### The Pentagram Map: A Discrete Integrable System

By Quang Nhat Le

November 20, 2013

#### Abstract

I'll define the pentagram map, a simple and projectively natural map on
the space of polygons. it can be extended to act on the space P_n of
twisted polygons. The beauty of this map is that it acts like translations
along tori. What can be simpler than that? Indeed, P_n can be foliated by
submanifolds, which can be given Euclidean structures such that the
pentagram map preserves the foliation and is simply a translation on the
leaves. This fact is obtained by constructing a Poisson structure on P_n
and enough smooth functions on P_n which are invariant under the
pentagram map.