Bi-Constructible pattern of weak and flavour mixing: implications for electroweak coupling constants

TL;DR

This paper proposes a geometric framework based on constructible polygons to model weak and flavour mixing, leading to predictions of mixing angles and gauge couplings, including the fine-structure constant, through elegant mathematical relations.

Contribution

It introduces a novel geometric approach using constructible polygons to explain weak and flavour mixing, unifying these phenomena with fundamental constants.

Findings

Reproduces quark and lepton mixing angles accurately.

Derives Weak--Quark--Lepton Complementarity relations.

Predicts the fine-structure constant using geometric ratios.

Abstract

We present semi-empirical evidence suggesting that weak and flavour mixing, at the most fundamental level, can be described in terms of the Euclidean geometry of regular polygons constructible with compass and straightedge, specifically, the pentagon and the heptadecagon, associated with Fermat primes -- a pattern referred to as Bi-Constructible. Our approach accurately reproduces quark and lepton mixing angles and offers indications that the Weinberg angle also fits naturally within this geometric framework. Concise Weak--Quark--Lepton Complementarity relations are derived. These findings suggest a semi-empirical unification pattern of weak and flavour mixing. The Standard Model gauge couplings g and g' admit elegant expressions involving the golden ratio, yielding a neat prediction for the fine-structure constant entirely in these terms.