Growth of Hyperbolic Cells

We consider a hyperbolic polygon in the unit disk $\{z:\, |z|<1\}$ with all its vertices on the unit circle $\{z:\, |z|=1\}$ and a growth process of such polygons when each $n$-gon generates an $n(n-1)$-gon by inverting itself across all of its sides. In this paper, we prove some general monotonicity results of inversion for convex hyperbolic $n$-gons and solve an extremal problem that, among all convex hyperbolic $4$-gons containing the origin, the inverted side length of the longest side of the given hyperbolic $4$-gon is minimal for the regular hyperbolic $4$-gon.