  Disordered hyperuniform packings are unusual amorphous states of two-phase
materials that are endowed with exotic physical properties. Such hyperuniform
systems are characterized by an anomalous suppression of volume-fraction
fluctuations at infinitely long-wavelengths, compared to ordinary disordered
materials. While there has been growing interest in such singular states of
amorphous matter, a major obstacle has been an inability to produce large
hyperuniform samples due to practical limitations of conventional methods. To
overcome these limitations, we introduce a general theoretical methodology to
construct perfectly hyperuniform packings in $d$-dimensional Euclidean space.
Specifically, beginning with an initial general tessellation of space that
meets a "bounded-cell" condition, hard particles are placed inside each cell
such that the packing fraction within the cell becomes identical to the global
packing fraction. We prove that the constructed packings are perfectly
hyperuniform in the infinite-sample-size limit and their hyperuniformity is
independent of particle shapes, positions, and numbers per cell. We employ two
distinct types of initial tessellations: Voronoi as well as sphere
tessellations. Beginning with Voronoi tessellations, we show that our algorithm
can remarkably convert extremely large nonhyperuniform packings into
hyperuniform ones in $\mathbb{R}^2$ and $\mathbb{R}^3$. Implementing our
theoretical methodology on sphere tessellations, we establish the
hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres
structures that possess optimal effective transport and elastic properties. In
addition, the tunability of our methodology offers promise for discovery of
novel disordered hyperuniform two-phase materials.
