Extended Kaliset

[tit_toinou]

Hi everyone.

I've manage to extend the Kaliset with a little maths.
Thinking about the "abs" function in terms of angles, we can generalize the folding :
We choose an integer q : q=2 is for applying the conjugate if Im(z)<0, q=4 is the traditional Abs (meaning we apply an opposite and conjugate if Re(z)<0 in addition to the other reflection q=2).

The idea is to equally "cut" the plan between q angles.
The only thing to do is : looping between the last one to the second one (if z is in the first zone angle we do nothing). If z is in this zone then we reflect z with the half line separating this zone with the previous one passing through the origin.
That's all !

The angles of theses half lines passing through the origin are : 2*pi*p/q for every p between 0 and q.
z is in the p "angle zone" <=> ( 2*pi*p/q < arg(z) < 2*pi*(p+1)/q <=> ( 0< arg(z)*q/(2*pi) - p < 1 )
arg(z) is between 0 and 2*pi, so add pi if it is between -pi and pi.
In terms of complex numbers, reflecting to the half line passing through the origin with an angle of A is done by the function : z -> conjugate(z)*e^(i*2*A).
In terms of angles this is just the function : theta -> 2*A - theta.

The results looks good  !
I have images but strange lines appears (due maybe to the use of cos and sin functions).
Anyway, for q=8 we can avoid cos and sin :
We apply the abs function, then if Im(z) > Re(z) we swap Re(z) and Im(z).

Note : q = a power of 2 tends to be better i think.

Voilà    .

[Kali]

I just tried the last technique you mentioned (abs+conditional swap), and I think the results are pretty cool... well done! 

[Alef]

Last one looks great. Could go for vampire movies;)

[DarkBeam]

The same principle used in "FoldingOct" formula