Effective Methods in Computational Synthetic Geometry

Abstract

We discuss algorithmic steps when dealing with realizability problems in discrete geometry, especially that of finding realizations for a given oriented matroid. After a brief introduction to known methods, we discuss a dynamic inductive realization method, which has proven successful when other methods did not succeed. A useful theorem in this context in the rank 3 case asserts that a one-element extension of a uniform rank 3 oriented matroid depends essentially just on the mutations involving that element. There are problems in computational synthetic geometry of course, where intuition must help. In this context we mention the application of the software Cinderella to automated deduction in computational synthetic geometry, when studying face lattices of polytopes.KeywordsEquivalence ClassSimplicial ComplexHalf PeriodAntipodal PointAutomate DeductionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.