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).

[2] J. R. Munkres,

*Elements of Algebraic Topology*, Menlo Park, CA: Addison-Wesley, 1984.