  A matroid is a machine capturing linearity of mathematical objects and
producing combinatorial structures. Matroid structure arises everywhere since
linearity is a ubiquitous concept. One natural way to obtain matroids is by
considering hyperplane arrangements, which give rise to convex polytopes called
matroid polytopes. Much research has been conducted on these three areas:
matroids, matroid polytopes, and hyperplane arrangements. However, substantial
gaps in our knowledge remain, and the correspondence diagram between those
areas needs to be more extensive. For instance, currently, there is no matroid
counterpart of a matroid subdivision, and only some matroid subdivisions are
associated with stable hyperplane arrangements. Moreover, we need a deeper
understanding of the face structure of a matroid polytope and how to glue or
subdivide base polytopes; the latter requires overcoming Mnev's universality
theorem. Another interesting question is whether the birational geometry of
hyperplane arrangements can be implemented over matroids. In this paper, we
develop a theory that integrates the three areas into a trinity relationship
and provide solutions to the aforementioned questions while answering as many
as possible.
