Revolutionizing our understanding of cubic irrationals through interactive visualization
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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:
Each method builds upon centuries of mathematical inquiry while introducing novel perspectives that redefine our understanding of cubic irrationals. |
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See mathematics in action with our suite of interactive tools
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Cubic Polynomial Explorer Visualize polynomial behavior and roots |
Projective Space Visualization Explore HAPD algorithm in action |
Matrix Trace Calculator Calculate and visualize trace patterns |
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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
Click to expand mathematical details
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:
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.
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.
For complex conjugate roots, we utilize the identity:
sin²(θ) = (1 - cos(2θ))/2
to detect periodicity through angular relationships.
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
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Experience the full interactive paper with a single click:
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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 |
Modern Browser Chrome, Firefox, Safari, Edge |
JavaScript Enabled in browser settings |
WebGL For 3D visualizations (optional) |
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}
}
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This work is licensed under the Creative Commons Attribution 4.0 International License (CC-BY 4.0). You are free to:
Under the following terms:
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We stand on the shoulders of giants:
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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
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We welcome contributions to enhance this interactive academic paper. If you're passionate about mathematics, visualization, or education:
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Contact us for collaboration or feedback: contact@example.com
