An interactive 3D visualization of the Collatz conjecture through the lens of the Tuple-based Transform algebraic framework.
The Collatz Amphora presents the famous 3n+1 problem through a novel mathematical framework that transforms abstract sequences into an intuitive 3D structure. Like ancient Greek amphoras that stored treasures, this mathematical vessel contains the fundamental rules governing Collatz behavior.
- Interactive 3D Environment: Navigate through a beautiful amphora-shaped structure
- Real-time Analysis: Process any number up to 10 billion and visualize its Collatz sequence
- Mathematical Classification: Automatic sequence type identification (A, B or C) based on the Tuple-based Transform
- Computational Optimization: Pre-calculated wormhole sequences save computational steps
- Demo Mode: Automatic exploration with camera rotation and random number generation
- Responsive Design: Works on desktop and mobile devices
This visualization is based on original research presenting the Tuple-based Transform, a novel algebraic framework that:
- Establishes convergence and cycle uniqueness for all Collatz sequences
- Proves the existence of exactly 42 unique mr classes
- Provides taxonomic classification into three geometric types
- Demonstrates finite wormhole networks providing stopping time reduction
| Color | Meaning |
|---|---|
| Green | Pre-calculated wormhole sequences (dictionary values) |
| Orange | User-computed values (before wormhole entry) |
| Red | Pseudocycle values (repeating patterns) |
| White | Convergence point (value = 1) |
| Yellow | Currently highlighted value during navigation |
- Three.js (r128) - 3D graphics and rendering
- MathJax (3.2.2) - Mathematical equation rendering
- Vanilla JavaScript - Core application logic
- CSS3 - Responsive UI design
- HTML5 - Structure and semantic markup
the-collatz-amphora/
├── the_collatz_amphora.html # Main application file
├── README.md # This file
└── LICENSE # CC-BY-NC-SA 4.0 license
- Input a number: Enter any positive integer (up to 10 billion) and click "Process"
- Navigate sequence: Use "Previous" and "Next" to step through the Collatz sequence
- Explore views:
- Mouse drag to rotate the amphora
- Mouse wheel to zoom in/out
- Use rotation controls for precise adjustments
- Learn more: Click "Legend", "Theory", or "About" for additional information
- Demo mode: Click "Demo" for automatic exploration
- Steps Analysis: Shows total computational steps in the sequence
- Efficiency Metrics: Displays steps saved using pre-calculated wormholes
- Type Classification: Identifies sequence geometry (Type A, B or C)
- Maximum Value Tracking: Highlights and analyzes peak values
- 42 Wormhole Curves: Each represents a fundamental Collatz pathway
- Amphora Structure: Geometric metaphor with pinnacle, body, and cylinder sections
- Sphere Positioning: Arc-length parameterization for uniform visual spacing
- Dynamic Highlighting: Real-time sphere selection and navigation
- Rotation: X, Y, Z axis controls plus mouse interaction
- Zoom: Smooth zooming with distance constraints
- Reset View: Return to optimal viewing position
- Responsive UI: Adapts to different screen sizes
- Computation: Handles numbers up to 10^10 efficiently
- Memory: Optimized sphere placement and rendering
- Rendering: 60fps on modern hardware with WebGL
- Network: Self-contained, no external dependencies (CDN only)
This tool is valuable for:
- Mathematical Research: Exploring Collatz sequence patterns
- Educational Purposes: Visual teaching of iterative processes
- Algorithm Analysis: Understanding computational complexity
- Data Visualization: Demonstrating 3D mathematical structures
This project is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
You are free to:
- Share — copy and redistribute the material
- Adapt — remix, transform and build upon the material
Under the following terms:
- Attribution — You must give appropriate credit
- NonCommercial — You may not use the material for commercial purposes
- ShareAlike — If you remix, transform or build upon the material, you must distribute your contributions under the same license
See LICENSE for full details.
Javier Hernández
- 📧 Email: 271314@pm.me
- 🔗 GitHub: @hhvvjj
- Mathematical Community - For decades of research on the Collatz conjecture
- Three.js Team - For the excellent 3D graphics library
This is a research project. If you're interested in:
- Mathematical extensions of the framework
- Performance optimizations for larger numbers
- Educational adaptations for different audiences
- Bug reports or feature suggestions
Please open an issue or contact me directly.
"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." — William Paul Thurston