Pentagram Spirals

Abstract

This paper introduces a projective geometry construction that relates to the pentagram map much in the way that logarithmic spirals relate to circles. The paper proves some elementary geometric and analytic properties of the construction, and raises some deeper questions related to integrable systems.

2000 AMS Subject Classification:

• 51A20

Keywords:

• pentagram
• integrability
• polygon iteration

Acknowledgments

I would like to thank Valentin Ovsienko and Sergei Tabachnikov, my usual collaborators on the pentagram map, for many discussions about the pentagram map and related areas of mathematics. This work was carried out during my sabbatical at Oxford in 2012–2013. I would especially like to thank All Souls College, Oxford, for providing a wonderful research environment. My sabbatical was funded from many sources. I would like to thank the National Science Foundation, All Souls College, the Oxford Maths Institute, the Simons Foundation, the Leverhulme Trust, the Chancellor's Professorship, and Brown University for their support during this time period.

This research was supported by N.S.F. grant DMS-0604426.

Notes

¹You can download this program at http://www.math.brown.edu/res/Java/SPIRAL.tar .