Homology Theory Simplicial Complex Builder

Build mode

Click empty space in the canvas to place a vertex (0-simplex). Rotate the view with a click-drag.

Betti numbers

Vertices

Edges

Faces

Examples

How it works

The complex you build is a set of 0-, 1- and 2-simplices (vertices, edges, filled triangles). Betti numbers are computed live from mod-2 boundary matrices: ∂1 maps each edge to its two endpoints, ∂2 maps each face to its three edges. Both are reduced with Gaussian elimination over GF(2) to get their ranks.

b0 = V − rank(∂1)      components
b1 = (E−rank∂1) − rank(∂2)   loops
b2 = F − rank(∂2)      cavities

Vertex mode — click empty space to add a point; drag to orbit the camera.
Edge mode — click two vertices in turn to connect them.
Face mode — click three vertices; missing edges among them are added automatically, then the triangle is filled.
Delete mode — click a vertex, edge or face to remove it (removing a vertex also removes everything attached to it).

Try it: a bare triangle (3 vertices, 3 edges, no face) has b1 = 1 — an unpatched hole. Fill the triangle and b1 drops to 0. A hollow octahedron (the "Sphere" preset) has b2 = 1 — a sealed cavity with nothing filling it in.

Vertex Selected Edge Filled face

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Homology Theory: Simplicial Complex Builder