notebooks

Weierstrass ℘ Playground

This repository contains an interactive Jupyter notebook that visualizes complex fields derived from the Weierstrass ℘ function on a rectangular lattice and overlays shared second-order trajectories.

Features

• Two-panel visualization:
  • Left panel: ℘(z) field
  • Right panel: ℘′(z) field
• Interactive trajectories following the second-order ODE: z’‘(t) = -℘(z(t)) z(t)
• Comprehensive controls:
  • Lattice parameters (p, q, truncation N)
  • Rendering options (grid resolution, contours, vector fields)
  • Color palette customization
  • Integration parameters
  • Dynamic particle management

Getting Started

1. Install required dependencies:

   pip install numpy matplotlib ipywidgets
   

2. Enable Jupyter widgets (for classic Jupyter):

   jupyter nbextension enable --py widgetsnbextension --sys-prefix
   

3. Open the notebook:

   jupyter notebook weierstrass_playground.ipynb
   

4. Run all cells and interact with the controls at the bottom.

Usage

1. Set lattice parameters: Adjust p, q (lattice periods) and N (truncation level)
2. Configure rendering: Set grid resolution, contour levels, and vector field display
3. Add particles: Define initial positions (z0) and velocities (v0) for trajectories
4. Integrate: Set time step (dt) and duration (T) for trajectory integration
5. Render: Click the “Render” button to generate the visualization
6. Save: Use “Save PNG” to export the current figure

Mathematical Background

The Weierstrass ℘ function is implemented using a truncated lattice sum for rectangular lattice Λ = ℤp + ℤiq:

℘(z) = 1/z² + Σ[1/(z-ω)² - 1/ω²]

where the sum runs over non-zero lattice points ω within the truncation bound N.

Trajectories are integrated using RK4 method with blow-up detection near poles and for large step sizes.

Examples

• Default settings (p=11, q=5, N=3) provide a good starting point
• Try different initial conditions for varied trajectory patterns
• Increase N for higher accuracy (but slower computation)
• Experiment with different lattice aspect ratios

Dependencies

• numpy: Numerical computations
• matplotlib: Visualization
• ipywidgets: Interactive controls

This site is open source. Improve this page.