[Paper Review] Glick's conjecture on the point of collapse of axis-aligned polygons under the pentagram maps

Zijian Yao|arXiv (Cornell University)|Oct 28, 2014

Mathematics and Applications10 references3 citations

TL;DR

This paper proves Glick's conjecture that the point of collapse of an axis-aligned polygon under the pentagram map is its center of mass, using projective geometry and cross-ratio invariants. The authors generalize the result to higher and lower-dimensional pentagram maps, introducing the mirror pentagram map to provide a geometric interpretation of the lower-dimensional case, with all results holding over ℝ and ℂ via projective invariance.

ABSTRACT

The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the point of collapse is always the center of mass of the axis-aligned polygon. In this paper, we answer Glick's conjecture positively, and generalize the statement to higher and lower dimensional pentagram maps. For the latter map, we define a new system -- the mirror pentagram map -- and prove a closely related result. In addition, the mirror pentagram map provides a geometric description for the lower dimensional pentagram map, defined algebraically by Gekhtman, Shapiro, Tabachnikov and Vainshtein.

Motivation & Objective

• To prove Glick's conjecture that the point of collapse of an axis-aligned polygon under the pentagram map is its center of mass.
• To generalize the collapsing behavior of the pentagram map to higher and lower dimensions.
• To define and study a new system—the mirror pentagram map—for lower-dimensional pentagram maps.
• To provide a geometric interpretation of the algebraically defined lower-dimensional pentagram map by Gekhtman, Shapiro, Tabachnikov, and Vainshtein.
• To establish projective invariance of the center of mass and the collapse point in the context of axis-aligned polygons.

Proposed method

• Use of projective transformations to align axis-aligned polygons with coordinate axes, enabling computation of the center of mass in ℝ².
• Definition of the center of mass as the average of vertex coordinates after projective normalization, proven independent of transformation choice.
• Application of cross-ratio invariants in ℙ¹ to track vertex evolution under the pentagram map, particularly for the lower-dimensional case.
• Introduction of the mirror pentagram map as a geometric realization of the lower-dimensional pentagram map, defined via cross-ratio conditions.
• Use of recursive application of the lower pentagram map T₁ on sequences of points with cross-ratio -1 to model the iteration process.
• Proof via induction and algebraic computation: showing that after n−1 iterations, all vertices converge to the centroid, matching the center of mass.

Experimental results

Research questions

• RQ1 Does the point of collapse of an axis-aligned polygon under the pentagram map coincide with its center of mass?
• RQ2 Can the pentagram map's collapsing behavior be generalized to higher and lower dimensions?
• RQ3 What geometric structure underlies the lower-dimensional pentagram map defined algebraically by Gekhtman et al.?
• RQ4 How do cross-ratios with value -1 characterize the evolution of vertices in the lower-dimensional pentagram map?
• RQ5 Is the center of mass of an axis-aligned polygon preserved under projective transformations and invariant under the pentagram map's iteration?

Key findings

• The point of collapse of an axis-aligned 2n-gon under the pentagram map is exactly its center of mass, confirming Glick's conjecture.
• After n−1 iterations, all vertices of the pentagram map on an axis-aligned polygon collapse to a single point, which equals the center of mass.
• The center of mass is well-defined and projectively invariant, regardless of the choice of projective transformation aligning edges with the coordinate axes.
• The mirror pentagram map provides a geometric realization of the lower-dimensional pentagram map, defined algebraically by Gekhtman et al.
• The lower pentagram map T₁ acts as a shift on sequences of points with cross-ratios equal to -1, leading to convergence at the centroid after n−1 steps.
• The final collapsed point has coordinates equal to the average of all original vertex coordinates, both in x and y directions.