; ; ; ; ; ;

(A15) – (B15) – (C15) – (D15) – (E15) – (F15) – (G15) – (A15);

Specifically, suppose we are given an $n$-gon $P$ in the projectivization ${\mathbb P}{\mathbb {V}}$ of a $3$-dimensional vector space ${\mathbb {V}}$. Lift the vertices of the polygon to vectors $V_i \in {\mathbb {V}}$, $i = 1, \dots, n$. Define an operator $G_P \colon {\mathbb {V}}\to {\mathbb {V}}$ by the formula $$\begin{aligned}
\label{eq:glick}
G_p(V) := n V - \sum_{i=1}^n \frac{V_{i-1}\wedge V \wedge V_{i+1}}{V_{i-1}\wedge V_i \wedge V_{i+1}}\,V_i,\end{aligned}$$ where all indices are understood modulo $n$. Note that this operator does not change under a rescaling of $V_i$’s and hence depends only on the polygon $P$. What Glick proved is that the limit point of successive images of $P$ under the pentagram map is one of the eigenvectors of $G_P$ (equivalently, a fixed point of the associated projective mapping ${\mathbb P}{\mathbb {V}}\to {\mathbb P}{\mathbb {V}}$).

We believe that the significance of Glick’s operator actually goes beyond the limit point. In particular, as was observed by Glick himself, the operator $G_P$ has a natural geometric meaning for both pentagons and hexagons. Namely, by Clebsch’s theorem every pentagon is projectively equivalent to its pentagram map 