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Chaos Patterns

Math Chaos

Chaos Patterns & Fractals

Section titled “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.

Section titled “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

Section titled “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

Section titled “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:

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

Implementation Snippet (C++)

Section titled “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

Section titled “Other Patterns”

Burning Ship Fractal

Section titled “Burning Ship Fractal”

Generated by the iteration:

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

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

Tinkerbell Map

Section titled “Tinkerbell Map”

A chaotic map defined by the following equations:

x_{n+1} = x_n^2 - y_n^2 + a*x_n + b*y_n
y_{n+1} = 2*x_n*y_n + c*x_n + d*y_n

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