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

TL;DR

This paper presents a new, simple proof of directional scaling symmetry in square and triangular lattices using Gaussian and Eisenstein integers, revealing infinitely many scale symmetries including those related to the golden ratio.

Contribution

It introduces a straightforward proof method for scale symmetry in lattices and shows the existence of infinitely many such symmetries, expanding understanding of lattice structures.

Findings

Existence of infinitely many scale symmetries in square lattices.

Explicit directions and scale factors related to these symmetries.

Inclusion of the golden ratio in the set of scale symmetries.

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.