Haros graphs: an exotic representation of real numbers

TL;DR

This paper introduces Haros graphs as a novel graph-theoretical representation of real numbers in the unit interval, revealing a hierarchical classification based on topological and entropic properties related to Farey trees.

Contribution

It presents a new graph-based framework for classifying real numbers using topological and entropic analysis, connecting number theory with graph theory and fractal geometry.

Findings

Entropy peaks at the reciprocal of the Golden number

Hierarchy of local maxima and minima relates to noble and rational numbers

Haros graphs provide an exotic, fractal-based classification of real numbers

Abstract

This paper introduces Haros graphs, a construction which provides a graph-theoretical representation of real numbers in the unit interval reached via paths in the Farey binary tree. We show how the topological structure of Haros graphs yields a natural classification of the reals numbers into a hierarchy of families. To unveil such classification, we introduce an entropic functional on these graphs and show that it can be expressed, thanks to its fractal nature, in terms of a generalised de Rham curve. We show that this entropy reaches a global maximum at the reciprocal of the Golden number and otherwise displays a rich hierarchy of local maxima and minima that relate to specific families of irrationals (noble numbers) and rationals, overall providing an exotic classification and representation of the reals numbers according to entropic principles. We close the paper with a number of conjectures and outline a research programme on Haros graphs.