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Geometric objects (like polytopes or arrangements of hyperplanes) carry two layers of information. First of all they are described by the coordinates of the parts involved. On the other hand there is also a combinatorial description that cares only about the relative position of the elements. This lecture is about the subtle interplay of coordinates and combinatorics. We introduce the "theory of oriented matroids" as the primary framework for the study. This theory allows us to get deep structural insight in topics like "polytope theory", "linear optimization", "automatic geometric theorem proving", "quasicrystals" and many more.