Cech versus Rips Triangle Hole

In topological data analysis, we often build simplicial complexes from a set of points in a metric space to study their shape. Two common constructions are the Čech complex and the Vietoris-Rips complex. Both depend on a radius parameter r.

experiment

Use the Cech versus Rips Triangle Hole module controls to explore how the complexes evolve:

1. Set the Triangle side to 1.
2. Set the Ball radius ratio to 1/2. Observe that edges appear in both complexes. The Rips complex immediately fills in the face because it's a clique complex, so its 1-dimensional homology (Betti 1) is 0.
3. Set the Ball radius ratio to 11/20. The Čech complex still has a hole (Betti 1 = 1) because the balls do not share a common intersection point.
4. Set the Ball radius ratio to 1/sqrt(3). Observe that the Čech complex finally fills the face, as the balls now share a common intersection point at the circumcenter. Both complexes now have Betti 1 = 0.