  We study hyperuniform properties in various two-dimensional periodic and
quasiperiodic point patterns. Using the histogram of the two-point distances,
we develop an efficient method to calculate the hyperuniformity order metric,
which quantifies the regularity of the hyperuniform point patterns. The results
are compared with those calculated with the conventional running average
method. To discuss how the lattice symmetry affects the order metric, we treat
the trellis and Shastry-Sutherland lattices with the same point density as
examples of periodic lattices, and Stampfli hexagonal and dodecagonal
quasiperiodic tilings with the same point density as examples of quasiperiodic
tilings. It is found that the order metric for the Shastry-Sutherland lattice
(Stampfli dodecagonal tilings) is smaller than the other in the periodic
(quasiperiodic) tiling, meaning that the order metric is deeply related to the
lattice symmetry. Namely, the point pattern with higher symmetry is
characterized by the smaller order metric when their point densities are
identical. Order metrics for several other quasiperiodic tilings are also
calculated.
