  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.
