Characterised by an anomalous suppression of large-scale density fluctuations, hyperuniform patterns are found in a vast array of systems, ranging from the molecular arrangements of amorphous ices to a certain type of jammed granular packings. In particular, designed hyperuniform materials are gaining more and more attention for their unique properties contrasting to their crystal counterparts. Thereby, it is of great scientific and technological importance to understand how hyperuniformity can arise and why it appears in a broad range of systems with different length and time scales.

In this study, we put our focus on hyperuniform systems generated by a robust mathematical model, called Lloyd’s algorithm. The algorithm manifests many-body interactions based on the geometric relations between near-neighbours. Regardless of initial conditions, we find that the resulting hyperuniform configurations consist majorly of points with six neighbours and a small fraction of them with five- or seven-fold symmetry, classified as topological defects. We pursue to uncover how these topological entities come to exist, and we delve into their geometric and topological characteristics. 

Further, we identify locally favoured motifs composed of defects as well as the dynamics of defect formation mechanism dictated by the Lloyd process. The universal defective hexagonal landscapes are found to be characterised by scale-free correlations between hexagonal domains. Motivated by the resemblance between the ordering transition through the Lloyd’s algorithm and rigidity or jamming transitions of soft matter systems, we explore surface minimisation principles and maximal disk packings of the cellular geometries of Lloyd-generated hyperuniform systems. We finish our work by providing a physics insight through eigenfrequency analysis of wireframe geometries of the obtained hyperuniform structures.

Our findings may cast light on our understanding of universality in systems exhibiting collective behaviours, self-organisation and critical phenomena, all of which are frequently observed not only in living systems, but also in non-living systems, including granular materials, colloidal systems and foams.