My idea is to iterate a function that is z_(n+1)=f(n,z_n), instead of z_(n+1)=f(z_n).
I tried this with a perturbation of the classic formula z-->z^2+c, btaining very nice effects with tan(x).
An example (code in c):

///////////////////
int z;
double tm;
double k1,s,zz;s=rrrr+iiii;

for (z=0;z (minor) maxit;z++)
{
zz=(double)(z);
k1=tan(5zzzzsqrt(s))/(2.0zzzzzz+250000)+1;
tm=rr;
rr=rrrr-iiii+rk1;
ii=2tmii+ik1;
s=rrrr+iiii;
if (s>10000000) break;
}

return z;
////////////////////////

this is the classica mand formula perturbed with the k1 value
some thing about the argument of tan(x)
the zz^2 is something that make the perturbation more frequent when iteration grows, making the perturbation 'constant' while zooming .The zz^3 in the following division is to make the effect of reducing the power of perturbation. You can play changing this 2 exponents, or other parameters.

My last 5 fractals had been with this formula (with some minor variations).
I'm not able to play with UF, or with textur, multilayer, ecc, i only like to play with programmming and mathematic, but i will be very happy if this can be used by UF master to do new images.....

Thanks
HG

Message edited on: 02/12/2006 14:15

There are several formula writers who subscribe to the UF list and I am sure they would be happy to help you. I would be happy to play with this formula and make images. I have tried to paste what you posted into an empty formula file, but it doesn't work in UF, or maybe I am doing something wrong. :( Mary

I don't use UF.... i only want to suggest a formula. The formula is written in c code, to use it in UF someone should traslate it in UF language. I used this formula directly in a program i write to do fractals.

I just posted your message to the UF mailing list. All the best, Jos.

Don't quite understand how your tangent example fits in with the z_n+1 = f(n, z_n) idea HG?

Here's a subprogram I wrote in BASIC to generate the classic Mandelbrot set z->z^2 + C

Private Sub Command1_Click()
Dim counter As Integer
Dim a As Double, b As Double, c As Double, d As Double
Rem z = a + ib
Dim creal As Double, cimag As Double
Rem c is the complex add-on point
a = 0: b = 0: c = 0: d = 0
Open "testmand.txt" For Output As #1
For cimag = -2 To 2 Step 0.005
DoEvents
For creal = -2 To 2 Step 0.005
a = 0: b = 0: c = 0: d = 0
counter = 0
While (Abs(a + b) < 4 And counter < 200)
c = a ^ 2 - b ^ 2 + creal
d = a * b * 2 + cimag
a = c: b = d
counter = counter + 1
Wend
If (Abs(a + b) < 4) Then
Print #1, "*";
Else
Print #1, " ";
End If
Next
Print #1, " "
Next
Close #1
End
End Sub

David

A lot of fractals formula use an iteration like
x-->f(x)
if you use x-->f(n,x) you can ontain 'more caos'.

In my formula if k1=1 you obtain classic mandelbrot/julia. In my example k1=1+epsilon, where epsilon is a perturbation that depend both from x and z (z, in my example, is n).
I use zz=double (z) do convert integer to double float.

I use a lot this idea in my fractals, not only as in the examples but also in other ways. I posted this formula because i think it produces very nice effects, quite unusual in fractals.

However i don't think you will obtain any nice effect with a txt based fractal :-). Also, with the parameter i give you in the example, nice spot and perturbation is only after a bit zooming, that you can find only with a graphic interface.

Message edited on: 02/17/2006 12:23

Yes, I agree fractals should be graphics :-) I write formula to output as text just to get a very rough idea of the general shape of the fractal.

David