Homology Theory: Simplicial Complex Builder

Build a simplicial complex by hand — vertices, edges and filled triangles — and watch its Betti numbers b0, b1 and b2 recompute in real time from mod-2 boundary-matrix ranks. b0 counts connected components, b1 counts independent 1-dimensional loops (holes not patched by a face), and b2 counts sealed cavities enclosed by faces with nothing filling their interior. Three built-in examples (a circle, a torus-like grid and a hollow octahedron) demonstrate each invariant directly.