  We investigate the structural and spectral properties of deterministic
aperiodic arrays designed from the statistically isotropic pinwheel tiling. By
studying the scaling of the cumulative integral of its structure factor in
combination with higher-order structural correlation analysis we conclude that
pinwheel arrays belong to the weakly hyperuniformity class. Moreover, by
solving the multiple scattering problem for electric point dipoles using the
rigorous Green's matrix theory, we demonstrate a clear transition from
diffusive transport to localization behavior. This is shown by studying the
Thouless number as a function of the scattering strength and the spectral
statistics of the scattering resonances. Surprisingly, despite the absence of
sharp diffraction peaks, clear spectral gaps are discovered in the density of
states of pinwheel arrays that manifest a distinctive long-range order.
Furthermore, the level spacing statistics at large optical density exhibits a
sharp transition from level repulsion to the Poisson behavior, consistently
with the onset of the wave localization regime. Our findings reveal the
importance of hyperuniform aperiodic structures with statistically isotropic
k-space for the engineering of enhanced light-matter interaction and
localization properties.
