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In a previous work [Scientific Reports 4, 6193(2014)] we proved the existenceof scale symmetry in square and triangular (thus honeycomb) lattices byinvestigating the functiony=\arcsin(\sin(2\pinx)), where the parameter iseither 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 ofGaussian and Eisenstein integers. More importantly, it can be proven that thereare infinitely many possibilities for scale symmetry in the square lattice, andone of them is even related to the golden ratio\varphi=(\sqrt{5}-1)/2 . Thedirections and the corresponding scale factors are explicitly specified. Theseresults might inspire the search of scale symmetries in other evenhigher-dimensional structures, and be helpful for attacking physical problemsmodeled on the square and triangular lattices.

Proof of Directional Scaling Symmetry in Square and Triangular Lattices with the Concept of Gaussian and Eisenstein Integers