EDIT: As of version 2.25, the variable name changed from zz to zlast.
The phoenix-fractal’s formula is different from others because it accesses a previous value of z of the orbit. The formula for the basic phoenix uses two parameters a and b (both complex numbers) and the formula looks like z(n+1) = z(n)^2 + a + b z(n – 1).
In Fractview, the last value of z is stored in the variable zlast, so we can easily enter this function (presets were fixed in version 2.19 to accept zlast in “function”). Yet, there are two constants, a and b while we only have p. In FractInt, this is solved by using the real part of the parameter for a and the imaginary for b. Thus, we obtain the following formula for the mandphoenix:
function: “sqr z + p.x + p.y zlast”
This gives us the following picture (I use the new “Mandelbrot Wikipedia”-preset; guess where it comes from)
It does not look that spectacular, but with zooming and shearing there are actually some hidden treasures in there, and contrary to what you read it is actually useful as a map for julia sets (with shearing).
The julia sets are very spectacular. With juliapoint = 0.566666, -0.5 you get these twin dragons:
And to prove my point that the mandelbrot variation of the phoenix fractal is actually useful as a map, here is a zoom around this julia point:
FractInt contains another variation of the phoenix fractal in which b is always set to -0.5 while the addend is now the complex parameter, hence the formula becomes
function: “sqr z + p – 0.5 zlast”
For juliapoint = 0.5666666, 0 we again obtain the twin dragons but you should investigate the bounds of this mandelbrot set. There are many nice, asymmetric julia sets hidden in here.
FractInt has another fractal type uses a similar formula (also accessing zlast), called “manowar”. Its function is
function: “sqr z + p + zlast”
The mandelbrot set itself is not that interesting, but it has nice julia sets like the following (juliapoint = -0.01/0). If the imaginary part is non-zero the julia set becomes asymetric.
There are lots of varieties of these kinds of fractals. One hobby of mine is to “burningshipify” fractals. Just replace some value by its absolute value. In the following I replaced zlast by “abs zlast” in manowar (function: “sqr z + p + abs zlast”) and obtained the following weird fractal:
Zooming into these spider-web-like structures is definitely a nice thing, and so are the julia-sets of this fractal.
I hope you enjoy the new Mandelbrot Wikipedia preset.
Some words on recent updates: My phone updated to Android 6 and I noticed that the new permission system did not allow saving images. This should be resolved now. The new Android SDK has some flaws and Renderscript is not that well-documented, so I apologize for problems with Android 5 in the latest build. I hope most issues are resolved now.
See you next time where I will explain the unity-fractal and the spider-fractal in this series. All the best.