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

Abstract: 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.