React Interactive Applications¶

This work is licensed under a Creative Commons Attribution 4.0 International License.

The React application suite provides high-performance, GPU-accelerated fractal visualizations directly in your web browser.

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Fractal Visualizations¶

Mandelbrot Set¶

The Mandelbrot set is defined as the set of complex numbers \( c \) for which the iteration:

\[ z_{n+1} = z_n^2 + c, \quad z_0 = 0 \]

remains bounded.

Features: - Real-time zoom and pan - Adjustable iteration count - Multiple color schemes - WebGL-accelerated rendering

Controls: - Click and drag to pan - Scroll to zoom - Adjust max iterations for detail

Julia Sets¶

Julia sets use the same iteration as the Mandelbrot set, but \( c \) is fixed and the initial value \( z_0 \) varies:

\[ z_{n+1} = z_n^2 + c \]

Features: - Interactive parameter selection - Real-time preview - Connection to Mandelbrot set visualization - Click on Mandelbrot to select Julia parameter

Diffusion Limited Aggregation¶

DLA simulates particle aggregation through random walks, producing fractal clusters with dimension \( D \approx 1.7 \).

Features: - Step-by-step or continuous simulation - Adjustable particle count - Sticking probability control - Export cluster data

Algorithm: 1. Start with a seed particle at center 2. Release particles at random boundary positions 3. Particles random walk until they stick to the cluster 4. Repeat to grow the structure

Barnsley Fern¶

The Barnsley fern is generated using an Iterated Function System (IFS) with four affine transformations.

Transformations:

Transform	Probability	Description
\( f_1 \)	0.01	Stem
\( f_2 \)	0.85	Main leaflet
\( f_3 \)	0.07	Left leaflet
\( f_4 \)	0.07	Right leaflet

Features: - Adjustable transformation parameters - Point count control - Real-time rendering - Export as image

Branching Architectures¶

Pythagoras Tree¶

A self-similar fractal tree where each branch generates two smaller branches forming a right triangle.

Features: - Adjustable branching angle - Iteration depth control - Color gradient by depth - Animation support

Mathematics: At depth \( n \), the side length is \( r^n \) where \( r = \cos\theta \) for angle \( \theta \).

L-System Trees¶

Lindenmayer systems (L-systems) use formal grammars to generate plant-like structures.

Example Rules:

Axiom: F
Rules: F → F[+F]F[-F]F

Features: - Custom axiom and rules - Adjustable angle and length - Multiple preset configurations - Step-through generation

Noise Generators¶

Pink Noise (1/f Noise)¶

Generates signals with power spectral density:

\[ S(f) \propto \frac{1}{f^\alpha} \]

where \( \alpha \approx 1 \) for pink noise.

Features: - Adjustable spectral exponent - Real-time audio playback - Waveform visualization - Spectrum analysis

Fractional Brownian Motion¶

Self-affine random process controlled by the Hurst exponent \( H \):

\( H > 0.5 \): Persistent (trending)
\( H = 0.5 \): Standard Brownian motion
\( H < 0.5 \): Anti-persistent (mean-reverting)

Features: - Hurst exponent slider - 1D trace and 2D surface generation - Export data for analysis

Mathematical Visualizations¶

Riemann Zeta¶

Visualization of the Riemann zeta function:

\[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \]

Features: - Complex plane exploration - Zero highlighting - Critical strip focus - Domain coloring

Riemann Zeta 3D¶

Three-dimensional surface plot of \( |\zeta(s)| \).

Features: - Interactive rotation - Adjustable domain - Multiple visualization modes - Pole detection

Conway's Game of Life¶

Cellular automaton with emergent complexity from simple rules.

Rules: 1. Live cell with 2-3 neighbors survives 2. Dead cell with exactly 3 neighbors becomes alive 3. All other cells die or stay dead

Features: - Pattern library - Custom pattern drawing - Speed control - Population statistics

Waves Visualization¶

Wave interference patterns demonstrating harmonic superposition.

Features: - Multiple wave sources - Frequency and amplitude control - Interference pattern visualization - Connection to Fourier analysis

Technical Details¶

Performance¶

WebGL 2.0 for GPU-accelerated rendering
Shader-based computation for Mandelbrot/Julia
Canvas 2D for simpler visualizations
React 18 with optimized rendering

Browser Compatibility¶

Browser	Support
Chrome 90+	Full
Firefox 88+	Full
Safari 14+	Full
Edge 90+	Full

Development¶

Source code is in react/src/pages/ organized by category: - fractals-1d-2d-3d/ - Core fractal visualizations - branching-architectures/ - Tree and plant structures - riemann-zeta-functions/ - Zeta function visualizations