TITLE: Deeper meanings of barycentric subdivision
QUESTION [15 upvotes]: I just want to ask if there is any deeper motivation or clear geometric "sense" behind the barycentric subdivision. Some friend asked me about this a few months ago, looking back the section at Hatcher, I still feel quite confused. I remember one friend told me combinatorically one can do this from posets back to posets, but this does not give me any way to "understand" it properly. In some books (Bredon, for example), the author use excision property as one of the axioms, I'm wondering "where they came from, why they make any sense?".

REPLY [3 votes]: The following paper includes a very detailed and elegant description of how to construct edgewise subdivisions that subdivide a $d$-simplex into $k^d\cdot d$-simplices all of the same volume and shape characteristics, for every integer $k\geq 1$
Edgewise Subdivision of a Simplex
H. Edelsbrunner and D. R. Grayson
DISCRETE AND COMPUTATIONAL GEOMETRY
Volume 24, Number 4, 707-719