Geometry of Matroids and Hyperplane Arrangements

There is a trinity relationship between hyperplane arrangements, matroids and convex polytopes. We expand it as resolving the complexity issue expected by Mnevs universality theorem and conduct combinatorializing so the theory over fields becomes realization of our combinatorial theory. A main theorem is that for n less than or equal to 9 a specific and general enough kind of matroid tilings in the hypersimplex Delta(3,n) extend to matroid subdivisions of Delta(3,n) with the bound n=9 sharp. As a straightforward application to realizable cases, we solve an open problem in algebraic geometry proposed in 2008.