On the density of shapes in three-dimensional affine subdivision

The affine subdivision of a simplex $\Delta$ is a certain collection of $(n+1)!$ smaller $n$-simplices whose union is $\Delta$. Barycentric subdivision is a well know example of affine subdivision(see ). Richard Schwartz(2003) proved that the infinite process of iterated barycentric subdivision on a tetrahedron produces a dense set of shapes of smaller tetrahedra. We prove that the infinite iteration of several kinds of affine subdivision on a tetrahedron produce dense sets of shapes of smaller tetrahedra, respectively.