For a different labeling the resulting formulas differ by a shift in indices. The choice of labeling, and more generally, the specific form of the above formulas will be of no importance to us. We will only use the following corollary. Consider a $1$-parametric group of densely defined transformations ${\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2) \to {\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ given by $$\begin{aligned}
\label{eq: scaling}
{R}_z\colon(x_i,y_i)\mapsto (x_i z, y_iz^{-1})\end{aligned}$$ These transformations are known as *scaling symmetries*.

The scaling symmetry $R_z \colon {\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2) \to {\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ on projective equivalence classes of twisted polygons commutes with the pentagram map for any $z \neq 0$.

The above formulas for the pentagram map in $x,y$ coordinates remain unchanged if all $x$ variables are multiplied by $z$ and all $y$ variables by are multiplied by $z^{-1}$.

This proposition was a key tool in the proof of integrability of the pentagram map. Namely, consider a (twisted or closed) polygon $P$ defined up to a projective transformation, and let $P_z$ be its image under the scaling symmetry. Then, since the pentagram map commutes with scaling and preserves the monodromy, it follows that the monodromy $M_z$ of $P_z$ (which does not 