9827

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.
If performance is slow (due to a large number of points), uncheck "compute homology" and change the other parameters.

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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.
References
[1] A. Hatcher, Algebraic Topology, New York: Cambridge University Press, 2002.
[2] J. R. Munkres, Elements of Algebraic Topology, Menlo Park, CA: Addison-Wesley, 1984.
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