Simplicial Homology of the Alpha Complex

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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.
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Contributed by: Richard Hennigan (March 2013)
Open content licensed under CC BY-NC-SA
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Details
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.
Permanent Citation
"Simplicial Homology of the Alpha Complex"
http://demonstrations.wolfram.com/SimplicialHomologyOfTheAlphaComplex/
Wolfram Demonstrations Project
Published: March 27 2013