A random simplicial complex (Linial-Meshulam model) is built by scattering points randomly and connecting them when they fall within a threshold radius — vertices become edges, then triangles (2-simplices), then tetrahedra. The resulting shape encodes topological data: its Betti numbers β₀ (connected components), β₁ (loops), β₂ (voids) change dramatically as the radius grows through a phase transition. At the critical radius r_c ≈ √(log n / πn), the complex becomes connected and higher homology appears and vanishes in rapid succession. This is the mathematical foundation of persistent homology and topological data analysis — understanding the shape of data clouds.