Star Neighborhoods in Double Barycentric Subdivision
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration illustrates skeleta of stars and the dual of the double barycentric subdivision of a 3-simplex.
[more]
Contributed by: Aleksandr Berdnikov (August 2018)
Open content licensed under CC BY-NC-SA
Details
A significant number of topological spaces are (or can be modeled by) simplicial complexes—spaces that are "glued" out of simplexes, attaching them face to face. Given a subcomplex of a complex , one would often like to consider a neighborhood of in that approximates fairly well. It is not always possible to build such a out of simplexes of ; For example, if , the only simplicial neighborhood of is the whole segment , which fails to capture the disconnectedness of .
However, there are such neighborhoods that are simplicial in a double barycentric subdivision of (i.e. each simplex is subdivided barycentrically twice). Given any subcomplex of , such a neighborhood is given by in , where is defined as the union of all open simplexes with a vertex in .
The Demonstration illustrates this concept with the 3-simplex as an example of . Each controller toggles the display of stars (in the double subdivision) of centers of corresponding elements. For example, selecting "vertices" and "edges" gives a neighborhood of a 1-skeleton of (i.e. of the collection of its 1-simplexes).
Barycentric subdivision is defined inductively. Subdivision of a point is the point itself. To define a subdivision of a simplex , take the subdivision of its boundary and take as the collection of simplexes with base the simplex in and vertex the barycenter of .
Snapshots
Permanent Citation