles of deep minibrots represent solutions of mind blowing high degree polynoms. We know the solutions, because we know the location of the circle, but we have no idea of the polynom! We know the answer but not the question :)
But that is perhaps another topic...

i think if people were honest when they talk about the true grail what they really mean is a truly aesthetic grail, not a truly mathematical grail.  basically a preconceived idea of sort of rotating a 2D mandelbrot to produce something like what karl just posted, which is visually geometrically pleasing.  in the time that ive been interested in fractals ive always thought the mandelbox is far more pure and grailish as a 3D fractal than any attempts at projecting a 2D mandelbrot into 3D will ever be, both in the simplicity of the math and the resultant beauty and complexity of the object it produces.  if the mset is the 2D grail, i think we already have a native 3D grail in the mandelbox.  that being said, its always exciting to see people trying to explore new territory  :)

this is exactly my opinion, the mandelbox could serve as THE 3d fractal holy grail, its simplicity is unbeaten, it does not even induce those althoug-not-so-complex "complex" numbers and works by just plain multidimensional real number mathematics

Mandelbox isn't so much explorable and haven't all that mathematical mumbo jumbo behind :hmh: Probably David could publicate this in some scientical journal, there are folks whou are recieving goverment grants and making publications for much less creative things about the fractals.

This thing are perfectly symmetrical, but maybe too much. Even with more bulbs it would be pretty boring in the respect of exploring and don't gives feeling about the mandelbrot set with all of the imperfect lightnings and seahorses.

The beauty of the 2D Mandelbrot is because we can colorize points based on how fast we can determine that they are not in the set. If we only look at the points that are inside the set, they form boring circles and a few cardioids.
So it might be that the idea of a 3D Mandelbrot is fundamentally wrong?

Seen in space, the mandelbrot set is all about "the spaces in between", like the empty space of a mould that would be filled with matter to form the actual product. What about mass and the real world instead of the math? We are also about ... 99% or something of empty space. The closer you look, the less there is. Also seen in time (iteration time) it's all about the journey, not the destination. In the burning ship iteration bands often end in nothing (no miniship or whateven, just singularity). But in the mandelbrot it all approaces the boundary (is this a real boundary ?!). So I would think that the DE method for the 3D mandelbulb is correct. I just don't understand how the formula in 3D was formed. So not very sure on the questions maybe not to be asked.

it seems to me that the boundary of the mset is more than just somewhat analogous to reality, but in fact perfectly analogous to reality, more 100% than 99%.  the closer you look at reality it seems to be comprised of the "boundaries" of the "orbits" of "energies".  some arrogant humans might like to argue that there is a point at which you cant go any smaller, such as the planck length, but would anyone really try to argue that it is simply a perfect "solid" at such a point?  a bunch of perfectly "solid" indivisible planck marbles?  also considering that the colloquial notion of "solid" is rendered all but entirely empty and meaningless in the context of defining reality as closer to what it really is, the interaction of the boundaries of the orbits of some mysterious cosmic energies that never die.

But I don't know if I think for example Steven Hawking is arrogant in this topic...

All evidence shows that things only occur in discrete chunks.  The mathematics of reals and the associated notions of infinity and infinitesimal are pure abstraction (imaginary things).  Logically they can't exist either because they imply an infinite amount of information per item...so an infinite amount of infinite.

just because he is intelligent does not mean he is not arrogant.  possessing intelligence beyond the average person does not somehow exempt you from the pitfalls of human emotions and desires such as arrogance and the desire to command authority.  indeed extreme intelligence often goes hand in hand with human failings like arrogance and the desire to command authority.  the hallmark of a truly wise individual is the capacity to say "I don't know," or "The following is my best guess, though we cannot know for sure at this time."

in other words "all the evidence" boils down to "Infinity does not exist because it wouldn't make sense to me," and the variation on that, "Infinity does not exist because I wouldn't be able to calculate it."  this perfectly and succinctly illustrates the unfortunate world view even lots of brilliant minds rigidly adhere to.

The notion of a smallest possible unit of measure does not imply solidity, just what it says - a smallest possible unit of measure.

Perhaps beyond that we may pass through to some other weird dimension like in Star Trek's "Subspace" but that does not exist in our Universe (it is outside our universe) so the concept of a smallest possible unit remains valid even though it is possible to pass beyond that limit.

But the only thing we can be certain about regarding the infinite is that we can never prove it one way or another, so we all have to either take a side and choose to believe it or not, or of course take the third option and choose not to decide.  

I choose to disbelieve the notion as anything more than an abstraction.


I see it more as "Infinity does not exist because it is inconsistent with Quantum Theory, Logic, and The Nature of Reality!"

I have long held that Infinity is a very useful Mathematical Construct, but it has no corollary in Reality.

Of course, it's possible that Quantum Theory is wrong and the Monster really does exist, but the two are mutually exclusive in my view.

good point! an important distinction.


indeed!  ideally activities such as "belief" and "taking sides" would occur outside of science.


the problem with this type of argument in my view is that it evolves from an underlying assumption that a fundamental, immutable property of reality and the universe is that it cannot possibly hold any properties which a homo sapien living on earth in the 21st century AD cannot comfortably understand and calculate with his maths.  if we were being perfectly honest, i think this belongs more in the aforementioned "belief" and "taking sides" range of activities which, again, would ideally occur outside of science.

Notions of infinity do make sense to me.  As an abstraction.  Notice I'm not really taking a physics model as a reasoning, that was just tossed that out as a bullet point.  I'm taking a pure mathematical one.  Information theory.  Infinity and infinitesimal require infinite information.  But more than that, by Cantor's proof most real numbers are irrational and therefore require infinite information each.

On the contrary!

It says that just because a homo sapien living on earth in the 21st century AD can comfortably understand and calculate something with his maths does not necessarily mean it exists   O0

We make calculations using infinity every day. And many of us believe in things that can never be subjected to such rigor  (like god, reincarnation, Santa Claus, Karma, a Sentient Universe/Planet, and even Ceiling Cat!). But merely being able to calculate something mathematically does not make it real.


Again, I beg to differ.

Science is all about "taking sides" and "belief systems" in order to devise and test new theories.

Without believing in their theories before they are proven, there would be precious little motivation to devote lifetimes to gathering supporting evidence for them.

In this respect Science and Religion have much in common.  To truly "Believe" in The Big Bang is every bit as much a leap of faith as belief in any supernatural being (or even Infinity) - with the main distinction being that Science always leaves room for new evidence to change our prevailing theories, while religion insists that nothing can ever change their beliefs.

yes i agree the same base motivations like self-promotion, self-aggrandizement, and tribalism that inspire and perpetuate things like religion do not magically disappear when it comes to humans attempting to engage science.  it is simply unfortunate that a selfless desire to know the universe regardless where the answers lead is not motivation enough, and exponentially more unfortunate are the implications this has regarding the merit and reliability of the human institution of science.  :-\

Not quite so. Here I removed all the colouring just the insides or outsides. Max iteration limit of 250 did the rest. Of coarse If you increase maxiter to 2500 you 'll get 3 small minibrots. This works almoust like colouring just it would be almoust impossible to manualy guess the border.

"But I don't know if I think for example Steven Hawking is arrogant in this topic..."
In wikipeda this argument is listed as logical fallacies 'appeal to authority'.

"yes i agree the same base motivations like self-promotion, self-aggrandizement, and tribalism that inspire and perpetuate things like religion do not magically disappear when it comes to humans attempting to engage science."
Well I read article about science in my country. It said, we have a lot of female scientists not becouse of equality but becouse in our country science have no such glory (and $). Probably even science works in the same natural / biological categories of alfa scientists.

p.s.
Maybe this could be turned into Folding Int Pow. One of the most popular formula in hybrids.

GROWL never noticed this until now! :fiery: Moved this into a more visible section... :D And implemented :beer:

There are many more formulas just one done btw!

A power 8 - with Mandelbulb's "rot of whole bulb" :D

 :horsie: Keep 'em coming Luca!  O0
(http://nocache-nocookies.digitalgott.com/gallery/16/141_24_08_14_4_33_18.jpeg)
^^^This also?

 :thumbsup1:

Hellooo ;)

Johan I must find out thise formulas ;)

Not sure I can get them all working? The one(s) I finished is very nice especially pow2 ;)

EDIT; Also I didn't use cabs() in my formula sqrt(a*a + b*b) but simply (a*a + b*b). Just testing cabs() and it looks a lot less nice actually :sad1: So I will keep old function ^-^

After a fight :D finally successful done the formula in 1st page in three flavors even if the only useful is probably pow8? :D

 :horsie: :horsie: :horsie:

It's got SPINES!!!!  :beer:;)

Stop claiming and render :order:

 :surrender:

Lots of weirdness on the inside of that p8 version.
Thank you David, and Luca for implementing!!!  :beer: :beer: :beer:

Next: Zoom in on that spine...  O0

...Spine!  :beer:

Looks very very nice :D Happy you find a new toy :angel1:

 :thumbsup1:

That's a very nice find!

:) Thank you guys!

Some spiral craziness in the Makin4D p2a  :beer: O0

 :beer: O0

 :beer: :beer: O0

Those details!!! :alien: wonderful Johan :beer:

It was a fun break from my other works.

I have found interesting shapes on the outside of it... my only concern is that it does not look fully connected...  :hurt: