# A pair of non-identical twins

A complex number $x + i y$ can be plotted in a complex plain where the $x$ coordinate is the real axis and the $y$ coordinate the imaginary.

Let’s consider the following iteration:

$z_{n+1} = z_{n}^2 + c\quad\quad\quad(1)$

where $z, c$ are complex numbers.

If (1) are started at $z_0 = 0$ for various values of $c$ and plotted in c-space, we have the Mandelbrot set:

When $c$ is held fixed and points generated by (1) are plotted in z-space, the result is the Julia set:

# Having a cow !

Let’s have a cow first:

There are more:

For example:

# Chaos Esthétique

Drawing fractal by hand is tedious at best and close to impossible at worst. With the help of a CAS program, all you need is a few lines of code with a for-loop operation.

The fractal structure in Fig. 1 is generated from Gumnowski-Mira chaos model

$\begin{cases} F(x)=a x+{{2(1-a)x^2} \over { 1+x^2}} \\ x_{n+1}=y_{n}+F(x_{n}) \\ y_{n+1}=-x_{n} +F(x_{n+1}) \end{cases}$

Fig.1 $a=0.31, (x_0, y_0) = (12, 0)$

Replace the term $x_n$ by $-bx_n$ with b = 0.9998, the result is shown in Fig. 2.

Fig.2 $a=0.7, (x_0, y_0) = (15, 0)$