View source for Barycentric subdivision

''of a geometric complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152901.png" />''

A complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152902.png" /> obtained by replacing the simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152903.png" /> by smaller ones by means of the following procedure. Each one-dimensional simplex (segment) is halved. On the assumption that all simplices of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152904.png" /> are already subdivided, the subdivision of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152905.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152906.png" /> is defined by means of cones over the simplices of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152907.png" /> with a common vertex that is the barycentre of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152908.png" />, i.e. the point with [[Barycentric coordinates|barycentric coordinates]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b0152909.png" />. The vertices of the resulting complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b01529010.png" /> are in a one-to-one correspondence with the simplices of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b01529011.png" />, while the simplices of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b01529012.png" /> are in such a correspondence with inclusion-ordered finite tuples of simplices from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015290/b01529013.png" />. The formal definition of a barycentric subdivision for the case of an abstract complex is analogous.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Lamotke,   "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR></table>