  In a previous work [Scientific Reports 4, 6193(2014)] we proved the existence
of scale symmetry in square and triangular (thus honeycomb) lattices by
investigating the functiony=\arcsin(\sin(2\pinx)), where the parameter is
either the silver ratio\lambda=\sqrt{2}-1 or the platinum ratio\mu=2-\sqrt{3}.
Here we give a new proof, simple and straightforward, by using the concept of
Gaussian and Eisenstein integers. More importantly, it can be proven that there
are infinitely many possibilities for scale symmetry in the square lattice, and
one of them is even related to the golden ratio\varphi=(\sqrt{5}-1)/2 . The
directions and the corresponding scale factors are explicitly specified. These
results might inspire the search of scale symmetries in other even
higher-dimensional structures, and be helpful for attacking physical problems
modeled on the square and triangular lattices.
