Last term, we learned about the Chow ring of a matroid, given by certain generators and relations. Backman, Eur and Simpson found a linear change of variables to new generators, called the simplicial generators, and found a basis for the Chow ring, called the nested basis. We'll review the wonderful compactification of a matroid and the corresponding Chow ring, and then we will explain what the simplicial generators and monomial generators mean, both geometrically and combinatorially. Next week, Calvin will explain how we use these tools to prove that Chow rings of matroids are Poincare duality algebras.