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

2014/10/20 by Cao Zexian, Zexian, Cao
Materials Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Nanocluster Synthesis and Applications #Quasicrystal Structures and Properties #Random Matrices and Applications #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1410.5139

arxiv created 2014/10/20 · openalex publication_date 2014/10/20 · arxiv updated 2014/10/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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 ratioλ=√(2)-1 or the platinum ratioμ=2-√(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 ratioφ=(√(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.

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