Fractal Related Discussion > Fractal Mathematics And New Theories

Mandelweird 3D burning ship

(1/1)

Alef:
Very ugly;) But maybe it is a fractal curiosity.
Probably conditionals can create some equivalent of abs function, a mirror.

Some time agou I did this:
4D
- Calculate hypercomplex number based 3D tricorn fractal.
- Calculate power 2 quaternion mandelbrot set.
- Then chose largest value in each axis.
- Add + C.

--- Code: ---[OPTIONS]
.DEScale = 0.5
.SIPow = 2
.Version = 9
.Double Height4D = 0
[SOURCE]
procedure MandelWeirdN(var x, y, z, w: Double; PIteration3D: TPIteration3D);
var
x1, y1, z1, w1, x2, y2, z2, w2: double;

begin

//========================================
x1:=x;
y1:=y;
z1:=z;
w1:=w;

//===============================
//Now goes algebra. Calculates two formulas

// hyper tricorn - hypercomplex number tricorn.
x2:= x1*x1 - y1*y1 - z1*z1 + w1*w1;
y2:= 2* z1*w1 - 2*x1*y1;
z2:= 2*x1*z1 - 2* y1*w1;
w2:= 2*x1*w1 + 2* y1*z1;

//==================================================================

//quaternion pow2
x := x1*x1 - y1*y1 - z1*z1 - w1*w1;
y := 2*x1*y1;
z := 2*x1*z1;
w := 2*w1*x1;

//=============================================================
//then choses larger value of the two

If (x2>x) then
x:=x2;

If (y2>y) then
y:=y2;

If (z2>z) then
z:=z2;

If (w2>w) then
w:=w2;

//========================================
// + C
x := x +   PIteration3D^.J1;
y := y +   PIteration3D^.J2;
z := z +   PIteration3D^.J3;
w := w +   Height4D;

end;
[END]
4D
TEST

* * *
By Edgar Malinovsky

--- End code ---

As in this thread: https://fractalforums.org/share-a-fractal/22/burning-ship-sunken-by-a-little
I don't want to steal Mr Rebooted thread with my fractal.

In 2D result must be the same.
(as orbits travell throught all 4 dimensions only if z<>0 as z and w are "entangled").

- calculate mandelbrot
- calculate tricorn
- chose largest value in the axis Y.

x1=   x*x - y*y

y2= - 2*x*y
y  =   2*x*y

If (y2>y) then
y1=y2
else
y1=y

x=x1+Cx
y=y1+Cy

There must be burning ship. This hypothesis is not tested. PC is burning becouse of weather, so maybe later. Conditionals here works as abs function. Anyway there is mutch more easy way to calculate Burning Ship.
This means Burnig Ship fractal is a combination of Mandelbrot set and Tricorn fractal. Now it seems so obvious.

In attachment Mandelbulb3D formula I used to render this.

Alef:
One can generate all abs based fractals in this way.

Iteration backbone, Mandelbrot set:

--- Code: ---x= real(z)
y= imag(z)

x1=   (x*x - y*y)
x2=   (x*x - y*y)

y2=    (2*x*y)
y1 =   (2*x*y)

IF (y2>y1)
y=y2
ELSE
y=y1
ENDIF

IF (x2>x1)
x=x2
ELSE
x=x1
ENDIF

z = x + flip(y)  + c

--- End code ---

Celtic 1:
x1=   (x*x - y*y)
x2=   -(x*x - y*y)

y2=    (2*x*y)
y1 =   (2*x*y)

Burning ship:
x1=   (x*x - y*y)
x2=   (x*x - y*y)

y2=    -(2*x*y)
y1 =   (2*x*y)

Buffalo:
x1=   (x*x - y*y)
x2=   -(x*x - y*y)

y2=    -(2*x*y)
y1 =   (2*x*y)

Celtic Tricorn:
x1=   (x*x - y*y)
x2=   -(x*x - y*y)

y2=    -(2*x*y)
y1 =   -(2*x*y)

Tricorn (obviously):
x1=   (x*x - y*y)
x2=   (x*x - y*y)

y2=    -(2*x*y)
y1 =   -(2*x*y)

JPG slightly smeared my pictures, but I think everyone here knows these fractals. Not shure where this can be used and what this adds to theory. Of corse abs with certain signs are same as conjugation function.

I tried combination of different powers and it don't works well, that made distorted fractals. However different coeffitients or sine, cosine or log functions can make something meaningfull, jet simmilar to existing fractals.

youhn:
It's dualisms^2  8)

Seems like mandelbrot set variations are like magic numbers (Pi etc), they turn up in seemingly different ways. Same for some math concepts, which can be derived using different methods and/or notations and still give the same result. Does it hint at something universal underlying this all, or does the universe naturally allow separate paths to the same goal?

Also wave and particle dualism comes to mind, as well the common practice in quantum mechanics to summarize all paths together to get a probability distribution. Sorry, drifting off a bit. I wish we could meaningfully visualize the calculation steps in 3D (projected 4D) fractals. Not like a buddhabrot, but more animated.

Alef:
I think something universal underlying that all, like science beyond standart model (so far realy none had figured it out, as no theories had been proven by experiments), or many proofs of Ferma theorem.

Here I think seperate methods in some obscure way create same function. But maybe it alsou says something about fundamental properties beyond these sets.

In picture Y1 sine modification.

Ultra Fractal formula ( a bitt slow maybe):

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Duality_Mandelweird.ufm:Duality {
; 1.Calculates 2 complex powers for each axis
; aka mandelbrot or tricorn.
; 2. chose largest value in the axis X and Y.
; https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelweird-3d-burning-ship/4290/msg29214#msg29214

;Edgar Malinovsky
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}

--- End code ---

Alsou UF formula is in attachment file.  Maybe someone can figure out some more canonical fractals out of this.

Alef:
Finaly I found this image shared by Deseptor. Kind of fits here.
https://fractalforums.org/index.php?action=gallery;sa=view;id=5823

--- Quote ---Description: I discovered recently that the Burning Ship fractal is just 2 Mandelbrots and 2 Mandelbars connected. I whipped up a formula in Ultra Fractal that helps illustrate this. Compare it to a Burning Ship. See the resemblance?
--- End quote ---

There are some simmilarities in owerall shape. But maybe alsou in more fundamental aspect that abs function is conj function in disguise, or maybe not.

Log function here for Y works like halfway to burning ship. Zooms on x=-1.6 on antenna where ships should be it shows a lot of distorted tricorns and mandelbrots and further zoom on distorted tricorn.