A Textbook Case of Pentagram Rigidity

Abstract

In this paper I will explain a rigidity conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. I will also establish a simple case of the conjecture, the one involving the $3$-diagonal map on a convex $8$-gon with $4$-fold rotational symmetry. This case involves a textbook analysis of a pencil of elliptic curves.