A Textbook Case of Pentagram Rigidity

TL;DR

This paper explores a rigidity conjecture linking pentagram maps and Poncelet polygons, demonstrating a specific case involving elliptic curves and symmetric polygons.

Contribution

It establishes a simple case of the conjecture using elliptic curve analysis and symmetry considerations.

Findings

Proves a specific case of the rigidity conjecture.

Connects pentagram maps with Poncelet polygons through elliptic curves.

Provides a geometric and algebraic framework for the conjecture.

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.