Scaling Laws and Topological Exponents in Voronoi Tessellations of Intermittent Point Distributions

TL;DR

This paper investigates Voronoi tessellations of scale-invariant fractal point distributions, revealing unique scaling laws and a new topological exponent that differ from natural cellular structures.

Contribution

It introduces a universal scaling law for scale-invariant tessellations and estimates a novel topological exponent through numerical analysis.

Findings

Average cell area grows faster than in natural structures.

A universal scaling law relating shape and size is verified.

A new topological exponent is identified and estimated.

Abstract

Voronoi tessellations of scale-invariant fractal sets are characterized by topological and metrical properties that are significantly different from those of natural cellular structures. As an example we analyze Voronoi diagrams of intermittent particle distributions generated by a directed percolation process in (2+1) dimensions. We observe that the average area of a cell increases much faster with the number of its neighbours than in natural cellular structures where Lewis' law predicts a linear behaviour. We propose and numerically verify a universal scaling law that relates shape and size of the cells in scale-invariant tessellations. A novel exponent, related to the topological properties of the tessellation, is introduced and estimated numerically.