The pentagram map, Poncelet polygons, and commuting difference operators

Research output: Contribution to journalArticlepeer-review

Abstract

The pentagram map takes a planar polygon to a polygon whose vertices are the intersection points of consecutive shortest diagonals of. This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if is a Poncelet polygon, then the image of under the pentagram map is projectively equivalent to. In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.

Original languageEnglish (US)
Pages (from-to)1084-1124
Number of pages41
JournalCompositio Mathematica
Volume158
Issue number5
DOIs
StatePublished - May 1 2022

Keywords

  • difference operator
  • pentagram map
  • Poncelet polygon

ASJC Scopus subject areas

  • Algebra and Number Theory

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