The Gale Transform and the Pentagram Map

Abstract

We prove that polygons under the higher dimensional pentagram map are projectively equivalent through the use of the Gale transform. Specifically, we demonstrate this for (k+3)-gons and (2k+2)-gons in a k-dimensional projective space. We begin by showing equivalence between historical definitions of the Gale transform, and then we propose a novel geometric definition of the Gale transform by extending Castelnuovo's approach for (2k+2) points. This new definition provides a nice geometric way to take the Gale transform of (k+s+2) points, as opposed to the standard linear algebra approach that is commonly used. Utilizing this new definition, we establish that the Gale transform and projective duality map commute. This is crucial in proving the projective equivalence of polygons under the higher-dimensional pentagram map.