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.