Symmetries of Icosahedral group and classification of G-circuits of length six

Muhammad Nadeem Bari

doi:10.17485/ijst/v13i27.728

Abstract

Objectives: To determine the exact number of equivalence classes of G-circuits of length q≥2:Methods/Statistical Analysis: To classify G-orbits of Q(√m)/Q containing G-circuits of length 6. Findings: The equivalence classes of G-circuits of length 6 is ten in number and determine the exact number of G-orbits and structure of G-orbits corresponding to each of ten equivalence classes of Gcircuits. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of these G-circuits with conditions on t. Applications/Improvements: We employ Symmetries of Icosahedral group to explore cyclically equivalence classes of Gcircuits and similar G-circuits of length 6 corresponding to each of these ten equivalence classes. This study helps us in classifying reduced numbers lying in PSL(2, Z)-orbits. These results are verified by some suitable example. Keywords: Rotational symmetries of icosahedral group; partition function; equivalence classes of G-circuits; reduced quadratic irrational numbers

Highlights

I{(cfoαγnF=n∈toa=aiQr+ncα√∗dQ(n2=√m∗ (:,n√a)+ahnc:√,e)brγnea>n∈=ddQ1Q∈a∗a2∗r(nec−√ddNn(n,√−c)an,1n∈)id
The idea to classify G-circuits of G-orbits on quadratic field by modular group, which is given in this paper, is new and original
We describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of these G-circuits with conditions on t