Uniform line fillings

TL;DR

This paper introduces a method for generating random, homogeneous, and rotationally invariant line fillings in arbitrary dimensions, useful for fabricating metamaterials and optical scattering samples.

Contribution

The authors present a novel algorithm for creating uniform line fillings in any dimension, with proven statistical properties and practical applications in fabrication.

Findings

The method produces homogeneous and isotropic line distributions.

The pore sizes follow a lognormal distribution.

The approach is applicable to 3D optical fabrication.

Abstract

Deterministic fabrication of random metamaterials requires filling of a space with randomly oriented and randomly positioned chords with an on-average homogenous density and orientation, which is a nontrivial task. We describe a method to generate fillings with such chords, lines that run from edge to edge of the space, in any dimension. We prove that the method leads to random but on-average homogeneous and rotationally invariant fillings of circles, balls and arbitrary-dimensional hyperballs from which other shapes such as rectangles and cuboids can be cut. We briefly sketch the historic context of Bertrand's paradox and Jaynes' solution by the principle of maximum ignorance. We analyse the statistical properties of the produced fillings, mapping out the density profile and the line-length distribution and comparing them to analytic expressions. We study the characteristic dimensions of the space in between the chords by determining the largest enclosed circles and balls in this pore space, finding a lognormal distribution of the pore sizes. We apply the algorithm to the direct-laser-writing fabrication design of optical multiple-scattering samples as three-dimensional cubes of random but homogeneously positioned and oriented chords.