Build the foundations of TDA: understand point clouds as data representations and simplicial complexes as the building blocks for topological analysis.
Topological Data Analysis begins with point clouds — collections of points sampled from some underlying space. The fundamental question is: can we recover the "shape" of the original space just from these discrete samples?
The answer involves building simplicial complexes — combinatorial structures that approximate the continuous space. Just as a mesh of triangles can approximate a curved surface, simplicial complexes provide a discrete representation of topological features like connectivity, loops, and voids.
Explore different point cloud shapes. The circle has one loop (H₁ = 1), the figure-8 has two loops (H₁ = 2), and the three clusters are disconnected (H₀ = 3). In custom mode, click to draw your own point cloud.
Points sampled from a circle
Expected Topology
H₀ = 1, H₁ = 1
A k-simplex is the simplest possible k-dimensional shape:
The key property is that every simplex includes all of its faces. A triangle includes its three edges and three vertices. A tetrahedron includes its four triangular faces, six edges, and four vertices. This nesting property is what makes simplicial complexes well-behaved for computing homology.
Switch between simplex dimensions to see their structure. Enable "Boundary" to see how each simplex's boundary is computed as an alternating sum of its faces. The boundary of a boundary is always zero (∂² = 0) — this is the foundation of homology.
A filled triangle with three vertices and three edges.
A simplicial complex is a collection of simplices that fit together nicely: if a simplex is in the complex, all its faces must also be in the complex, and any two simplices can only intersect along a common face.
A triangle with all its edges and vertices included. Every face of every simplex is present.
A triangle missing one edge, or two triangles that share only a vertex from the interior of an edge.
The boundary operator ∂ maps each k-simplex to an alternating sum of its (k-1)-dimensional faces:
The crucial property is ∂² = 0: the boundary of a boundary is empty. This is because each (k-2)-face appears twice with opposite signs and cancels out. This nilpotency is what allows us to define homology groups.