Simplicial Homology of the Alpha Complex
This Demonstration generates a random set of points and a corresponding simplicial complex, which is a topological space connecting those points. Computing the homology of a complex is a technique from algebraic topology to find groups that describe how the complex is connected.[more]
If performance is slow (due to a large number of points), uncheck "compute homology" and change the other parameters.[less]
For a given set of vertices , the -complex is a simplicial subcomplex of the Delaunay triangulation parameterized by . For any simplex , we have that if the pairwise distances between vertices in that simplex are all less than the given . That is, for all . This Demonstration generates a random set of planar points; you can vary to see how the complex changes. The simplicial homology groups and their corresponding Betti numbers are topological invariants that characterize the -dimensional "holes" in the complex. For example, gives the number of connected components, is the number of "tunnels," and gives the number of closed-off spaces with volume (however, in this Demonstration, the complex is planar, so remains trivial).
For more information on homology (and algebraic topology in general), see the following.
 A. Hatcher, Algebraic Topology, New York: Cambridge University Press, 2002.
 J. R. Munkres, Elements of Algebraic Topology, Menlo Park, CA: Addison-Wesley, 1984.