  Shelling orders are a ubiquitous tool used to understand invariants of cell
complexes. Significant effort has been made to develop techniques to decide
when a given complex is shellable. However, empirical evidence shows that some
shelling orders are better than others. In this article, we explore this
phenomenon in the case of matroid independence complexes. Based on a new
relation between shellability of dual matroid polytopes and independence
complexes, we outline a systematic way to investigate and compare different
shellings orders. We explain how our new tools recast and deepen various
classical results to the language of geometry, and suggest new heuristics for
addressing two old conjectures due to Simon and Stanley. Furthermore, we
present freely available software which can be used to experiment with these
new geometric ideas.
