🧮 Solving Hermite's Problem: Interactive Academic Paper 🧮

Revolutionizing our understanding of cubic irrationals through interactive visualization

💡 Overview

This repository contains an immersive mathematical experience that transforms complex theoretical concepts into interactive visual explorations. We present three groundbreaking methods for solving Hermite's Problem related to the characterization of cubic irrationals: 🔍 HAPD Algorithm: A projective space approach for detecting periodicity in cubic irrationals that provides geometric insights previously unattainable. 📊 Matrix Approach: Utilizing companion matrices and trace sequences to reveal patterns in the continued fraction expansion of cubic irrationals. 📐 Modified sin²-Algorithm: An adaptation for cubic irrationals with complex conjugate roots, extending the theoretical framework to the complex domain. 🧮 Subtractive Algorithm: A numerically stable variation of the HAPD algorithm that maintains precision even with large coefficients. Each method builds upon centuries of mathematical inquiry while introducing novel perspectives that redefine our understanding of cubic irrationals.

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." — William Paul Thurston A visual representation of our approach

✨ Interactive Features

See mathematics in action with our suite of interactive tools

Cubic Polynomial Explorer Visualize polynomial behavior and roots	Projective Space Visualization Explore HAPD algorithm in action	Matrix Trace Calculator Calculate and visualize trace patterns
Sin² Algorithm Demo Visualize complex root periodicity	Subtractive Algorithm Demo Explore numerical stability	Mathematical Notation Helper Interactive glossary of concepts

Try It Now: Launch Interactive Paper

🧠 The Mathematics

Click to expand mathematical details

The Fundamentals of Hermite's Problem

Hermite's Problem asks about the periodicity of continued fraction expansions for cubic irrationals. Given a cubic irrational α that satisfies:

ax³ + bx² + cx + d = 0

The continued fraction expansion can be represented as:

α = a₀ + 1/(a₁ + 1/(a₂ + 1/...))

Our work provides a complete characterization of when this expansion becomes periodic, using three complementary approaches:

HAPD Algorithm: Projective Geometric Approach

We map the problem to projective space P² where:

[xₙ, yₙ, zₙ]ᵀ = M^n [x₀, y₀, z₀]ᵀ

Periodicity is detected through invariant subspaces of the transformation matrix M.

Matrix Trace Sequence Detection

For companion matrix A of the cubic polynomial, we analyze the sequence:

Tr(A^n) = α^n + β^n + γ^n

Periodicity emerges in patterns of this trace sequence.

Modified sin²-Algorithm

For complex conjugate roots, we utilize the identity:

sin²(θ) = (1 - cos(2θ))/2

to detect periodicity through angular relationships.

🗂️ Repository Structure

hermitesproblem
├── githubpages/            # Web version with interactive elements
│   ├── index.html          # Entry point for the interactive paper
│   ├── paper-viewer.html   # Enhanced paper viewing experience
│   ├── css/                # Styling for the interactive elements
│   └── js/                 # JavaScript for the interactive tools
├── arxiv_submission/       # LaTeX source code for arXiv submission
│   ├── main.tex            # Main LaTeX document
│   └── figures/            # Static figures for the paper
└── figures/                # Shared visualization resources
    ├── algorithms/         # Algorithm visualization resources
    └── interactive/        # Resources for interactive elements

🚀 How to Use

🌐 Online Experience (Recommended) Experience the full interactive paper with a single click: Visit https://bbarclay.github.io/hermitesproblem Navigate through sections using the sidebar menu Interact with visualizations to deepen understanding Explore algorithm demos with custom parameters Toggle between paper view and interactive mode

💻 Local Installation For offline access or development: # Clone the repository git clone https://github.com/bbarclay/hermitesproblem.git # Navigate to the project directory cd hermitesproblem # Open in browser cd githubpages open index.html # or paper-viewer.html # Optional: Run a local server python -m http.server 8000 # Then visit http://localhost:8000

📋 Requirements

Modern Browser Chrome, Firefox, Safari, Edge

JavaScript Enabled in browser settings

WebGL For 3D visualizations (optional)

📝 Citation

If you find this work useful for your research on Hermite's Problem or cubic irrationals, please cite:

@article{hermite_problem2025,
  author    = {Brandon Barclay},
  title     = {Solving Hermite's Problem: Three Novel Approaches for Complete Characterization 
               of Cubic Irrationals},
  year      = {2025},
  journal   = {arXiv preprint},
  url       = {https://arxiv.org/abs/xxxx.xxxxx},
  keywords  = {Hermite's Problem, cubic irrationals, continued fractions, 
               number theory, interactive mathematics}
}

⚖️ License

This work is licensed under the Creative Commons Attribution 4.0 International License (CC-BY 4.0). You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose Under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made.

👏 Acknowledgments

We stand on the shoulders of giants:

Charles Hermite Original problem formulation

Mathematical Community Prior work on continued fractions

Open Source Community Libraries that power our visualizations

Libraries used: MathJax, JSXGraph, Desmos, Bootstrap, Prism.js

🤝 Contribute

Join the Mathematical Revolution We welcome contributions to enhance this interactive academic paper. If you're passionate about mathematics, visualization, or education: ⭐ Star this repository to show your support 🍴 Fork the repo and submit pull requests with improvements 🐛 Report issues or suggest new features 🔍 Review the code and help improve quality 📚 Extend the documentation with additional examples 🎨 Create new visualizations for complex concepts

Contact us for collaboration or feedback: contact@example.com

Let's make mathematics more accessible and engaging together!