Spatial Structure Factors of Metallic-Mean Vogel Arrays: Empirical High-Resolution Diffraction and Robustness Properties

This empirical computational study presents a systematic high-resolution Two-Dimensional Fast Fourier Transform (2D FFT) spatial momentum evaluation of planar point sets generated by the first five metallic means (golden, silver, bronze, copper, and nickel ratios) in the Vogel parameterization. Arrays of up to N=1597 points are examined under ideal packing, radial hollowing, extreme sparsity (N=34), and random site removal (~35%) evaluated on a 1000x1000 Cartesian grid. The resulting momentum-space intensity maps exhibit a universal central peak of intensity $N^{2}$ together with metallic-mean-dependent families of concentric diffraction rings and depleted annular regions. The experiments demonstrate that the ring topology remains recognizable under substantial random deletion and under complete removal of the geometric interior. This confirms that the dominant spectral features are controlled by the global angular irrationality rather than by local density. The results provide a reproducible numerical atlas for evaluating distinct diffraction channels, with applications for photonic, phononic, or plasmonic metamaterial design.