A way to make certain fractal formulas easy to understand

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Offline greentexas

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« on: February 24, 2018, 03:41:43 AM »
I am primarily concerned with stardust4ever's variations: I'm guessing 90% of my fractal-related time involves either the Mandelbrot set or one of those ABS variations.

It was inspired by this: https://fractalforums.org/fractal-mathematics-and-new-theories/28/naming-scheme-for-abs-variant-fractals/818

GHN (General Hybridization Notation) works as follows:

The formula f(x,y) is the same as z = abs(zy)x/y + z0. One of the members of fractals invented a folding function, which is applied here. x is the power, y is the number of "folds".

Let Mx be shorthand for zx + z0. This is short for Mandelbrot.
Let Bx be shorthand for abs(z)x + z0. This is the same as f(x,1). This is short for Burning Ship.
Let Fx be shorthand for abs(zx) + z0. This is the same as f(x,x). This is short for Buffalo.

If no x is given, it's assumed to be 2.

These capital letters are redundant, but they could speed things up.

If you have two pieces of a fractal (call them f1 and f2), then [f1,f2] is real(f1) + i*imag(f2).

The complex conjugate of any fractal in GHN is x' for a given fractal x.

Here's an example:

The 4th BS Partial Real Mandelbar is [B4,M4]'. We can break this down to a regular formula you can use in a program like Ultra Fractal:
(Real(B4) + Imag(M4)*i)' which is the same as:
(Real(B4) - Imag(M4)*i) which is the same as:
Real(abs(z)4 + z0) - Imag(z4 + z0)*i which is the correct formula if you stick z = to the beginning.

All of Stardust4ever's twelve 2nd fractals are close to trivial with GHN. You can also raise a bracketed expression to a power and make a composition.

f1.f2 means to do f2 to z and then f1, since you start inside and move out, as in parentheses. You put an exclamation point before f1 or f2 if you want to ignore z0.

What does the 4th Celtic Imaginary Quasi Perpendicular / Heart become?

I found out that iterating the 2nd order Perpendicular Buffalo without a z0 followed by a 2nd order Buffalo with a z0 gives this fractal. Here is what the GHN for this fractal is as well as the formula:

The Perpendicular Buffalo is a hybrid of the Buffalo and Perpendicular Burning Ship, or [F,[M,B]], while the Buffalo is just F. Since the P-Buffalo comes first, we have F.[F,[M,B]]. (remember, f1 goes inside). Since the  Perpendicular Buffalo has no z0, the complete GHN formula for this fractal is:


Let's now convert this to a regular formula:

F.![F,(real(z2 + z0) + imag(abs(z)2 + z0)*i)]
F.!(real(abs(z2) + z0) + imag(real(z2 + z0) + imag(abs(z)2 + z0)*i)*i)
F.(real(abs(z2)) + imag(real(z2) + imag(abs(z)2)*i)*i)
abs((real(abs(z2)) + imag(real(z2) + imag(abs(z)2)*i)*i)2) + z0. (This happens because the formula to the right of the period acts like z.)

You can also use z0, +, -, *, /, all of the trig functions, etc. but they usually aren't necessary.

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/a-way-to-make-certain-fractal-formulas-easy-to-understand/923/