Uniform line fillings

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.

arXiv comment10 pages, 12 figures; v3: restructured paper, more references, more graphs

References in corpus (8)

• Multiple scattering of classical waves: from microscopy to mesoscopy and diffusion
• Quantum-Secure Authentication with a Classical Key
• Eigenvalue distributions of large Euclidean random matrices for waves in random media
• Two-dimensional disorder for broadband, omnidirectional and polarization-insensitive absorption
• Direct laser writing of three dimensional network structures as templates for disordered photonic materials
• Asymmetric Cryptography with Physical Unclonable Keys
• Failure and Uses of Jaynes' Principle of Transformation Groups
• Lognormal Distribution of Cosmic Voids in Simulations and Mocks

Cited by in corpus (4)

• A gravity-independent powder-based additive manufacturing process tailored for space applications
• Deterministic and controllable photonic scattering media via direct laser writing
• Mirror Symmetry in three-dimensional Multiple-Scattering Media
• Experimental tests of Bertrand's question and the Duhem-Quine problem