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Example Notebooks

Complexplorer provides comprehensive tutorial notebooks and application examples to help you master complex function visualization.

Tutorial Notebooks

The examples/notebooks/ directory contains 8 progressively structured tutorial notebooks (~2.5 hours total).

1. Getting Started (10 min)

File: examples/notebooks/01_getting_started.ipynb

Your first introduction to complexplorer:

  • Installation verification
  • Basic domains: Rectangle, Disk, Annulus
  • Enhanced phase portraits with auto_scale_r=True
  • 2D plotting with plot()
  • Standard demo function: f(z) = (z - 1) / (z**2 + z + 1)

Prerequisites: None - start here!

2. Advanced Domains (15 min)

File: examples/notebooks/02_domains_advanced.ipynb

Master domain set operations (v2.0 feature):

  • Union: domain1 | domain2
  • Intersection: domain1 & domain2
  • Difference: domain1 - domain2
  • Complex compositions (flower patterns, swiss cheese)
  • Custom domain design

Prerequisites: Notebook #1

3. PyVista 3D Visualization (20 min)

File: examples/notebooks/03_pyvista_3d.ipynb

High-performance 3D visualization:

  • plot_landscape_pv() - analytic landscapes
  • Inline mode (notebook=True) vs interactive (notebook=False)
  • Pair plots: pair_plot_landscape_pv()
  • Performance comparison: 15-30x faster than matplotlib

Key decision: Use notebook=True for development (non-blocking), notebook=False for publication quality.

Prerequisites: Notebook #1

4. Colormaps Comprehensive (25 min)

File: examples/notebooks/04_colormaps_comprehensive.ipynb

Complete colormap reference:

All 13 families: - Phase (basic + enhanced) - Chessboard / PolarChessboard - LogRings - PerceptualPastel (OkLCh-based, v2.0) - Isoluminant - CubehelixPhase - AnalogousWedge (v2.0) - DivergingWarmCool (v2.0) - InkPaper (v2.0) - EarthTopographic (v2.0) - FourQuadrant (v2.0) - Sinebow - CustomCmap

Use case guide included!

Prerequisites: Notebook #1

5. Accessibility & CVD (20 min)

File: examples/notebooks/05_accessibility_cvd.ipynb

CRITICAL for publications!

Color vision deficiency testing:

  • CVD statistics: ~8% of males affected
  • Simulation with colorspacious library
  • Test protanopia, deuteranopia, tritanopia
  • CVD-friendly colormaps: CubehelixPhase, PerceptualPastel
  • Design guidelines for inclusive figures

Requirement: pip install colorspacious

Prerequisites: Notebook #4

6. Riemann Sphere (30 min)

File: examples/notebooks/06_riemann_sphere.ipynb

Extended complex plane ℂ ∪ {∞}:

  • Stereographic projection math
  • 2D hemisphere charts with riemann_chart()
  • 3D sphere with riemann_pv() (PyVista)
  • Essential singularities: exp(1/z)

Prerequisites: Notebooks #1, #3

7. Modulus Scaling (25 min)

File: examples/notebooks/07_modulus_scaling.ipynb

Control 3D height mapping:

10+ scaling modes: - none - direct modulus - constant - flat (phase only) - arctan - smooth bounded (recommended) - logarithmic - emphasize poles/zeros - linear_clamp, adaptive, power, sigmoid, ... - custom - user-defined function

Parameter exploration for fine control.

Prerequisites: Notebooks #3, #6

8. STL Export for 3D Printing (30 min)

File: examples/notebooks/08_stl_export.ipynb

Transform math into physical art:

  • OrnamentGenerator workflow
  • Modulus scaling selection
  • Domain restrictions (stability)
  • Bisection workflow (PrusaSlicer/Cura) - RECOMMENDED
  • Gluing tips: cyanoacrylate vs epoxy
  • Post-processing: sanding, painting

Why bisect? No supports, easier printing, multi-color options.

Prerequisites: Notebooks #6, #7

Application Notebooks

The examples/applications/ directory contains 4 real-world application examples.

1. FFT/DFT Matrices (15 min)

File: examples/applications/app_01_fft_matrices.ipynb

Discrete Fourier Transform visualization:

  • DFT matrix: \(W_N^{jk}\) where \(W_N = e^{-2\pi i/N}\)
  • 2×4 grid gallery: N = 32, 64, 128, 256, 512, 1024, 2048, 4096
  • Circular symmetry from roots of unity
  • Extensions: IDFT, Hadamard, Toeplitz matrices

Mathematical context: Linear algebra, signal processing, FFT algorithm.

Prerequisites: Notebook #1

2. Special Functions (25 min)

File: examples/applications/app_02_special_functions.ipynb

Classical special functions in ℂ:

  • Gamma function: Poles at 0, -1, -2, ...
  • Riemann zeta: Critical strip, non-trivial zeros (Riemann Hypothesis!)
  • Bessel functions: Entire functions, wave equations
  • Elliptic functions: Doubly periodic (Weierstrass ℘)
  • Singularity classification: Poles, zeros, essential singularities

Mathematical context: Number theory, physics, analysis.

Prerequisites: Notebooks #1, #6

3. Conformal Mappings (25 min)

File: examples/applications/app_03_conformal_maps.ipynb

Angle-preserving transformations:

  • Möbius transformations: Linear fractional, circles → circles
  • Joukowsky transform: Circles → airfoils (aerodynamics!)
  • Exponential/logarithm: Strip → plane
  • Power functions: Angle multiplication
  • Composition: Build complex from simple

Uses pair_plot() to show domain → codomain transformation.

Applications: Fluid dynamics, electrostatics, cartography.

Prerequisites: Notebooks #1, #2

4. Complex Dynamics & Fractals (35 min)

File: examples/applications/app_04_complex_dynamics.ipynb

Iterative systems and chaos:

  • Newton's method: Fractal basin boundaries (z³-1, z⁵-1)
  • Julia sets: Iteration of z² + c
  • Mandelbrot set: Parameter space catalog
  • Escape time coloring: Fractal visualization
  • Chaos theory: Sensitive dependence, self-similarity

Mathematical context: Dynamical systems, fractal geometry, holomorphic dynamics.

Prerequisites: Notebooks #1, #4

Quick Navigation

Goal Notebook
Complete beginner Start with #1, follow sequence
Just want colormaps Jump to #4
Need 3D/PyVista #3, then #6-7
3D printing #8 (requires #6-7)
Check accessibility #5 (CRITICAL before publishing!)
Real-world examples Applications folder

Running the Notebooks

Install Dependencies

# Core dependencies (required)
uv pip install -e "."

# With all optional features (recommended)
uv pip install -e ".[all]"

# Minimal (excludes PyVista)
uv pip install -e ".[dev]"

Launch Jupyter

jupyter notebook examples/notebooks/

Or use JupyterLab: 

jupyter lab examples/notebooks/

Optional: CVD Simulation

For accessibility testing (Notebook #5): 

uv pip install colorspacious

Tips for Learning

  1. Tutorial sequence (8 notebooks): Follow in order for comprehensive understanding
  2. Applications: Pick based on interest after tutorials
  3. Interactive scripts: Try examples/interactive_showcase.py for hands-on exploration

Estimated Time

  • Tutorials: ~2.5 hours total
  • Applications: ~1.5 hours total
  • Combined: ~4 hours for complete coverage

Using the Standard Demo Function

Most notebooks use: 

def f(z):
    return (z - 1) / (z**2 + z + 1)

This function has: - Two poles (denominators zeros) - One zero (numerator zero) - Interesting structure without being too complex

PyVista Modes

During development/learning: 

cp.plot_landscape_pv(..., notebook=True, show=True)
- Non-blocking - Inline display - Good enough quality

For publications: 

cp.plot_landscape_pv(..., notebook=False, show=True)
- High quality - External interactive window - Blocks until closed

Troubleshooting

Import Errors

"No module named complexplorer": 

cd ../..  # Go to repo root
uv pip install -e ".[all]"

"No module named pyvista": 

uv pip install pyvista

"No module named colorspacious": 

uv pip install colorspacious

PyVista Issues

Blank output in Jupyter: - Use notebook=True for inline - Or use notebook=False and check external window

Low quality inline: - This is expected with notebook=True - Use notebook=False for high quality - Or run as script instead of notebook

Performance

Slow 3D rendering: - Use PyVista functions (*_pv) not matplotlib - Reduce resolution parameter - Ensure PyVista is properly installed

Contributing

Found an error or have an improvement?

  1. Open an issue on GitHub
  2. Submit a pull request
  3. Follow the notebook style guide in CONTRIBUTING.md

Ready to start? Open 01_getting_started.ipynb and begin your journey!