The cosmic spiderweb: equivalence of cosmic, architectural and origami tessellations

For over 20 years, the term 'cosmic web' has guided our understanding of the large-scale arrangement of matter in the cosmos, accurately evoking the concept of a network of galaxies linked by filaments. But the physical correspondence between the cosmic web and structural engineering or textile 'spiderwebs' is even deeper than previously known, and also extends to origami tessellations. Here, we explain that in a good structure-formation approximation known as the adhesion model, threads of the cosmic web form a spiderweb, i.e. can be strung up to be entirely in tension. The correspondence is exact if nodes sampling voids are included, and if structure is excluded within collapsed regions (walls, filaments and haloes), where dark-matter multistreaming and baryonic physics affect the structure. We also suggest how concepts arising from this link might be used to test cosmological models: for example, to test for large-scale anisotropy and rotational flows in the cosmos.

Bayesian analysis of deformed tessellation models

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

Statistics of cross sections of Voronoi tessellations

In this paper, we investigate relationships between the volumes of cells of three-dimensional Voronoi tessellations and the lengths and areas of sections obtained by intersecting the tessellation with a randomly oriented plane. Here, in order to obtain analytical results, Voronoi cells are approximated to spheres. First, the probability density function for the lengths of the radii of the sections is derived and it is shown that it is related to the Meijer G function; its properties are discussed and comparisons are made with the numerical results. Next, the probability density function for the areas of cross sections is computed and compared with the results of numerical simulations.