Fig. 3 | Nature Communications

Fig. 3

From: Universal hidden order in amorphous cellular geometries

Fig. 3

Evidence supporting the universal effectively hyperuniform final states in 3D. The structure factor S(k) is shown before (a) and after (b) application of Lloyd iterations. Different symbols represent different initial stochastic models, including anisotropic and hyperfluctuating systems, and including point processes (PP) and sphere packings (SphP), see Methods section. For the final configurations, S(k) collapses within statistical errors and indicates effective hyperuniformity, in the inset the point cloud (red) shows the unbinned scattering intensity for all allowed wavenumbers for the largest sample (Binomial PP); for log-log plots, see Supplementary Figs. 8 and 9. Frequency distributions f(e) of cell energy densities e (see Methods section) are shown before (c) and after (d) application of Lloyd iterations; for a semi-log plot, see Supplementary Fig. 6. The distributions of energies after Lloyd iterations have mean 〈e〉 ≈ 1.008 × eBCC and FWHM 0.01 × eBCC. In the final universal configurations, there is a significant proportion (≈ 3%) of cells with a lower energy density than the crystalline optimal BCC structure. The error bars represent the standard error of the mean

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