Dragon's Home

[Pauldelbrot]

The dragons gather at the shore of a strange sea.

There are four layers: the minibrot interior and dragons have Juli-angle coloring in oranges and the outside has three separate gradients applied. One is the basic purple-to-pink-to-maroon-to-white gradient, a second is high frequency wave pattern HSL-added to bring out the fine structural details near the dragons, and the third is identical to the first, except it has an alpha channel and is on top. That one serves to suppress the effects of the second layer in high-iteration areas where it would otherwise just produce noise.

Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.

As usual, all of today's images have 2048x1536 versions easily found after clicking on them.

Detailed statistics:
Name: Dragon's Home
Date: February 19, 2009
Fractal: Supernova Mandelbrot set
Location: c-plane, a = 0.51803398, b = -0.001679 - 0.163504i
Depth: Shallow
Min Iterations: 2133
Max Iterations: >10,000,000
Layers: 4
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 10 minutes
Calculation time: 18 hours (2GHz dual-core Athlon XP)

[cKleinhuis]

cool julia worms ! and i also can spot minibrots !

great work! 

[Pauldelbrot]

Thanks.

[bib]

As usual, great images from our Supernova specialist
This one is my new desktop background, I would have named it "Attack of the Elephants' parasites"

[Dinkydau]

I agree, very nice work.

[bib]

Another Supernova image, inspired by Pauldelbrot's findings:

[Pauldelbrot]

Wow, thanks.

I see someone's been busy implementing the formula I posted for some fractal renderer or another. If you need any pointers go ahead and ask.

[Dinkydau]

Amazing image, bib, I think it's even better than Pauldelbrot's image (sorry Paul , but still nice work)

[JackOfTraDeZ]

nice pix but what is "supernova" ?

[Pauldelbrot]

nice pix but what is "supernova" ?

Quote from: me, months ago

One day I thought it would be nifty if the lake shapes in Antimatter fractals were more varied. In particular, if they could have petals. So I decided to make zero stop being always a superattracting fixed      point.                                                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                                                                                       
Antimatter is the family of mappings z². Supernova just adds an innocent little "+ b" to that, not unlike how Nova was derived from the Newton's Method Julia fractal whose attractors were the roots of z³ - 1.                                                                                                                                                                                                                                                                                             
                                                                                                                                                                                                                                                                                                       
That innocent little "+ b" unleashes a veritable flood of exotic new forms. Of course the system remains capable of producing the full gamut of interesting dynamics, including Herman rings. The addition of a constant does not change the partial derivative with respect to z, so the critical points remain the same. However, only infinity is a permanent attractor now; zero superattracts only for b = 0 and for other    b-values is an "interesting" critical point. The fixed point moves away from zero, though it still attracts for small b and small enough c.                                                                                                       
                                                                                                                                                                                                                                                                                                       
In fact, the c-plane develops a Mandelbrot set looking like the original, but scaled and rotated depending on b, as viewed using the critical point zero as the lens. The most interesting images, however, come from studying one of the "usual" critical points for Antimatter, within the region near the border of the (now-"ghostly") Zero Mandelbrot. In these regions, the characteristic "lakes" in the centers of seahorses and similar structures implode as the attractor bifurcates and eventually disintegrates.                                                                                                                                                                             


(Apologies for the slightly wacky-looking quoting; the quote tag seems to screw up LaTeX display for some reason, necessitating a workaround.)

[bib]

I forgot to add some details, it was late in the night

Detailed statistics:
Name: Space Lace
Date: August 28, 2009
Fractal: Supernova Mandelbrot set
Location: b-plane, a = -0.841142 + 0.802824i, c = -1.54762 + 0.40476i
Depth: 7354
Max Iterations: 100000
Layers: 3 (Smooth, triangle Inequality Average, and Gaussian Integer)
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 1 hour
Calculation time: 11 minutes (2.66GHz quad-core Intel i7 920)

[bib]

Wow, thanks.

I see someone's been busy implementing the formula I posted for some fractal renderer or another. If you need any pointers go ahead and ask.

Ultrafractal is really easy to do that it took me 5 minutes to implement. My next animations will be based on this formula.
 I am visually (not mathematically) studying the dynamics of both C-plane and B-plane fractals in order to choose some interesting dynamic features to show. Stay tuned!

Supernova-Cplane {


init:
  z = @start
loop:
  z = @b+@a*z^@power * (z-#pixel) / (1 - #pixel*z)
bailout:
  |z| <= @bailout
default:
  title = "Supernova C-plane"
  center = (-0.5, 0)
 
  param start
    caption = "Starting point"
    default = (0,0)
  
  endparam
  param power
    caption = "Power"
    default = (2,0)
  
  endparam
  param a
        caption ="a*"
        default = (1,0)
  endparam
  param b
        caption ="+b"
        default = (1,0)
  endparam

  float param bailout
    caption = "Bailout value"
    default = 4.0
    min = 1.0

  endparam

}

Supernova-Bplane {

init:
  z = @start
loop:
  z = #pixel+@a*z^@power * (z-@c) / (1 - @c*z)
bailout:
  |z| <= @bailout
default:
  title = "Supernova B-plane"
  center = (-0.5, 0)

  param start
    caption = "Starting point"
    default = (0,0)

  endparam
  param power
    caption = "Power"
    default = (2,0)
  
  endparam
  param a
        caption ="a*"
        default = (1,0)
  endparam
  param c
        caption ="c"
        default = (1,0)
  endparam
  float param bailout
    caption = "Bailout value"
    default = 4.0
    min = 1.0

  endparam
}

[Nahee_Enterprises]

    The dragons gather at the shore of a strange sea.
       .....  Calculation time: 18 hours (2GHz dual-core Athlon XP)

I missed this image before.  Glad I came back and started looking over some of the past few postings.

It appears you have read, and based this image upon, the "DragonRiders of Pern" series by Anne McCaffrey.  They always had a problem with "worms" falling from the sky.