This interactive visualization explores the mathematical hierarchy underlying Topological Data Analysis (TDA)—a framework that extracts robust structural features from high-dimensional data. Beginning with a discrete point cloud embedded in ℝ², we construct increasingly abstract representations that reveal fundamental geometric invariants.

The Vietoris-Rips parameter ε governs the formation of our simplicial complex: for each pair of points within Euclidean distance ε, we add a 1-simplex (edge). When three points are pairwise connected, we fill in the 2-simplex (triangle). This construction yields a Vietoris-Rips complex Rips(X,ε), providing a combinatorial approximation to the underlying topological space.

The visualization demonstrates three distinct geometric structures, each preserved by progressively weaker equivalence relations:

• Riemannian structure (Metric View): The complete geometric data, including the metric tensor g_ij. Preserved by isometries φ: (M,g) → (M',g') where φ*g' = g. Invariants include geodesic distances, curvature tensors, and volume forms.
• Smooth structure (Smooth View): The differentiable manifold structure, forgetting the metric but retaining C^∞ compatibility. Preserved by diffeomorphisms. The "Exotic Structure" toggle illustrates Milnor's discovery of exotic 7-spheres—manifolds homeomorphic but not diffeomorphic to S⁷.
• Topological structure (Topological View): The coarsest structure, preserving only continuity. Characterized by homeomorphism invariants: the Betti numbers β_k = rank(H_k) measuring k-dimensional holes. Here, β₀ counts connected components and β₁ counts 1-dimensional cycles.

This hierarchical decomposition—from rigid Riemannian geometry through smooth manifolds to flexible topology—exemplifies the power of categorical thinking in mathematics. In applications to biological data, where noise and measurement uncertainty dominate, topological invariants provide stable signatures of global structure. Rather than asking "what is the precise conformation?" we ask "what is the persistent homology?"—capturing essential connectivity patterns that survive perturbations.