Abstract
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.
Highlights
Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology
An efficient way to dynamically generate a CVT is via Lloyd iterations[39] which evolve a given initial set of points by iteratively replacing each point with the centre of mass of its Voronoi cell[40], see Fig. 1
This corresponds to a gradient descent algorithm[34], a standard energy minimisation procedure that in general converges to a random minimum in the potential energy surface when starting from different initial conditions
Summary
Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. 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. 1234567890():,; Hyperuniformity[1] is a geometric concept to probabilistically characterise the structure of ordered and disordered materials It is defined as the anomalous suppression of density fluctuations on large length scales. Excellent examples include the Kelvin minimal-cell-interface-area problem[27] and Kepler spherepacking problem[28,29]
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