First I must say that this thread derives from
this one, in which I presented a simple formula that produces what Sam Monnier's, creator of "Ducks", calls "space-filling patterns", or sometimes "Ducky-Thalis" patterns (I prefer "kaleidoscopic patterns"). That was after reading what Sam wrote on
another thread about how this patterns are created, and as I like a lot the patterns from Ducks formula, I wanted to try another ways for getting this kind of shapes. I saw there was a relationship with Mandelbox so I wanted to try with abs function and the mandelbox fold. But first I tried the most simpler version of that... just taking the absolute real values of x and y, then dividing by the squared modulus. No 'if' conditions, just abs+inversion, and it turned to be the simplest method for this kaleidoscopic patterns yet, that also has a 3D version.
The complex number analog of my 2D formula turned to be
z=abs(z)^-1+c - Syntopia explains well the analogy in my previous thread, and called it a "power -1 burning ship". So I decided to try with other powers, and I discovered that even the well-known "burning ship" fractal, has this patterns on it's insides... and it seems that nobody have noticed that before... why? I think the coloring method is the answer... as the patterns doesn't diverge, an inner coloring must be used... and the patterns doesn't show up or become evident with any coloring.
I found that
exponential smoothing works perfectly on this. I know of this coloring technique as it's included in the standard set of formulas on Ultrafractal, but I'm using my own version... it's similar but it computes the magnitude only of Z, and it's pretty simple:
At each iteration, do this: (|z| means the squared module of z in UF)
z=abs(z)^2+c (or whatever formula you want to use)
mold=m
m=sqrt(|z|)
sum=sum+exp(-1/abs(mold-m))
Then the color index is picked depending on the value of 'sum'.
I really like how this coloring works. It gives good effects with low iterations and with certains patterns (glows, blur, sometimes like a DOF effect), and with the correct coloring settings, produces a cool 3D feel on the images. The color density and number of iterations must be fine-tuned for this.
Well, the fact is that after this encounter with the burning ship, I begin to explore another fractals using this method of adding an abs function and coloring with exp.smoothing, and I will be posting some of my findings in this thread. I will use always the default gradient from UF5 and no postprocessing, so don't expect too much artistic work here because the idea is just to show the great variety of patterns and structures I found. (Even when there are some similarities that most of this formulas shares, each one have unique kind of shapes and patterns somewhere)
I also invite you to post here your own findings and artworks using this formulas
One thing I forgot to mention, is that in Mandelbrot mode you get the variety of patterns but most are streched/distorted and very chaotic... you can find cool images, though, but the nicer ones are Julias, and you can use the Mandelbrot as a map for a preview of what kind of patterns you will get with the Julia values corresponding at that neighborhood of points.
The regular burning ship fractal will be the first series of images, with three variations... abs(z)^2+c, abs(z^2)+c and abs(z^2+c).
Enough talking, this are the first images:
The last is in Mandelbrot mode, and it shows a "miniship" inside the patterns area. I raised iterations and coloring density to show that inside the miniship there is a mini-set of patterns too. I have looked at this closely; patterns are not the same of the main fractal, and there's also "mini-mini-ships" inside... this goes on and on as more iterations are made.
I'll post more images later...