Chaos Patterns

Math Chaos

Chaos Patterns & Fractals

This collection explores various mathematical patterns that emerge from simple iterative processes and chaotic systems, ranging from biological-looking ferns to abstract fractal sets.

Featured Patterns

Barnsley Fern

An Iterated Function System (IFS) fractal that closely resembles a natural fern, created using affine transformations.

Burning Ship Fractal

A fractal similar to the Mandelbrot set but generated with absolute values, resulting in a distinct “burning ship” appearance.

Sierpinski Gasket

A classic fractal based on a triangle with a recursive subdivision into smaller triangles.

Tinkerbell Map

A discrete-time dynamical system that exhibits chaotic behavior and forms a unique attractor shape.

Deep Dive: Barnsley Fern

The Barnsley fern is a classic example of how complex, organic-looking structures can be generated from just four simple linear equations.

The Algorithm

The fern is constructed using a Chaos Game approach with four different affine transformations ( $f_1$ to $f_4$ ), each chosen with a specific probability:

Stem generation ( $f_1$ ): Maps a point to the Y-axis (1% probability).
Successive leaflets ( $f_2$ ): Handles the main body of the fern (85% probability).
Left-hand leaflet ( $f_3$ ): Creates the left branches (7% probability).
Right-hand leaflet ( $f_4$ ): Creates the right branches (7% probability).

Implementation Snippet (C++)

// Choose transformation based on random probability
int p = rand() % 100;
if (p == 1) {
    a = f1(a); // Stem
} else if (p < 86) {
    a = f2(a); // Body
} else if (p < 94) {
    a = f3(a); // Left branch
} else {
    a = f4(a); // Right branch
}

Other Patterns

Burning Ship Fractal

Generated by the iteration:

z_{n+1} = (|\text{Re}(z_n)| + i|\text{Im}(z_n)|)^2 + c

This variation of the Mandelbrot set calculation creates a striking symmetry along the real axis.

Tinkerbell Map

A chaotic map defined by the following equations:

\begin{align} x_{n+1} &= x_n^2 - y_n^2 + a \cdot x_n + b \cdot y_n \\ y_{n+1} &= 2 x_n y_n + c \cdot x_n + d \cdot y_n \end{align}

Common parameters used are $a=0.9, b=-0.6013, c=2.0, d=0.5$ .