The density of shapes in three-dimensional barycentric subdivision
Let \(S\) be an \(n\)-dimensional simplex. A subdivision of \(S\) into \((n+1)!\) partial \(n\)-simplices is called barycentric if it is defined (by induction on \(n\)) in the following way: For \(n= 0\), \(S\) is a point and its own barycentric subdivision. And if \(S'\) is one of the simplices of such a subdivision of \(S\), then \(S'= \text{conv}(v\cup F')\), with \(v\) as barycenter of \(S\), and \(F'\) as one of the simplices in the barycentric subdivision of one of the top-dimensional faces \(F\) of \(S\). The author considers the infinite process of iterated barycentric subdivisions of a tetrahedron \(T\) in 3-space, i.e., after getting the first sequence of partial tetrahedra \(T_1',\dots, T_{(n+1)!}'\), any \(T_j'\) from there is again barycentrically subdivided into simplices \(T_{j1}',\dots, T_{j(n+1)!}'\), and so forth. He shows that this infinite process yields a dense set of shapes of smaller tetrahedra, where ``shape'' means a simplex modulo similarities.
scientific article; zbMATH DE number 2078216