# The transition from order to disorder in Voronoi Diagrams with applications

## TL;DR

This paper investigates the transition from order to disorder in Voronoi diagrams by perturbing seed points, introducing a new scaling relationship, and applying the model to physical and cosmological structures.

## Contribution

It introduces a novel relationship modeling the scaling of the Kiang function with a geometrical parameter in Voronoi tessellations.

## Key findings

- Successful modeling of local structures in ammonia using the Voronoi approach.
- Application of the model to cosmic void volumes.
- Demonstration of the transition from order to disorder in tessellations.

## Abstract

The transition from ordered to disordered structures in Voronoi tessellation is obtained by perturbing the seeds that were originally identified with two types of lattice in 2D and one type in 3D. The area in 2D and the volume in 3D are modeled with the Kiang function. A new relationship that models the scaling of the Kiang function with a geometrical parameter is introduced. A first application models the local structure of sub- and supercritical ammonia as function of the temperature and a second application models the volumes of cosmic voids.