🎬 Animated Proof Gallery

Mathematical Proofs Come Alive --- Powered by Manim

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Rigorous mathematics deserves more than static ink on paper.

This repository contains animated visualizations of key theorems and proof techniques spanning Topological Data Analysis, Fixed Point Theory, Multi-Objective Optimization, Banach Algebras, and Support Vector Machine Classification. Every animation is written in Manim Community Edition and is fully reproducible from source.

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Gallery Preview

Proof	Domain	Key Idea
Persistent Homology	Topological Data Analysis	Vietoris--Rips filtration and persistence barcodes
Banach Fixed Point Theorem	Functional Analysis	Contraction mappings and cobweb iteration
Pareto Front Construction	Optimization	Dominance, non-dominated sets, and the Pareto frontier
Banach Algebra Spectrum	Abstract Algebra / Analysis	Norms, spectral radius, and invertibility
SVM Maximum Margin	Machine Learning Theory	Optimal separating hyperplane and support vectors

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Proof Catalog

1. Persistent Homology

File: proofs/persistent_homology/scene.py

Visualizes the core pipeline of persistent homology:

• A random 2D point cloud is sampled.
• Balls of increasing radius grow around each point (Vietoris--Rips construction).
• Edges appear when balls overlap; triangles fill when three-way intersections form.
• A persistence barcode diagram is drawn, showing birth and death of topological features.

Mathematical context: Persistent homology tracks how topological features (connected components, loops, voids) appear and disappear across a filtration of simplicial complexes built from data. The resulting barcodes provide a multi-scale signature of the shape of data.

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2. Banach Fixed Point Theorem

File: proofs/fixed_point_theorems/scene.py

Animates the classical Banach contraction mapping theorem on the real line:

• A contraction mapping f: [0,1] -> [0,1] is plotted alongside y = x.
• Starting from an arbitrary initial point, the cobweb iteration is traced.
• The iterates visibly converge to the unique fixed point.
• The contraction constant and convergence rate are displayed.

Mathematical context: If (X, d) is a complete metric space and T: X -> X satisfies d(Tx, Ty) <= q * d(x, y) for some q in (0,1), then T has a unique fixed point, and every Picard iteration converges to it.

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3. Pareto Front / Multi-Objective Optimization

File: proofs/optimization/scene.py

Demonstrates Pareto optimality in bi-objective optimization:

• A scatter of candidate solutions is plotted in objective space.
• Dominated solutions are identified and dimmed.
• Non-dominated (Pareto-optimal) solutions are highlighted.
• The Pareto front curve is traced through the non-dominated set.

Mathematical context: A point x* is Pareto optimal if there is no feasible x such that f_i(x) <= f_i(x*) for all objectives with strict inequality for at least one. The set of all Pareto-optimal points forms the Pareto front.

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4. Banach Algebra Spectrum

File: proofs/banach_algebra/scene.py

Visualizes fundamental concepts in Banach algebra theory:

• A Banach space norm ball is drawn in an abstract 2D representation.
• The spectrum of an element is displayed in the complex plane.
• The spectral radius is shown as a bounding circle.
• Invertibility regions are highlighted with the resolvent set.

Mathematical context: In a unital Banach algebra A, the spectrum sigma(a) of an element a is the set of lambda in C for which lambda e - a is not invertible. The spectral radius formula states r(a) = lim ||a^n||^{1/n} = inf ||a^n||^{1/n}.

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5. SVM Maximum Margin Classifier

File: proofs/svm_classification/scene.py

Builds intuition for the Support Vector Machine optimization problem:

• Two-class data is plotted in 2D feature space.
• Several candidate separating hyperplanes are drawn.
• The maximum-margin hyperplane is identified.
• Support vectors are highlighted and the margin band is shaded.

Mathematical context: The SVM solves min (1/2)||w||^2 subject to y_i(w . x_i + b) >= 1. The optimal hyperplane maximizes the geometric margin 2/||w|| between classes. Only the support vectors (points on the margin boundary) determine the solution.

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How to Run

Render a Single Proof

# Low quality (fast preview, 480p @ 15fps)
manim render -ql proofs/persistent_homology/scene.py PersistentHomologyScene

# High quality (production, 1080p @ 60fps)
manim render -qh proofs/persistent_homology/scene.py PersistentHomologyScene

Render All Proofs

for scene_dir in proofs/*/; do
    manim render -qh "${scene_dir}scene.py"
done

Export to GIF

manim render -qh --format=gif proofs/persistent_homology/scene.py PersistentHomologyScene
# Output: media/videos/scene/1080p60/PersistentHomologyScene.gif

Extract Frame Sequences (for Beamer embedding)

mkdir -p frames/persistent_homology
ffmpeg -i media/videos/scene/1080p60/PersistentHomologyScene.mp4 \
       -vf "fps=30" \
       frames/persistent_homology/frame_%04d.png

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Dependencies

Python

Package	Version	Purpose
manim	>= 0.18	Animation engine (Community Edition)
numpy	>= 1.24	Numerical computation
scipy	>= 1.11	Scientific computing utilities

System

Tool	Purpose
ffmpeg	Video/frame conversion
pdflatex	Beamer slide compilation (optional)
Python 3.10+	Runtime

Quick Install

pip install -r requirements.txt

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Repository Structure

animated-proof-gallery/
├── README.md
├── LICENSE
├── requirements.txt
├── proofs/
│   ├── persistent_homology/
│   │   └── scene.py          # Vietoris-Rips filtration + persistence barcodes
│   ├── fixed_point_theorems/
│   │   └── scene.py          # Banach contraction mapping + cobweb diagram
│   ├── optimization/
│   │   └── scene.py          # Pareto front construction
│   ├── banach_algebra/
│   │   └── scene.py          # Spectrum and spectral radius visualization
│   └── svm_classification/
│       └── scene.py          # Maximum margin hyperplane + support vectors
└── media/
    ├── gifs/                  # Exported GIF animations
    └── videos/                # Rendered MP4 videos

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Contributing

Contributions are welcome. To add a new animated proof:

1. Create a new directory under proofs/ with a descriptive name.
2. Add a scene.py file containing one or more Manim Scene subclasses.
3. Ensure the animation renders cleanly with manim render -qh.
4. Update this README with a description of the proof.
5. Open a pull request.

Please follow the existing code style: descriptive class names, consistent color palettes, clear mathematical labels, and animations between 10--20 seconds.

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Citation

If you use these animations in a talk, lecture, or publication, please cite:

@misc{joshi2026animatedproofgallery,
  author       = {Joshi, Milan Amrut},
  title        = {Animated Proof Gallery: Mathematical Proofs Come Alive},
  year         = {2026},
  publisher    = {GitHub},
  howpublished = {\url{https://github.com/mlnjsh/animated-proof-gallery}},
  note         = {Manim Community Edition visualizations of theorems in TDA,
                  optimization, functional analysis, and machine learning}
}

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License

This project is licensed under the MIT License. See LICENSE for details.

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Built by Prof. Milan Amrut Joshi
mlnjsh@gmail.com · github.com/mlnjsh

Mathematics · Topology · Optimization · Machine Learning