Communications in Mathematical Physics

, Volume 299, Issue 2, pp 409–446

The Pentagram Map: A Discrete Integrable System

  • Valentin Ovsienko
  • Richard Schwartz
  • Serge Tabachnikov

DOI: 10.1007/s00220-010-1075-y

Cite this article as:
Ovsienko, V., Schwartz, R. & Tabachnikov, S. Commun. Math. Phys. (2010) 299: 409. doi:10.1007/s00220-010-1075-y


The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \({\mathbb Z}\) into \({{\mathbb{RP}}^2}\) that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Valentin Ovsienko
    • 1
  • Richard Schwartz
    • 2
  • Serge Tabachnikov
    • 3
  1. 1.CNRS, Institut Camille Jordan, Université Lyon 1Villeurbanne CedexFrance
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA