The combinatorics of the universe

The mirror symmetry phenomenon of physical string theory predicts that certain varieties come in pairs $(X,Y)$ that should determine `the same physics'.

Essentially all known examples of mirror pairs come from convex polytopes. More precisely, they arise as hypersurfaces in projective toric varieties, and mirror duality comes from duality of polytopes.

I will sketch this construction due to Batyrev. Along the way, I will mention words like reflexive polytope, Hilbert basis of a rational cone, and unimodular triangulation; while I try to avoid scary things like moment map, large complex structure limit, and alike. If time permits, I will outline a combinatorial model for the more recent Strominger-Yau-Zaslow interpretation of mirror symmetry.

I would like to assume that people have seen a convex polytope before, at least from afar.