  In this paper we investigate numerically an instance of the problem of
G-closure for two-dimensional periodic metamaterials. Specifically, we consider
composites with isotropic homogenized elasticity tensor, obtained as a mixture
of two isotropic materials, focusing on the case of a single material with
voids. This problem is important, in particular, in the context of designing
small-scale structures for metamaterials in the context of additive
fabrication, as this type of metamaterials makes it possible to obtain a range
of material properties using a single base material. We demonstrate that two
closely related simple parametric families based on the structure proposed by
O. Sigmund attain good coverage of the space of isotropic properties satisfying
Hashin-Shtrikman bounds. In particular, for positive Poisson ratio, we
demonstrate that Hashin-Shtrikman bound can be approximated arbitrarily well,
within limits imposed by numerical approximation: a strong evidence that these
bounds are achievable in this case. For negative Poisson ratios, we numerically
obtain a bound which we hypothesize to be close to optimal, at least for
metamaterials with rotational symmetries of a regular triangle tiling.
