Combinatorial aspects of the pentagram map

The pentagram map is a projectively natural iteration on polygons in the projective plane. Introduced by R. Schwartz about 20 years ago, it was recently proven to be a completely integrable system. I shall discuss three combinatorial aspects of this topic: monodromy invariants of the map and strange relations between them for inscribed polygons; the relation to 2-frieze patterns, a structure similar to frieze patterns of Coxeter and Conway; and new configuration theorems of projective geometry, somewhat similar to the Pappus and Pascal theorems.