• May 18, 2021, 03:46:00 PM

### Author Topic:  Mandelbrot 3D: Mandelnest  (Read 3853 times)

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#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #105 on: May 04, 2021, 12:14:00 PM »
Alef: I’ve seen your achievements on DEVIANT ART, you’re an accomplished artist, congratulations. I don’t know how you get these beautiful images by mixing quaternions with my Mnest  , but it’s very successful, and I am honoured to contribute to art  , even if my main motivation is mathematical and philosophical.

#### Alef

• Fractal Friar
• Posts: 140
• a catalisator / z=z*sinh(z)-c^2
##### Re: Mandelbrot 3D: Mandelnest
« Reply #106 on: May 04, 2021, 03:27:46 PM »
Thanks
I'm not so mutch an artist (at least not so good), I mostly interested in algorithms and then make images on these.
There quaternions aren't used. I used a method what worked on quaternions.

//Fold (slightly different than of mandelbox)
X := X +  abs(x- FoldX) - abs(x+ FoldX)  ;
y := y +  abs(y- FoldY) - abs(y+ FoldY)  ;
z := z +  abs(z- FoldZ) - abs(z+ FoldZ)  ;
//Scale
x:=x* Scale;
y:=y* Scale;
z:=z* Scale;
Mandelnest (or quaternion) formula

It is then multiplied and conjugated. Idea behind this was that most of users in 3D for artistic purpose prefare box like formulas. Maybe this contributes some small insights in mathematics, sutch us revealing conjugate character, but its more for the visual purpose. Anyway mandelnest I think is superiour to mandelbulb in expansion of mandelbrot set to 3 dimenions.
a catalisator / z=z*sinh(z)-c2

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #107 on: May 04, 2021, 04:13:23 PM »
@Alef: Anyway mandelnest I think is superiour to mandelbulb in expansion of mandelbrot set to 3 dimenions.
Thanks for the explanation. Yes, it is possible that the Mnest algorithm is more general than Mbulb one, in the sense that it allows to generate both Mbrot 3D and dual  real Mbar 3D. It allows a generalization to any dimension easily also. The idea of working with non-Euclidean (pseudonorms) is independent of Mnest, even if it is well adapted to it, and is perhaps also a promising idea to exploit for other types of fractals more generally, that you inderstood I beleave.
But my intuition tells me that young Mnest hasn’t revealed all his secrets yet, so I’m still working on it...
« Last Edit: May 04, 2021, 04:48:43 PM by Jeannot »

#### Alef

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• a catalisator / z=z*sinh(z)-c^2
##### Re: Mandelbrot 3D: Mandelnest
« Reply #108 on: May 04, 2021, 07:22:02 PM »
Mandelbulb is less simmetrical. There is something called "perpendicular mandelbrot" which is just distorted mandelbrot set of XZ axis of mandelbulb. (Formula of perpendicular mandelbrot is z= z2 + c and then abs(x)-y ) This is kind of hacking to "real 3d mandelbrot".
Instead mandelnest for power 3 generates something what could be called platonic solid as a main bulb. All the cutouts are 2D tricorn fractals. There is no trickery to get intended shape.

It's what you consider pseudonorms. This
https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions
is not euclidian but probably is not considered pseudo.

These works mostly nicely. It just sometimes needs slight adjustment of render quality. Probably astroid 2/3 would destroy fractal, I had not tried. Wikipedia gives links to mutch more strange spaces.  " Lp spaces form an important class of Banach spaces" And then euclidian space is p=2 subclass.

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #109 on: May 04, 2021, 08:35:20 PM »
@Alef: ...is not euclidian but probably is not considered pseudo.
Of course, I do not confuse pseudonorm and Pnorm witch are notions different.
On Internet I found definitions variable for pseudonorm... I use of this flexible term for want of anything better to disign different exotic norms that can be used simply in programmation, and not only Pnorm (see my page above on this post, for example I proposed "cardioid 3D").
If I stick to the definition below, it seems to me that Pnorms can be included among the generic term of pseudonorms, like any norms, but maybe I’m wrong...
1-p(a v) = |a| p(v), (positive homogeneity or positive scalability)
2-p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity)
In fact, the nuance with a norm is that a pseudonorm is allowed to assign zero length to some non-zero vectors.
But I agree with you, talking simply about non-Euclidean norms might be more adequate. Or if you have a better term no problem.
« Last Edit: May 05, 2021, 10:31:43 AM by Jeannot »

#### Alef

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• a catalisator / z=z*sinh(z)-c^2
##### Re: Mandelbrot 3D: Mandelnest
« Reply #110 on: May 05, 2021, 06:23:12 PM »
Different norms is one my favorit lets say fractal instrument. These can be applied to mandelbulb, here in mandelnest but alsou to the mandelbox with sphere fold. (To the majority of 3D fractals as most of them rely on radius. Unlike 2D which mostly don't use it.) In 2D some formulas use abs(z) instead of proper |z| in bailout conditions what just changes colour pattern not the fractal. And it don't require very sophisticated formulas.

I think p=4 or p=8 gives kind of advanced looks like of smartphones or streamlined inter city busses.

In wikipedia I just found F-norm (of p=0) and a lots of exotic spaces I don't understand. There are some unexplored possibilities and need for testing.
Zero lenght to non-zero vector (what in my opinion could be called pseudo-) could pose problem to renderer program if it is periodicaly encountered outside of the fractal.

I used this to distort mandelbulb
Code: [Select]
 //inverse of geometric mean, calculates necesary root value invpower:= power(intPowerX *intPowerY *intPowerZ,-0.33333333333333333333); temp:=  intPower( abs(x),intPowerX) + intPower( abs(y),intPowerY) +intPower( abs(z),intPowerZ); r  := power( temp ,invpower);inversepower= (Px * Py * Pz )1/3
temp = |x|Px  + |y|Py + |z|Pz
This is owerkill and pseudo- as I chose geometric mean becouse it gave more better results not becouse it's mathematical.
« Last Edit: May 05, 2021, 06:39:13 PM by Alef »

#### Jeannot

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• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #111 on: May 05, 2021, 07:34:45 PM »
@Alef: Instead mandelnest for power 3 generates something what could be called platonic solid as a main bulb
I do not know if Mnest has anything to do with the solids of Plato (I would not fall into the esoteric mysticism of some...), but this mathematical entity probably has close links with the symmetries of coordinates system. Here is the key I believe to put order in all this Mnest fractal menagerie (canonical acos-cos and asin-sin or not, already known Mbrot or Mbar R2 and unknown, beautiful or not, all have their place in the puzzle).

And ok, to be more mathematically rigorous I would no longer use the term "pseudonorms", not really appropriate, rather non-Euclidean norms or norms at all. But as I was saying, there is still a lot to discover with Mandelnest numbers and unfortunately I won’t have time to go into the interesting issue of exotic norms with you right now... maybe later.
« Last Edit: May 06, 2021, 09:39:36 AM by Jeannot »

#### Jeannot

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• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #112 on: May 06, 2021, 10:39:06 AM »
Exploring non canonical forms of Mnest : acos acos asin + cos sin sin, I found this Julia one (-0.75, 0,0) for POWER 2:
« Last Edit: May 06, 2021, 10:50:07 AM by Jeannot »

#### Jeannot

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• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #113 on: May 07, 2021, 04:51:38 AM »
Mnest Julia: the unit vector U: (1/√3, 1/√3, 1/√3) defines the main axis of symmetry for Mnest. So the Julia points chosen on this cube axis generate interesting Julia fractals with 100% cubic symmetry. Examples below: (0.425 , 0.425 , 0.425) P10 and P12 acos-cos.

#### Jeannot

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• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #114 on: May 09, 2021, 08:26:41 AM »
Hello,
I tried a variant of Mnest: using atan-tan instead of acos-cos...
The result is surprising   below an exemple P15 (as for acos-cos le high powers give the best ones)

#### Alef

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##### Re: Mandelbrot 3D: Mandelnest
« Reply #115 on: May 09, 2021, 11:09:03 AM »
I tried cosine version. Power 3 is unsighty but have the same tricorns in cutouts. With larger power it becomes better. Indeed like bird's nests.

Jeannot: I will test tangent version. I think higher power versions shows less interesting 2d fractals and have too mutch bulbs;) It seems tangent version have a lot of noise but on increased power it kind of "normalises".

Mandelbrot set power = 7 and Julia 0,0,1 Power = 7 norm of Lp=4
« Last Edit: May 10, 2021, 12:10:50 AM by Alef »

#### Alef

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##### Re: Mandelbrot 3D: Mandelnest
« Reply #116 on: May 09, 2021, 11:16:18 AM »
I tried mandelnest algorithm mixing sine X, sine Y, cosine Z. (arcsine, arcsine, arccosine; sine, sine, cosine). Now it have tricorn in cutout Z=0 but mandelbrot sets in cutouts x=0 and y=0.

Same goes for cosine x, cosine y, sine z (power 3) fractal but it is less sighty. I think this is best 3D mandelbrot one could have. (in power 3)

It's about a simmetry.
This is not strict terminology but power3 tricorns have 4 fold simmetry aka 4 corners. Power 3 mandelbrot have 2 simmetry axis and 2 corners.
Geometricaly it would be impossible to have a figure so that 3 cutouts along canonical axis would be power 3 mandelbrot sets. Hovewer 3 power = 3 tricorn fractal allows this.

It's hard to imagine in head without visual engineering background. One can cut from a paper 3 power = 3 mandelbrot sets and 3 power = 3 tricorns. Or just imagine it in palms or papers. And then try to put them together along XYZ axis. I guess, you could only put together 3 (power3) tricorns or 2 (power 3) mandelbrots and 1 (power) 3 tricorn.

3D figure consisting of 3 (power 3) mandelbrot sets is impossible!!!
There must be some heavy mathematics behind this. How the algerbra corresponds  3D geometry.

I tested power 3 Mandelbulb cutouts along axis. They alsou are a tricorn, a mandelbrot and a mandelbrot but the resulting fractal is different, stalks are obscured, less (platonic) simmetry. Power 3 quaternion mandelbrot set is just a rotation surface, 2 mandelbrot sets and a circle.
« Last Edit: May 10, 2021, 12:05:52 AM by Alef »

#### Alef

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• Posts: 140
• a catalisator / z=z*sinh(z)-c^2
##### Re: Mandelbrot 3D: Mandelnest
« Reply #117 on: May 09, 2021, 11:27:18 AM »
There is an interesting property lets call it a "power 5 simmetry".

Power = 5 mandelnest fractal is identical for all of the legit combinations. Both sine, cosine and combined versions at that power generates identical figures. Alsou all of the cutouts along axis are power 5 mandelbrot sets. This must something to do with simmetry. Power 5 mandelbrot sets have 4 fold angle simmetry like power 3 tricorn. You can make 3D figure out of them easely. (placeing them on axis).

Power 7 fractals again is different.

Alsou power 5 Mandelbulb formulas generates fractals which are simmilar but not identical, but all with power 5 mandelbrot sets in cutouts. However power 5 Mandelbulbs is far from idealy simmetric.

This must be the reason behind no REAL 3D (power 2) mandelbrot set.
Identical perpendicular cutouts along axis must have 4 fold simmetry for them to be allowed to coexist in 3D. Power 5 Mandelbrot set have this simmetry (like power 3 tricorn). So power 5 mandelnest is idealy simmetric (like platonic solids).
Throught I must admit I don't know mutch about mystical properties of platonic solids;)

Edited:
This fractal have interesting property. In cutouts on axis, say in XY plane:
power 3 - tricorn (4 bulbs, power 3 tricorn)
power 5 - mandelbrot (4 bulbs, power 4 mandelbrot)
power 7 - tricorn (8 bulbs)
power 9 - mandelbrot (8 bulbs)
power 11 - tricorn (12 bulbs)
power 13 - mandelbrot (12 bulbs)
...

There is certain harmony of spheres in this algorithm.
« Last Edit: May 09, 2021, 11:58:57 PM by Alef »

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #118 on: May 11, 2021, 11:16:12 AM »
@Alef: I think higher power versions shows less interesting 2d fractals and have too mutch bulbs
The low powers are more representative by the cuts but are less sophisticated and often with less clear details, the high powers are too repetitive (too many bulbs as you say) indeed. So for me the most interesting are the average powers around 8 or 10, as for Mbulb.
Mnest’s interest is not only in generalizing Mbrot 2D and finding the Mbrot or Tricorn 2D cuts, the middle powers have a beauty and characteristics of their own that I personally prefer.
But I find your rendering of low powers really top, and the cuts shows us that Mbulb2D or Tricorn2D are everywhere

It's about a simmetry
In fact it’s more a question of dissymmetry than symmetry
-the maximum symmetry  (cube or octahedron, I do not believe that the other polyhedron of Plato intervene) where each faces of the cube are identical is those of the two canonical acos-cos forms (odd powers) and especially asin-sin (odd or even).
-if we add heterogeneity on the acos-cos formulas mixed with asin-sin, we introduce a first dissimetry, but we keep the 2 opposite faces of the cube identical for odd powers. This minimal dyssimetry seems due to the fact that we keep the acos-cos and asin-sin duality (reciprocal functions).
-but if you introduce additional heterogeneities such as asin-cos or acos-sin the opposite sides of the cube are no longer identical.
But dissymmetric fractals can also be graphically interesting, and deserve our attention, because an axial symmetry (1,1,1) remains...

However, the fact that that we do not recognize all the powers of MbulbR2 and MbarR2 in MnestR3 seems to indicate that something is missing and that the Mnest system as defined is incomplete although wider. This is the subject of my current investigations: to complete the picture if I can.
« Last Edit: May 11, 2021, 12:38:17 PM by Jeannot »

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #119 on: May 11, 2021, 11:46:46 AM »
Canonical Mnest inventory (continued): power 4 identity card.

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