TY - JOUR

T1 - Uniform line fillings

AU - Marakis, Evangelos

AU - Velsink, Matthias Christiaan

AU - Corbijn Van Willenswaard, Lars J.

AU - Uppu, Ravitej

AU - Pinkse, Pepijn W.H.

N1 - 14 pages, 9 figures.

PY - 2019/4/25

Y1 - 2019/4/25

N2 - 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's solution by the principle of maximum ignorance. We analyze 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 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.

AB - 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's solution by the principle of maximum ignorance. We analyze 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 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.

KW - cond-mat.soft

KW - physics.comp-ph

UR - http://www.scopus.com/inward/record.url?scp=85064867242&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1809.01490

U2 - 10.1103/PhysRevE.99.043309

DO - 10.1103/PhysRevE.99.043309

M3 - Article

C2 - 31108721

VL - 99

SP - 043309

JO - Physical review E: covering statistical, nonlinear, biological, and soft matter physics

JF - Physical review E: covering statistical, nonlinear, biological, and soft matter physics

SN - 2470-0045

IS - 4

M1 - 043309

ER -