### Kleinian expansion to supershapes

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#### Alef

• Fractal Phenom
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• catalisator of fractals

#### Kleinian expansion to supershapes

« on: November 10, 2018, 03:58:46 PM »
Is it possible to expand Kleinian fractal to Lp norm? (JosKn-KleinIFS.m3f)

I had an idea. Mostly becouse there are so mutch renders of this formula and it contains lots of spheres. I believe there is radius involved and if it's so then it should be able to calculate it by
https://en.wikipedia.org/wiki/Lp_space

in general it is:
radius = ( (abs(x))^p + (abs(y))^p + (abs(z))^p )^(1/p)

Of corse if could be optimised, if p = 1 then radius=abs(x)+abs(y)+abs(z) should be way faster.

This is universal math Lp norm works in just every 3D formula I experimented with. Say in mandelbulb. Just that p = 1 often are more noisy and simple. But some p = 4 often are more beautifull than standart version. Jet using p=1 spherefold I was able to generate egyptan pyramid fractal.

p=30 mandelbulb. More often puting p=8 norm into mandelbulb I got box with flat surfece at bottom.

There must be some mathematical equalities between p norm and result.

I could not realise becouse I simply have no idea about this type of fractals, I don't have a book Indra's Pearls, in JosKn-KleinIFS.m3f do not have readable formula and I'm not used to fragmentarium code. (Or maybe I'm too laisy. Realy I want to spend more time on rendering fractals and art not programming and math.) But someone could try.

Gielis supershape is trigonometric expansion of supershapes. I wanted to put it at the core of Mandelbulb. But since it is patented and I don't like trigonometry I tried to aprouch to it by shape I called superseed:

invpower= (PowerX *PowerY *PowerZ )^(-1/3)
radius= (abs(x)^sCurvatureX + abs(y)^sCurvatureY + abs(z)^sCurvatureZ)^invpower

It's pretty OK, but I m not shure about mathematicaly corect formula for invpower value. This power based mean seems to generated the best and most stable mandelbulbs. But if power values are different mandelbulb allways have some additional features and on some combination could become strange. What means formulas is not correct.
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#### Sabine62

• Fractal Frogurt
• Posts: 459

#### Re: Kleinian expansion to supershapes

« Reply #1 on: November 10, 2018, 08:24:21 PM »
I do not understand most of maths  , but might have some helpful links:

http://www.josleys.com/articles/Kleinian%20escape-time_3.pdf

This whole thread below is about Jos Leys' Kleinian group limit set, in that particular message yo can find a (Fragmentarium) .frag with the formula, which is readable in a text editor
http://www.fractalforums.com/3d-fractal-generation/an-escape-tim-algorithm-for-kleinian-group-limit-sets/msg98310/#msg98310
To thine own self be true

#### Alef

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#### Re: Kleinian expansion to supershapes

« Reply #2 on: November 16, 2018, 04:51:18 PM »

In simple it just a replacement for
r = square root of ( x^2 + y^2 z^2 )
which is round sphere

It can be made diamond shape (faster)
r = abs(x) +  abs(y)  +  abs(z)

Or cell phone producer favored shape (slower)
r = 8th root of ( x^8 + y^8 + z^8 )

I was thinking that it could look great in "Kleinian group limit set". So far it worked in all formulas I tested.

#### Sabine62

• Fractal Frogurt
• Posts: 459

#### Re: Kleinian expansion to supershapes

« Reply #3 on: November 17, 2018, 12:39:01 AM »
Alef, I have tried your Diamond setting, but the current DE is no good with that and render gets really slow...
So probably not a good idea for the Kleinian...

#### Alef

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#### Re: Kleinian expansion to supershapes

« Reply #4 on: November 20, 2018, 05:50:31 PM »
Then probably this thing would not work with kleinian.

Sometimes power=1 version (diamond) creates fractal very noisy. What was a case for mandelbulb. Not shure why, maybe it's becouse of thines or something. Mathematicaly it must be faster than standart.

Larger power versions - supersphere is more stable. So far it allways worked. But of corse render becomes significantly slower.

#### Sabine62

• Fractal Frogurt
• Posts: 459

#### Re: Kleinian expansion to supershapes

« Reply #5 on: November 20, 2018, 07:55:58 PM »
Maybe you should have a look at fragmentarium after all
Your r= abs(z.x) +  abs(z.y)  +  abs(z.z); (^1) in a mandelbulb ^8
Nice results when used like this just before the end of the iteration

#### Alef

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#### Re: Kleinian expansion to supershapes

« Reply #6 on: November 22, 2018, 04:09:40 PM »
Nice fractal.

Mandelbulb3D simply works on my PC. Fragmentarium locks. It's video card. Fragmentarium maybe have slightly more advanced special effects and more of programming freedom.

I can't remember this shape. Does  r= abs(z.x) +  abs(z.y)  +  abs(z.z); was not used in all formula radiuses? For me it generated too mutch noise and some very ugly mandelbulb.

Power 4 - power 8 radius mostly just looks more advanced;) This is why cell phone manufacturers use supershapes. In M3D v1.9 boxbulb formula is IQ-mandelbulb with power 4 radius. But I don't have image to share. I can just recomend to test

Maybe in fragmentarium there even are special operator to calculate radius with all power version. I 'm not shure. Once I looked at code and it had something what looks as that.

#### Alef

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#### Re: Kleinian expansion to supershapes

« Reply #7 on: November 26, 2018, 02:15:37 PM »
Slightly worst renders
p=1 and p=3 bulbs and respective "spheres".

This bulb formulas probably is worst than the one used above. I used basic bulb and it have some antenna on top.

#### Alef

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#### Re: Kleinian expansion to supershapes

« Reply #8 on: November 26, 2018, 02:22:03 PM »
p=1,2,4 sphere and bulb. Looks quit a rigid. Just I have no idea what root use for powers 1,2,4. I used power based mean value as it looked best.

I thing this version of radius will work in maby different formulas. But p=1 almoust allways is worst.

Code: [Select]
Mandelbulb3Dv18{g....Uy....u1...w....I6...U9nN1RiDY..jgW/3ELb6zDGTo9XU97MwXML1wke9BuzeoFBJgospyD................................rBMTCw1bqz1........A./.................U4/2...wD...Uz6...........M.1/....2.a0...g.....EC.....k6IwgtIzWpD/..........m/dkpXm1.....z.EnAnYD16..0..........wz.................................U0.oAnAr1...sD...../...z1...kDVoHBeloldx1..........m8rSJH7SIdjPy4/8ULB9uHZeNk.gRvMzgyQjAcMP0qDxBTXMYh7EunhJE7VWh7MzKU.T3.PrCqDU....yXE0/.......2....sD.6....sD..G..................................oAnAt1...sD....z...........................................A....k1......83iyz1..........QLMy/.U..6.P....Y0...EB....x2...k3....F....6/...s21....SN5A...U.a0KyRLs4rPMRYoWyx7lz16.0c..0nBr...........U6.cXcESLfz1...........228.UEkw...Ub96aAIVz9.1se7Umvxz0...........RU0.E190............s/...................E.2c..zzzz............00...................2./8.kzzzD............8..................../EU0.wzzz1....................................RpR1UvNPcvW0/oJrB.ibhVib53r.ER.YPituCXM.RpR1UTbfvtNd2oJrB.kMbrSfmHr.ER.McV4uRDH/RpR1.AqthrE65oJrB.7TTTrzJRr.ER.6fgmSyQs/RpR1UTbfvt..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................E....2..F2E.....I....M....UG7FpL3poLCxJFg3aSj7IRg74................................................6./........0E........kz1........6./........0E........U.2........../.........E..........2.............................................................................}{Titel: Lp=8_Bulb8th}
Code: [Select]
Mandelbulb3Dv18{g....Uy....u1...w....I6...kqym8tzgmzz0LD0ga6GUwDpQX15YcQdyH1FkYT9scZz.LD0ga6GUwD................................rBMTCw1bqz1........A./.................U4/2...wD...Uz6...........Q.1/....2kC/...z.....E4.....k6IwgtIzWpD/..........m/dkpXm1.....z.EnAnYD16..0..........wz.................................U0.oAnAr1...sD...../...z1...kDrqj9YryfMx1tekSibb82zO9G55.zwLlDqwOfE1zSLvHs9TnmPo5JzMdAcg4D0joDul8ZY2BXPwfARfWKPtuGzmtD0OKKlFpDU....yH1V........2....sD.6....sD..G..................................oAnAt1...sD....z...........................................7....k1......83iyz1..........QLMy/.U..6.P....Y0...EB....x2...k3....F....6/....J2....SF52...U.a0KyRLs4rPMRYoWyx7lz16.0c..0nBr...........U6.cXcESLfz1...........228.UEkw...Ub96aAIVz9.1se7Umvxz0...........RU0.E190............s/...................E.2c..zzzz.............0...................2./8.kzzzD............8..................../EU0.wzzz1....................................RpR1UvNPcvxDUCuc1.ibhVibw/ELRr.sSq4uyZc.RpR1UTbfvtXl0.aJ5.yRiibb0AELRr.srtuSiLJ/RpR1.Aqthr7YpB.I5.kMbrS98LELRr..XRSvhev/RpR1Emrrrx..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................E....2..F2E.....I....M....UG7FpL3poLJxpIp/LNmBLNZFqLA/5............................................6./........0E........U.2........6./........zD........I.2.............................................................................................................}{Titel: Lp=8_circle}

Well, exept here. These piramids were craeted as p=1 spherefold shapes and they are messy:

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