Vietoris-Rips filtration with barcode diagram — topology across scales
Point cloud (drag ε slider to grow balls)
Persistence barcode — H₀ (blue) and H₁ (pink)
Persistence: as ε grows, balls around points merge (H₀ bars die when components connect) and loops form then fill (H₁ bars born when a cycle appears, die when it's filled).
Long bars = robust topological features. Short bars = noise. The barcode is stable: small data perturbations → small bar changes.