Sequence-Generated Images

Thorn Fractal

Andrew Graff’s Thorn Fractal (otherwise known as Secant Sea) is a fractal on the Cartesian plane. It requires two parameters $c_x$ and $c_y$ . Every point on the plane $(x_0, y_0)$ is iterated through the equations:

$x_{n+1} = \frac{x_n}{cos(y_n)} + c_x$

$y_{n+1} = \frac{y_n}{sin(x_n)} + c_y$

Like the Mandelbrot Set, if this goes beyond a certain limit (which it often does), you color it based on how long it takes to break the limit. Otherwise, it is colored black. At the right,

$c_x = -0.1, c_y = 0.327$

Curlicue Fractals

Curlicue Fractals are fractals that are generated by an irrational parameter $s$ . It is generated by defining two sequences $\theta_n$ and $\phi_n$ , where:

$\theta_{n+1} = (\theta_n + 2*\pi*s)$ $mod$ $(2*\pi)$

$\phi_{n+1} = \theta_n + \phi_n$ $mod$ $(2*\pi)$

For every iteration of $\phi_n$ , a line is drawn with a specific length and the angle as $\phi_n$ . A next line is then started at the end of the first, and it repeats indefinitely. At the left, $s = 3.698187994...$

Popcorn Map

Created by Clifford Pickover, the Popcorn Map is a one parameter map, and is generated by taking lots of random initial points, and putting them through a sequence repeatedly, then plotting the results. The sequences are:

$x_{n+1} = x_n - h*\sin(y_n + tan(3*y_n))$

$y_{n+1} = y_n - h*\sin(x_n + tan(3*x_n))$

At the left, $h = 0.005$

Kaneko Map

The Kaneko Map is a two-parameter map. It is generated by taking two initial values $x_0 = 0.1$ , $y_0 = 0.1$ and plotting every iteration:

$x_{n+1} = a*x_n + (1-a)*(1-d*y_n^2)$

$y_{n+1} = x_n$

At the right,

$a = 0.18451, d = 1.69574$

Lyapunov Fractal

A Lyapunov Fractal is an image generated by the Lyapunov exponent of logistic maps with certain initial conditions. First, a sequence $S$ is required, which is constructed of $A$ ‘s and $B$ ‘s. The sequence can be any length (and it loops over). Then, for every point $(a, b)$ in a certain region (usually $[0, 4]$ and $[0, 4]$ ), we define $r_n = a$ if $S_n = A$ , and otherwise $r_n = b$ . Once this is done, we have the logistic map $x_{n+1} = r_n*x_n*(1 - x_n)$ . Once this is done, we are able to take the Lyapunov exponent of the logistic map. This can be expressed as $\lambda = \displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \log|r_n*(1 - 2*x_n)|$ . Once this is done, you can color $(a, b)$ as a blue color if $\lambda > 0$ , and yellow if $\lambda < 0$ .