Growth of Hyperbolic Cells

TL;DR

This paper studies the geometric growth process of hyperbolic polygons in the unit disk, proving monotonicity results and identifying the extremal configuration for minimal inverted side length among convex hyperbolic quadrilaterals.

Contribution

It introduces a novel hyperbolic polygon growth process and establishes key monotonicity and extremal properties for convex hyperbolic polygons.

Findings

Monotonicity results for inversion of convex hyperbolic polygons.

The regular hyperbolic 4-gon minimizes the inverted side length among all convex hyperbolic 4-gons containing the origin.

Abstract

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.