topology-sonarv1.5

Why this project?

Topological Data Analysis (TDA) has become one of the most powerful tools in modern data science, used to detect hidden structures in everything from protein folding to financial markets. Yet persistent homology — its core algorithm — remains locked behind Python libraries and academic papers. Nobody has made an intuitive, interactive browser tool where you can draw point clouds and watch topology emerge in real-time with audio feedback. This project bridges that gap.

What is it?

An interactive persistent homology explorer. You draw or generate point clouds on the canvas, then watch as growing circles reveal the hidden topological structure: connected components (H⁰) merge together and loops/holes (H¹) appear and disappear. Each feature is tracked across a "filtration" parameter (radius), visualized as animated barcodes and persistence diagrams, and sonified so you can hear the topology of your data.

How does it work?

• Vietoris-Rips Filtration: For each pair of points, compute the distance. Sort all edges by length and add them one by one as radius increases.
• H⁰ (Connected Components): Union-Find data structure tracks which points are connected. Each merge kills a component — birth at r=0, death at merge time.
• H¹ (Loops/Holes): When an edge connects two already-connected points, it creates a cycle. Spanning-tree path tracking identifies independent cycles. Triangles that fill in cycles cause them to die.
• Sonification: H⁰ births map to bass tones, H¹ births to higher harmonics. Feature persistence (lifespan) controls note duration and volume. Longer-lasting features sound louder — "significant" topology resonates more.

How to use it

• Draw points: Click or drag on the Point Cloud canvas to add points manually.
• Generate presets: Select a shape (Circle, Torus, Figure-8, etc.) from the dropdown, adjust point count with the slider, then click "Generate Points".
• Play filtration: Click ▶ Play to animate the filtration. Watch circles grow around points and edges form between nearby pairs.
• Pause/Reset: Use ⏸ Pause to freeze the animation, ↺ Reset to restart from r=0.
• Speed control: Adjust the Speed slider (1×–20×) for faster or slower exploration.
• Audio: Click 🔇 Audio to enable sonification. Feature births produce tones; persistent features sound louder and longer.
• Read barcodes: Green bars = connected components, Purple bars = loops. Bar length = feature persistence (longer = more topologically significant).
• Read diagrams: Each dot plots (birth, death). Distance from diagonal = persistence. Points far from the diagonal are "significant" topological features.