The pentagram map and Y-patterns

TL;DR

This paper explores the pentagram map, a geometric transformation on polygons, using cluster algebra techniques to derive explicit formulas for its iterations, advancing understanding of its algebraic structure.

Contribution

It introduces a novel approach by applying cluster algebra machinery to explicitly compute the iterates of the pentagram map.

Findings

Explicit formulas for the iterates of the pentagram map

Connection between the pentagram map and cluster algebras

Enhanced understanding of the algebraic structure of the pentagram map

Abstract

The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its "shortest" diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.