1. Choose a small complex
The app computes Betti numbers using $\beta_p=\dim C_p-\operatorname{rank}(\partial_p)-\operatorname{rank}(\partial_{p+1})$.
2. Geometry
Black dots are vertices, dark edges are $1$-simplices, and shaded regions are filled $2$-simplices.
3. Boundary matrices
4. Betti numbers
Interpretation: $\beta_0$ counts connected components, $\beta_1$ counts independent loops, and $\beta_2$ counts enclosed voids.
5. Vietoris--Rips graph from points
Change the scale $r$. Edges appear when the distance between two points is at most $r$. For this two-dimensional point set, the displayed graph gives a simple visual approximation of $\beta_0$ and graph loops.
6. Point-cloud geometry
7. Independent-study tasks with answers
Task A. Use the triangle boundary example. Explain why $\beta_1=1$.
Answer: There is one cycle and no filled triangle whose boundary kills it.
Answer: There is one cycle and no filled triangle whose boundary kills it.
Task B. Use the filled triangle example. Explain why $\beta_1=0$.
Answer: The triangular cycle is the boundary of a $2$-simplex, so it is zero in homology.
Answer: The triangular cycle is the boundary of a $2$-simplex, so it is zero in homology.
Task C. Use the two-triangles example. Which loop remains?
Answer: The unfilled triangle remains; the filled triangle does not count as a hole.
Answer: The unfilled triangle remains; the filled triangle does not count as a hole.
Similar practice. Use the square cycle. Predict $\beta_0$ and $\beta_1$, then check.
Answer: The square is connected and has one loop, so $\beta_0=1$ and $\beta_1=1$.
Answer: The square is connected and has one loop, so $\beta_0=1$ and $\beta_1=1$.