Preamble

scipy.spatial.Delaunay provides a way of calculating the Delaunay tesselation in n dimensional space. I am interesting in calculating the Euler characteristic of the resulting graph, which entails computing the number of vertices, edges, and faces.

Vertices

The number of vertices is easily accessed with scipy.spatial.Delaunay().vertices.shape[0].

Edges & Faces

If I recall correctly, the tesselation is composed of simplices, forming a simplicial complex. If my simplicial complex was composed of a single simplex then the number of edges would be the number of columns of scipy.spatial.Delaunay().vertices choose 2. And likewise there are ways of counting the number of n-faces (a generalized notion that I think includes edges) when there is only a single simplex. But since I have a simplicial complex, there are likely to be shared edges and faces (and n-faces)...

Question

How can I calculate the Euler characteristic from an instance of scipy.spatial.Delaunay?