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A Biomorph Explained

In this context, a Biomorph is refered to as a deviation of Julia sets, -- which are of interest in the so-called fractal geometry, a branch of chaos theory -- colored in a special fashion. A Julia set Jc, where c is a complex number, is defined as
    J(c)={z : |phi_n(z)| < 10 for all natural n}
where
    phi_n=phi_{n-1}(z)^p+c, phi_0(z)=z

z and c are complex numbers.

For this screen saver p is an integer satisfying -10≤p≤10.

In words, a Julia set is the set of points whose orbits stay bounded (here: remain in the disk at the origin with radius 10). An orbit is the sequence of iterates φ0(z), φ1(z), ...

In fact, orbits display three kinds of behaviours: either they are bounded, diverge to infinity or are a subset of the border of the Julia set.

This is especially easy to see for the Julia set J0, where the constant c=0, which actually is the unit disk. Any point z within the unit disk, i.e. |z|<1, has an orbit that converges to the origin since |zn|=|z|n tends to 0 as n→∞, i.e. stays bounded and hence belongs to the Julia set. Conversely, if |z|>1, then |zn|→∞ as n→∞, which exactly means that these points do not belong to the Julia set. And, finally, if |z|=1 then |zn|=1 for all n. These points form the border ∂Jc of the Julia set.

For constants c other than 0 (and actually -2, for which the Julia set is simply an interval) the Julia set Jc is much more complicated and lead to those fascinating pictures you see while running the screensaver.

If you want to learn more about Julia sets, you may want to have a look at this.
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