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Difference between revisions of "Barycentric subdivision"

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''of a geometric complex $K$''
 
''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|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.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Lamotke,  "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Lamotke,  "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR>
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</table>

Latest revision as of 11:10, 16 April 2023

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.

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=53824
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article