Ducky fractals with conjugate only

[thomas314]

Happy eastern everybody
I've found a way to create those lovely ducky patterns, but without any abs function.

It's quite easy (UF-Code):

Code:

;Inside your iteration loop before fractalformula

    if (imag(z)>0)
      z=conj(z)
    endif
;Very well known formula    
    z=z^p+c

Bailoutvalue is
Outsidecoloring is none
Insidecoloring for the examples are Statistics and Exponential Smoothing with Iteration skipping

Some examples:

Mandel1 p=0.5/0.5i  center=-1.65/0i  magnification=1

Julia1    seed=-1.05/0.415i


Mandel2 p=0.75/0.5i  center=0/0i  magnifcation=1

Julia2    seed=-1.85163/1.07058i


Mandel3 p=-1/1i  center=0/0i magnification=1

Julia3    seed=0.42/-1.165i


I've had much fun exploring duckies or talis-pattern created with abs... And that lead me to the question: Are there other ways(i found one) to fold a
fractal to get ducky like patterns? I suspect that symmetrie and equal coverage are important for this purpose.
What do you think?

   

[DarkBeam]

You are just replacing abs with his definition.
abs(x) is if x<0 x = -x
conj (m,n) = m - i*n
so now you can find out how it works

[Kali]

Nice images

You can try also the "rotated circle inversion", no abs function at all

The simpler form could be:

Code:

if (|z|<1)
  z=z/|z|
endif
z=z*m+c

where m is a complex number, and it works with Julias (picking a constant for c)

[Kali]

A quick example, with values  m=(0.00,0.90) & c=(0.25,0.50)

It normally requires more iterations than other "ducky" formulas, and also a bit of patience in finding the right values that leads to interesting results...

[s31415]

Hi,

To get ducky patterns, you only need to combine a mirror symmetry with some conformal transformation. Your conditional use of the conj function is exactly a mirror. Kali's circle inversion is a mirror operation about a circle, and can be conformally mapped to a standard mirror symmetry. Using the function "abs" is a bit less general, as it creates two perpendicular mirror symmetries (as it takes the absolute value of both the real and the imaginary part of the complex number).

You can also generalize the mirror symmetry to higher order symmetries (technically, to higher order dihedral groups). Here is a picture using an algorithm with four intersecting mirrors.
http://algorithmic-worlds.net/expo/work.php?work=20110526-kd8
You can see that the squarish figure in the center admits four approximate axes of symmetry (approximate because the symmetry is broken if you go far enough from the center).

Sam

[Kali]

Hi Sam. Nobody's could explained better than the creator/discoverer of Ducks formula 

BTW, I was wondering how to achieve a mandelbrot set of this kind of patterns without any stretching. Or maybe not strictly a Mandelbrot set, mathematically speaking, but a set of continuous different varying patterns along the plane, so you could explore the fractal and find the unstretched patterns just by looking around and zooming, without picking julias or changing any parameters... Any clues?

We could maybe call this the Ducky Grial of fractals

Edit: I forgot to mention that the method I described above (rotated circle inversion julia), can produce a certain amount of variation on the type of patterns along the plane, but I'm still not satisfied... also, just in cause I didn't explain myself well, I'm talking about variations on the kind of patterns it shows... off course any julia of a ducky formula shows a huge amount of different arrangements of the patterns but keeping the similarity of the specific kind of pattern the parameters produces.

[thomas314]

Hi,
thank you all for your kind and very interesting answers

DarkBeam

abs(x) is if x<0 x = -x

This make me thinking of:

Code:

    
   if (real(z)<0)
      z= -real(z)+flip(imag(z)) ;flip exchanges the parts of z
    endif

Kali
I'm curious what your rotated circle inversion formual can create

We could maybe call this the Ducky Grial of fractals grin

That would be cool to have an infinite plane filled with dense unstretched patterns as Mandelbrot 
But i would missing the switching feature in UF  (Maybe the julias will then be stretched?)

s31415

To get ducky patterns, you only need to combine a mirror symmetry with some conformal transformation.

Indeed, that is the rule (or axiom or recipe?)  i was looking for. Now i know the right direction to looking forward

[s31415]

BTW, I was wondering how to achieve a mandelbrot set of this kind of patterns without any stretching. Or maybe not strictly a Mandelbrot set, mathematically speaking, but a set of continuous different varying patterns along the plane, so you could explore the fractal and find the unstretched patterns just by looking around and zooming, without picking julias or changing any parameters... Any clues?

We could maybe call this the Ducky Grial of fractals

Edit: I forgot to mention that the method I described above (rotated circle inversion julia), can produce a certain amount of variation on the type of patterns along the plane, but I'm still not satisfied... also, just in cause I didn't explain myself well, I'm talking about variations on the kind of patterns it shows... off course any julia of a ducky formula shows a huge amount of different arrangements of the patterns but keeping the similarity of the specific kind of pattern the parameters produces.

Hi Kali,

I don't know about the conditions that should be put on the variation of a parameter so that the resulting pattern is not deformed. I'm starting to read seriously about complex dynamics, so maybe I'll get some clues... If I do I'll share it here for sure.

Best,

Sam

[Alef]

Nice result. Throught flip of real value should be the same as real value *1i.  Maybe should test, what is faster;)

[Alef]

z/|z|  is http://en.wikipedia.org/wiki/Unit_vector in complex number plane.

I think, infinite pattern set should hardly be posible without repeating functions like sinus. Simplest possible abs formula, the burning ship fractal, generates patterns only in quadrants where there are some negative C values, imaginary or real. Probably positive from abs + negative from pixel reveals fractal nature of numbers. When it gets too far from 0, differences between positive and negative values, escaping to infinity fastly and slowly increases in both side, hence streching.


p.s.
Try exponent smoothing were negative exponents are calculated not of cabs (z) but of cabs (z + complex parameter). Adding 1, -1, i, -i , maximum 2,  helps to reveal much more patterns.

[Alef]

Experimenting with equation I got a pretty strange but interesting formula.
z=abs(z)/ (z+1+C) +C

It have nice paterns in m-brot mode, but without some z based colour function it shows absolutelly nothing  (it don't have any iteration differences, nor diverges nor converges). Julias looks strange, but they have patterns inside. Maybe just a bitt streched.

Direct coloured:


Palette coloured:

[Alef]

Nice images

You can try also the "rotated circle inversion", no abs function at all

The simpler form could be:

Code:

if (|z|<1)
  z=z/|z|
endif
z=z*m+c

where m is a complex number, and it works with Julias (picking a constant for c)

Here is corrected version of circlefold. This calculates modulus of z only once, and don't have division by zero problem. Something similar like of mandelbox fold.

Code:

      modulus = |z|
      IF (modulus ==0)
      z=0
      ELSEIF (modulus<1)
      z=z/modulus
      ENDIF

[Alef]

With this I got nice formula:

Code:

z=z+@D
      modulus = |z|
      IF (modulus ==0)
      z=0
      ELSEIF (modulus<1)
      z=z/modulus
      ENDIF
z=z*c*@scale+c

in single line
z= scale* circlefold ( z+D )*c +c

With "if |z|<1 then z=z/|z|" formula fails to initialise when D=0 and z=0, so my circlefold are preferable. Maybe this formula could have additional name, Sefir (space filling rotation), as it is rotated.  Alsou very similar:
z= scale* circlefold ( z+D )/c +c

[Alef]

With formulas i posted (z=scale*circlefold(z+D)*c+c and z=scale*circlefold(z+D)/c+c) I got this

Here is the result.



kaliset_sefer4_storm {
::8xGe7jn2NSVTztNIUw7eG/fgR3jNCBC52hDtT7l+1l07eIIkNJIQBQx2dyP+C6Ln2OpT8NYf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==
}




Kaliset_Sefer4_tentacles {
::kG6S8jn2NP1yyuJMM09Zm8P4h9hYIQIt3xLuda30Xb6HAjiRQcixmabSu0v+KDJZufC3lW6I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}


Kaliset_sefer3_rotated_triangles {
::1Gvx3hn2NSVTPuJMQ07RK/Hs4+m4vAMtyH22eqVtX6PAkXwk4sGbqxs7m/9dggJrq0KVu53M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=
}

[element90]

Now that the discussion has mentioned circle fold, I thought I'd share a picture combining a Mandelbrot and "ducky" type patterns.



This is produced with a circle fold but instead of folding in it folds out, i.e. any point inside the circle is reflected outside the circle otherwise it is left alone. If the centre of the circle is left at the origin when used with the standard Mandelbrot formula you don't get ducky patterns in fact you don't get much at all, so the centre of the circle has to be moved. Here is a summary of the parameters for this picture.

Formula: Mandelbrot

z = transform(z)
z = z^2 + c

Transform: circle fold out, diameter 2 centred at -1.3 + 0i
Complex plane transform: none
Initial values of z: c
Image centre: -0.189938 - 0.718423i
Long side: 0.022725
Rotation about image centre: 300 degrees
Bailout condition: norm(z) > 16
Maximum iterations: 100
Colour selection: outer for bailout otherwise inner
Outer colouring: iteration
Inner colouring: absolute log of average magnitude

Saturn and Titan has circle fold in, circle fold out and circle reflect transforms. Initially I implemented these as simple inversions but it turns out that when inverting a complex number the result is not on the same vector as the original value, it is often in an entirely different sector. Saturn has in addition inverse fold in, inverse fold out and inverse reflect.

The circle fold in, circle fold out, inverse fold in and inverse fold out all produce ducky patterns, good ducky patterns are produced with low iteration counts, if the iteration count is increased the ducky patterns become just texture but Mandelbrot islands and Tricorns appear with circle folds and only Mandelbrots appear with inverse folds (I have no proof that this always the case).

Here are some Mandelbrots and Tricorns, in this case c has also be transformed by a "unit circle inversion" centred at 0.0 + 0.7i. When I produced these pictures Titan hadn't been updated to produce the parameter summaries so the descriptions are not in the same format and terminology tended to vary. The transform "unit circle inversion" is circle reflect.



Formula: Mandelbrot

z = transform(z)
z = z^2 + c

Transform: apply Transform 1
Transform 1: circle fold in, diameter 2 centred at 0 + 0i
Complex plane transform: unit circle inversion centred at 0 + 0.7i
Initial values of z: transformed c
Image centre: -2.670267 + 0.855936i
Long side: 3.203797
Rotation about image centre: 0 degrees
Bailout condition: norm(z) > 16
Maximum iterations: 250
Colour selection: outer for bailout otherwise inner
Outer colouring: not applicable, all points are inner
Inner colouring: absolute log of magnitude, fractal dimension