This chapter continues the discussion of general topics related to oriented matroids. More precisely, the geometric, algebraic and topological topics treated here are related to realizable oriented matroids, i.e., real matrices, point configurations and hyperplane arrangements. Nevertheless, the general point of view of oriented matroids seems to be relevant in many cases, and some understanding of these topics is important for a balanced view of oriented matroids within mathematics.
Real hyperplane arrangements
Arrangements of hyperplanes in Rd arise as fundamental objects in various mathematical theories: from inequality systems in linear programming, from facets of convex polytopes, from reflection groups in Lie theory, from geometric search in computational geometry, from questions in singularity theory, to name a few. Real hyperplane arrangements have also been studied for a long time by discrete geometers, particularly with respect to their combinatorial structure, that is, how they partition space.
In Section 1.2 it was explained how a hyperplane arrangement A gives rise to an oriented matroid M(A), and it follows from the discussion there (see also Section 1.4) that hyperplane arrangements correspond bijectively to realizable oriented matroids(up to reorientations). Here we will take a second look at hyperplane arrangements. Some basic definitions will be reviewed, the translation from geometric language to oriented matroid terminology will be explained in a few cases, and a theorem about the number of simplicial regions will be shown, which illustrates that the combinatorial behavior of oriented matroids can differ in the realizable and unrealizable cases.
Four variants of linear arrangements are presented in the following definition. In Chapter 5 we will encounter a generalization to topologically deformed “pseudohyperplanes” and “pseudospheres”, and arrangements of such.