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 Author Topic: True 3D Mandelbrot ?  (Read 10995 times) Description: Yet another try at the "true 3D" Mandelbrot 0 Members and 1 Guest are viewing this topic.
David Makin
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Posts: 2286

 « Reply #30 on: July 23, 2014, 02:39:38 AM »

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

Posts: 1174

 « Reply #31 on: July 25, 2014, 06:24:53 PM »

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

Pow 4

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.

Parameter files included.
 grail2Test.par (6.21 KB - downloaded 98 times.) « Last Edit: July 25, 2014, 06:40:42 PM by Alef » Logged

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Alef
Fractal Supremo

Posts: 1174

 « Reply #32 on: July 28, 2014, 04:51:15 PM »

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).

Bunch of parameters, of this and some other zooms on the stalks.
 3Dgrail.par (10.95 KB - downloaded 94 times.) « Last Edit: July 28, 2014, 04:59:10 PM by Alef » Logged

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David Makin
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Fractal Senior

Posts: 2286

 « Reply #33 on: July 31, 2014, 07:57:51 PM »

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

Code:
r     i    j     k
r    r     i    j     k
i    i    -r    k    -j
j    j     k    0     0
k    k    -j    0     0

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

Code:
j      k
j   +/-r   +/-i
k   +/-i   +/-r

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:

Code:
r     i    j     k
r    r     i    j     k
i    i    -r    k    -j
j    j     k   -r     i
k    k    -j    i  +/-r

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....
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David Makin
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Fractal Senior

Posts: 2286

 « Reply #34 on: July 31, 2014, 11:15:09 PM »

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

z^2 x-y sliced

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

z^2 full view

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

z^4 x-z sliced

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

z^4 x-y sliced

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

z^4 full view

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

z^8 x-z sliced

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

z^8 x-y sliced

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

z^8 full view

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

2nd version:

z^2 x-z sliced

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

z^2 x-y sliced

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

z^2 full view

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

z^4 x-z sliced

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

z^4 x-y sliced

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

z^4 full view

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

z^8 x-z sliced

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

z^8 x-y sliced

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

z^8 full view

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

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Alef
Fractal Supremo

Posts: 1174

 « Reply #35 on: August 01, 2014, 06:32:50 PM »

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

Code:
r    i    j    k
r    r    i    j    k
i    i   -r    k   -j
j    j    k    0    0
k    k   -j    0    0

Code:
blablablabla

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

previous parameter of the above, now much cleaner throught still a slightly strange.

Old coordinates from another formula worked fine, throught it's a bitt messy.

Zoom on its stalk, nicer.

Old parameter of very deep zoom on some stalk.

pow4. Looks unclean becouse of small details but there are some noticable angles.

pow8

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.
 DavidsGrail.zip (1.5 KB - downloaded 101 times.) « Last Edit: August 01, 2014, 06:39:07 PM by Alef » Logged

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Alef
Fractal Supremo

Posts: 1174

 « Reply #36 on: August 01, 2014, 06:49:49 PM »

Not shure what it 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.

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
 « Last Edit: August 01, 2014, 06:55:13 PM by Alef » Logged

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David Makin
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Fractal Senior

Posts: 2286

 « Reply #37 on: August 02, 2014, 02:59:31 AM »

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

z^2 x-y sliced

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

z^2 full view

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

z^4 x-z sliced

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

z^4 x-y sliced

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

z^4 full view

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

z^8 x-z sliced

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

z^8 x-y sliced

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

z^8 full view

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

Sine version:

z^2 x-z sliced

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

z^2 x-y sliced

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

z^2 full view

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

z^4 x-z sliced

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

z^4 x-y sliced

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

z^4 full view

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

z^8 x-z sliced

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

z^8 x-y sliced

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

z^8 full view

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

Sine+Cos combined version:

z^2 x-z sliced

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

z^2 x-y sliced

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

z^2 full view

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

z^4 x-z sliced

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

z^4 x-y sliced

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

z^4 full view

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

z^8 x-z sliced

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

z^8 x-y sliced

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

z^8 full view

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.
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jehovajah
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Posts: 2749

May a trochoid in the void bring you peace

 « Reply #38 on: August 02, 2014, 08:20:42 AM »

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.
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David Makin
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Posts: 2286

 « Reply #39 on: August 02, 2014, 10:20:38 PM »

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....
 « Last Edit: August 02, 2014, 10:28:06 PM by David Makin » Logged

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KRAFTWERK
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Fractal Senior

Posts: 1439

Virtual Surreality

 « Reply #40 on: August 04, 2014, 09:51:05 AM »

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...
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David Makin
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Fractal Senior

Posts: 2286

 « Reply #41 on: August 05, 2014, 11:56:54 PM »

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.
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Roquen
Iterator

Posts: 180

 « Reply #42 on: August 06, 2014, 10:03:52 AM »

What software are we using for these?
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Roquen
Iterator

Posts: 180

 « Reply #43 on: August 11, 2014, 10:40:50 AM »

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

Code:
r     i    j     k
r    r     i    j     k
i    i    -r    k    -j
j    j     k    0     0
k    k    -j    0     0

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

$
\left(a_2 e+i a_3 e+a_0+i a_1\right) \left(b_2 e+i b_3 e+b_0+i b_1\right)
a_2 b_2 e^2+i a_3 b_2 e^2+i a_2 b_3 e^2-a_3 b_3 e^2+a_2 b_0 e+i a_3 b_0 e+i a_2 b_1 e-a_3 b_1 e+a_0 b_2 e+i a_1 b_2 e+i a_0 b_3 e-a_1 b_3 e+a_0 b_0+i a_1 b_0+i a_0 b_1-a_1 b_1
$

Replace the rule of dual number (e2=0)
$
a_2 b_0 e+i a_3 b_0 e+i a_2 b_1 e-a_3 b_1 e+a_0 b_2 e+i a_1 b_2 e+i a_0 b_3 e-a_1 b_3 e+a_0 b_0+i a_1 b_0+i a_0 b_1-a_1 b_1
$

Replace e -> j and ie -> k
$
a_2 b_0 j-a_3 b_1 j+a_0 b_2 j-a_1 b_3 j+a_3 b_0 k+a_2 b_1 k+a_1 b_2 k+a_0 b_3 k+a_0 b_0+i a_1 b_0+i a_0 b_1-a_1 b_1
$

Ugg..did all of this in mathematica and the order got moved around.
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kram1032
Fractal Senior

Posts: 1863

 « Reply #44 on: August 11, 2014, 12:27:08 PM »

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.
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