  The hyperuniformity concept provides a unified means to classify all perfect
crystals, perfect quasicrystals, and exotic amorphous states of matter
according to their capacity to suppress large-scale density fluctuations. While
the classification of hyperuniform point configurations has received
considerable attention, much less is known about the classification of
hyperuniform heterogeneous two-phase media, which include composites, porous
media, foams, cellular solids, colloidal suspensions and polymer blends. The
purpose of this article is to begin such a program for certain two-dimensional
models of hyperuniform two-phase media by ascertaining their local
volume-fraction variances $\sigma^2_{_V}(R)$ and the associated hyperuniformity
order metrics $\overline{B}_V$. This is a highly challenging task because the
geometries and topologies of the phases are generally much richer and more
complex than point-configuration arrangements and one must ascertain a broadly
applicable length scale to make key quantities dimensionless. Therefore, we
focus on a certain class of two-dimensional periodic cellular networks,
periodic and disordered/irregular packings, some of which maximize their
effective transport and elastic properties. Among the cellular networks
considered, the honeycomb networks have a minimal value of $\overline{B}_V$
across all volume fractions. On the other hand, among all packings considered,
the triangular-lattice packings have the smallest values of $\overline{B}_V$
for the possible range of volume fractions. Among all structures studied here,
the triangular-lattice packing has the minimal order metric for almost all
volume fractions. Our study provides a theoretical foundation for the
establishment of hyperuniformity order metrics for general two-phase media and
a basis to discover new hyperuniform two-phase systems by inverse design
procedures.
