N-th Roots of Unity Visualization

An interactive mathematical visualization that plots the n-th roots of unity on the complex plane and dynamically connects them using modular multiplication to reveal beautiful geometric patterns.

🔬 Mathematical Foundation

Core Concept

The n-th roots of unity are complex numbers that satisfy the equation z^n = 1. They can be expressed using Euler's formula:

z_k = e^(2πik/n) = cos(2πk/n) + i·sin(2πk/n)

Where:

• k = 0, 1, 2, ..., n-1 (indices of the roots)
• n is the total number of roots
• Each z_k represents a point on the unit circle in the complex plane

Pattern Generation

The visual patterns are created by connecting each root z_k to z_{(k × multiplier) % n}, where:

• multiplier is a dynamic parameter that determines the connection pattern
• The modulo operation ensures we stay within the valid range of roots
• Different multipliers create different geometric patterns (cardioids, roses, etc.)

🎨 Visual Features

Interactive Controls

• n (Number of Roots): Controls how many points are placed around the unit circle
• Multiplier: Determines the connection pattern between points
• Animation: Automatically evolves the multiplier to show pattern transitions
• Display Options: Toggle between polar and Cartesian coordinate labels

Pattern Examples

• multiplier = 2: Creates a cardioid (heart-shaped curve)
• multiplier = 3: Forms a nephroid (kidney-shaped curve)
• multiplier = 4: Generates a three-cusped epicycloid
• Prime multipliers often create the most interesting patterns

🛠️ Implementation Options

Web Version (Recommended)

# Tech Stack
- Next.js 15 & TypeScript
- Tailwind CSS
- KaTeX (for math rendering)
- HTML5 Canvas API

Advantages:

• Interactive and responsive
• Easy to share and deploy
• Cross-platform compatibility
• Real-time parameter adjustment

🚀 Getting Started

Prerequisites

• Node.js 18+ (for web version)
• Basic understanding of complex numbers and trigonometry

Installation

Web Version

git clone https://github.com/khabzox/times-table-visualizer.git
cd times-table-visualizer
npm install
npm run dev

🎯 Key Learning Outcomes

Mathematical Concepts

1. Complex Numbers: Understanding polar and Cartesian representations
2. Euler's Formula: The relationship between exponentials and trigonometry
3. Modular Arithmetic: How remainder operations create cyclic patterns
4. Unit Circle: Trigonometric functions and their geometric interpretations

Programming Skills

1. Mathematical Visualization: Translating equations into visual representations
2. Interactive UI: Creating responsive controls for parameter manipulation
3. Animation: Smooth transitions and real-time updates
4. Performance Optimization: Efficient rendering of complex geometric patterns

📊 Pattern Gallery

Famous Curves Generated

• Cardioid (n=200, multiplier=2): Heart-shaped curve from circle rolling around another circle
• Nephroid (n=200, multiplier=3): Two-cusped curve, envelope of circles
• Deltoid (n=200, multiplier=4): Three-cusped hypocycloid
• Astroid (n=200, multiplier=5): Four-cusped hypocycloid

Interesting Observations

• When gcd(multiplier, n) = 1, the pattern connects all points
• Prime multipliers often create the most symmetric patterns
• Large values of n create smoother, more continuous curves
• Certain multiplier ratios create self-similar fractal-like structures

🔧 Advanced Features

Customization Options

• Color Schemes: Different palettes for various mathematical themes
• Line Styles: Solid, dashed, gradient, or animated connections
• Export Options: Save patterns as SVG, PNG, or mathematical data
• Multiple Layers: Overlay different multipliers for complex compositions

Educational Mode

• Step-by-Step Visualization: Show how each connection is calculated
• Formula Display: Real-time display of mathematical expressions
• Interactive Tutorials: Guided exploration of key concepts
• Quiz Mode: Test understanding of pattern predictions

📚 Further Reading

Mathematical Background

• "Visual Complex Analysis" by Tristan Needham
• "The Geometry of Complex Numbers" by Hans Schwerdtfeger
• Online resources on cyclotomic polynomials and Galois theory

Related Visualizations

• Spirographs and mathematical art
• Fourier series and epicycles
• Mandelbrot and Julia sets
• Lissajous curves and parametric equations

🤝 Contributing

We welcome contributions that enhance the mathematical accuracy, visual appeal, or educational value of this project. Areas of particular interest:

• Additional curve families and pattern types
• Performance optimizations for large n values
• Educational content and tutorials
• Mobile-responsive design improvements

📄 License

This project is open source and available under the MIT License. Feel free to use, modify, and distribute for educational and research purposes.

🙏 Acknowledgments

Inspired by the mathematical beauty revealed through computational visualization and the rich history of complex analysis and geometric pattern generation.

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"Mathematics is the art of giving the same name to different things." - Henri Poincaré

Transform mathematical abstractions into visual discoveries with this interactive exploration of the n-th roots of unity!