Burning Ship Fractals and Friends: Continued
Hello, this is 108Hassium.
Last week, I posted this article.
In that article, I originally planned to introduce "things related to the Burning Ship fractal that don't have names," but the introduction of the named ones alone ended up being quite voluminous.
So, this time, I will introduce the unused material from last week's article.
Burning Ship Transformation
The Burning Ship fractal can be expressed using complex numbers as the "Mandelbrot set of $(|\text{Re}(z)|+i|\text{Im}(z)|)^2+c$" (where $\text{Im}(z)$ and $\text{Re}(z)$ are the imaginary and real parts of $z$).
In this article, I will use the following definition for the term "Mandelbrot set."
The set of all complex numbers $c$ for which the following complex sequence does not diverge to infinity is defined as the "Mandelbrot set of $f(z,c)$."
$z_{n+1}=f(z_{n},c)$
However, unless otherwise specified, $z_0=0$.
Also, in this article, I will call the transformation $z\rarr |\text{Re}(z)|+i|\text{Im}(z)|$ the "Burning Ship transformation" and denote it as $B(z)$.
So, let's take a look at the Mandelbrot sets of $f(B(z),c)$ for various functions $f(z,c)$.
The Mandelbrot set of $B(z)^n+c$ seems to have only one axis of line symmetry when $n$ is odd, and becomes asymmetric when $n$ is even.
The Mandelbrot set of a linear function usually results in a shape that is not interesting at all, but by applying the Burning Ship transformation, it becomes somewhat interesting.
The function $\frac{c}{z^2-1}+1$, which I have introduced several times as a personal favorite, seems to show a different kind of interest when the Burning Ship transformation is applied.
The Julia set of $${B(z)^n+c}$$ is line-symmetric with respect to the real and imaginary axes (this seems to be a property common to all Julia sets of functions of the form $${f(B(z))}$$), and I did not find any particular characteristics due to differences in degree.
Split-complex numbers
Numbers expressed as $${x+yj}$$, where $${j}$$ is a non-real number satisfying $${j^2=1}$$ and $${x,y}$$ are real numbers, are called split-complex numbers.
For split-complex numbers, just as with complex numbers, a Mandelbrot set can be defined, and the calculation formula for the split-complex version of the Mandelbrot set for $${z^2+c}$$ can be expressed using only real numbers as follows.
$${\begin{cases}x_{n+1}=x_n^2+y_n^2+a\\y_{n+1}=2x_ny_n+b\end{cases}}$$
Note that in this article, in addition to the definition of the Mandelbrot set using complex functions mentioned above, the following definition using real functions is also used.
The set of all pairs of real numbers $${(a,b)}$$ such that the following sequence does not diverge to infinity is defined as the "Mandelbrot set of $${(f(x,y,a,b),g(x,y,a,b))}$$ ".
$${\begin{cases}x_{n+1}=f(x_n,y_n,a,b)\\y_{n+1}=g(x_n,y_n,a,b)\end{cases}}$$
However, unless otherwise specified, $${x_0=y_0=0}$$.
Split-complex numbers themselves have almost nothing to do with the Burning Ship fractal, but there are members of the Burning Ship fractal family that share similar characteristics with these fractal figures.
Factorization
As you know, the part $${x^2-y^2}$$ that appears in the Mandelbrot set formula can be factored as $${(x-y)(x+y)}$$.
I wondered if I could discover new members of the Burning Ship fractal family by utilizing this.
Others
The Mandelbrot set of $${(x^2-xy+a,y^2+xy+b)}$$ looks somewhat similar to the Burning Ship fractal, despite the formula not looking similar and not using an absolute value function.
The Julia set has two axes of symmetry, just like the one for $${(x^2-y^2+a,2|xy|+b)}$$, but for some reason it is tilted at an odd angle (probably 22.5°).
In this Mandelbrot set, you can see shapes similar to the Mandelbrot set of $${z^2+c}$$ and the Mandelbar on the left side.
It no longer has anything to do with the Burning Ship fractal, but I'm introducing it as a bonus because I feel a connection to it somehow.
Postscript
☟Continuation?