Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.

This paper describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the socalled Minkowski tensors. Minkowski tensors are generalizations of the wellknown scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The paper further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic formalism more readily accessible for future application in the physical sciences.

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

In particulate soft matter systems the average number of contacts Z of a particle is an important predictor of the mechanical properties of the system. Using x-ray tomography, we analyze packings of frictional, oblate ellipsoids of various aspect ratios α, prepared at different global volume fractions ϕ g . We find that Z is a monotonically increasing function of ϕ g for all α. We demonstrate that this functional dependence can be explained by a local analysis where each particle is described by its local volume fraction ϕ l computed from a Voronoi tessellation. Z can be expressed as an integral over all values of ϕ l :The local contact number function Z l ðϕ l ; α; XÞ describes the relevant physics in term of locally defined variables only, including possible higher order terms X. The conditional probability Pðϕ l jϕ g Þ to find a specific value of ϕ l given a global packing fraction ϕ g is found to be independent of α and X. Our results demonstrate that for frictional particles a local approach is not only a theoretical requirement but also feasible. The average number of contacts Z that a particle forms with its neighbors is the basic control parameter in the theory of particulate systems known as the jamming paradigm [1,2], where Z is a function of the difference between the global volume fraction ϕ g and some critical value ϕ J . For soft, frictionless spheres (a practical example would be an emulsion) this is indeed a good description [3] because additional contacts are formed by the globally isotropic compression of the particles which also increases ϕ g . However, in frictional granular media such as sand, salt, or sugar the control of ϕ g is not achieved by compression but by changing the geometric structure of the sample; if we want to fill more grains into a storage container we do not compress them with a piston, but we tap the container a couple of times on the counter top.But if Z and ϕ g are not simultaneously controlled by a globally defined parameter such as pressure, the idea of a function Zðϕ g Þ runs into an epistemological problem: contacts are formed at the scale of individual particles and their neighbors. At this scale the global ϕ g is not only undefined, it would even be impossible for a particle scale demon to compute ϕ g by averaging over the volume of the neighboring particles. The spatial correlations between Voronoi volumes [4-6] would require it to gather information from a significantly larger volume than the direct neighbors.To date, only two theoretical approaches have studied Z from a local perspective: Song et al. [7] used a mean-field ansatz to derive a functional dependence between Z and the Voronoi volume of a sphere. This ansatz has recently been expanded to arbitrary shapes composed of the unions and intersections of frictionless spheres [8,9]. Second, Clusel et al. [10,11] developed the granocentric model which predicts the probability distribution of contacts in jammed, polydisperse emulsions. The applicability of the granocentric model to frictional discs has been sho...