The Pentagram Map

Work supported by an Alfred P. Sloan Doctoral Dissertation Fellowship. We consider the pentagram map on the space of plane convex pentagons obtained by drawing a pentagon’s diagonals, recovering unpublished results of Conway and proving new ones. We generalize this to a ‘‘pentagram map’’ on convex polygons of more than five sides, showing that iterated images of any initial polygon converge exponentially fast to a point. We conjecture that the asymptotic behavior of this convergence is the same as under a projective transformation. Finally, we show a connection between the pentagram map and a certain flow defined on parametrized curves.