Topological data analysis: find loops in point clouds by tracking when holes appear and disappear
Point Cloud + Rips Complex
Persistence Diagram
H0: ? components | H1: ? loops
Vietoris-Rips filtration: connect any two points within distance ε, fill triangles when all edges present.
As ε grows, components merge (H0 dies) and loops form then fill (H1 born/dies).
Persistence diagram: each dot = one topological feature (birth, death). Far from diagonal = long-lived = real topology.
Blue dots = H0 (components), orange = H1 (loops).