Three-dimensional Random Voronoi Tessellations: From Cubic Crystal Lattices to Poisson Point Processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces.

💡 Research Summary

The paper investigates how three canonical cubic crystal lattices—simple cubic (SC), body‑centered cubic (BCC), and face‑centered cubic (FCC)—evolve into a completely random Poisson–Voronoi tessellation when their lattice points are displaced by spatially homogeneous Gaussian noise. The noise intensity is expressed by a dimensionless parameter α = σ/a₀, where σ is the standard deviation of the displacement and a₀ the lattice constant. α = 0 corresponds to the perfect crystal, while α → ∞ reproduces a homogeneous Poisson point process.

Using large‑scale Monte‑Carlo simulations (≈10⁶ cells per α value, with many independent realizations), the authors compute for each tessellation the distribution of the number of faces f, cell surface area A, volume V, total edge length P, and the isoperimetric quotient Q = 36πV²/A³ (a measure of sphericity). They also fit the empirical probability density functions (pdfs) of all these quantities with two‑parameter gamma distributions.

Key findings:

1. Topological stability – The BCC lattice, whose Voronoi cell is a truncated octahedron (14 faces), remains topologically invariant for any finite α; its face count does not change until the noise becomes comparable to the lattice spacing. In contrast, SC (cube, 6 faces) and FCC (rhombic dodecahedron, 12 faces) are topologically unstable: even infinitesimal noise (α ≈ 0.05) immediately alters the face count, leading to a rapid increase of ⟨f⟩.

2. Metric behavior – For weak noise (α ≲ 0.5) the mean cell area grows quadratically with α for BCC and FCC, i.e., ⟨A⟩ ∝ α², while SC exhibits a minimum of ⟨A⟩ at an optimal noise level α ≈ 0.3. When α exceeds about 0.5 the three perturbed tessellations become statistically indistinguishable: ⟨f⟩ ≈ 15.5, ⟨A⟩ ≈ 5.3, ⟨V⟩ ≈ 1 (in units of the original lattice cell). For α > 2 the statistics converge to the known Poisson‑Voronoi limits (⟨f⟩ ≈ 15.54, ⟨Q⟩ ≈ 0.66, etc.).

3. Gamma‑distribution modeling – The pdfs of f, A, V, P, and Q are all accurately described by two‑parameter gamma distributions across the whole α range. This confirms earlier 2‑parameter models and provides a compact statistical description of the whole family of tessellations.

4. Anomalous area–volume scaling – When all cells are pooled, the relationship between mean area and mean volume follows ⟨A⟩ ∝ ⟨V⟩^{β} with β ≈ 1.67 (≈5/3), markedly larger than the naïve geometric expectation β = 2/3. This “anomalous scaling” originates from large shape fluctuations: cells with many faces are more spherical (higher Q) and obey a scaling closer to β = 1.5, while cells with few faces deviate more strongly. By conditioning on a fixed face number, the exponent drops toward the classical value, showing that the anomaly is largely driven by the mixture of different topologies.

5. Physical implications – BCC’s topological robustness makes it a natural model for compact, mechanically stable structures; FCC’s area minimum at finite noise suggests a link to surface‑energy minimization; SC’s extreme sensitivity illustrates how small disorder can destroy simple cubic order. The universal gamma‑fit and the identified scaling laws could be exploited in materials design (e.g., optimizing pore geometry), in data compression schemes that rely on Voronoi quantization, and in the statistical description of porous or granular media.

In summary, the work provides a systematic, quantitative bridge between ordered crystal Voronoi tessellations and the fully random Poisson‑Voronoi limit, highlighting both topological and metric transitions, the adequacy of gamma‑distribution modeling, and the presence of non‑trivial area–volume scaling that depends on cell topology. These insights enrich the theoretical toolbox for anyone employing Voronoi tessellations in physics, materials science, or applied mathematics.