Pentagram Rigidity for Centrally Symmetric Octagons

Abstract

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.

The algebraic dynamics of the pentagram map

“…Many special classes of twisted n-gons have more idiosyncratic dynamics, including closed polygons [34], Poncelet polygons [19], and axis-aligned polygons [13]. Most recently, Schwartz has established a remarkable pentagram rigidity conjecture for the 3-diagonal map on centrally symmetric octagons [38].…”