Difference between revisions of "Barycentric subdivision"
From Encyclopedia of Mathematics
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| − | ''of a geometric complex | + | {{TEX|done}} |
| + | ''of a geometric complex $K$'' | ||
| − | A complex | + | A complex $K_1$ obtained by replacing the simplices of $K$ by smaller ones by means of the following procedure. Each one-dimensional simplex (segment) is halved. On the assumption that all simplices of dimension $\leq n-1$ are already subdivided, the subdivision of any $n$-dimensional simplex $\sigma$ is defined by means of cones over the simplices of the boundary of $\sigma$ with a common vertex that is the barycentre of the simplex $\sigma$, i.e. the point with [[Barycentric coordinates|barycentric coordinates]] $1/(n+1)$. The vertices of the resulting complex $K_1$ are in a one-to-one correspondence with the simplices of the complex $K$, while the simplices of the complex $K_1$ are in such a correspondence with inclusion-ordered finite tuples of simplices from $K$. The formal definition of a barycentric subdivision for the case of an abstract complex is analogous. |
Revision as of 16:20, 15 April 2014
of a geometric complex $K$
A complex $K_1$ obtained by replacing the simplices of $K$ by smaller ones by means of the following procedure. Each one-dimensional simplex (segment) is halved. On the assumption that all simplices of dimension $\leq n-1$ are already subdivided, the subdivision of any $n$-dimensional simplex $\sigma$ is defined by means of cones over the simplices of the boundary of $\sigma$ with a common vertex that is the barycentre of the simplex $\sigma$, i.e. the point with barycentric coordinates $1/(n+1)$. The vertices of the resulting complex $K_1$ are in a one-to-one correspondence with the simplices of the complex $K$, while the simplices of the complex $K_1$ are in such a correspondence with inclusion-ordered finite tuples of simplices from $K$. The formal definition of a barycentric subdivision for the case of an abstract complex is analogous.
Comments
References
| [a1] | K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968) |
How to Cite This Entry:
Barycentric subdivision. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Barycentric_subdivision&oldid=14992
Barycentric subdivision. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Barycentric_subdivision&oldid=14992
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article