Building the Butterfly Fractal: The Eightfold Way

TL;DR

This paper models the Hofstader butterfly fractal using an octonary tree structure with unimodular matrices, linking it to topological quantum numbers and generalizing Pythagorean triplet trees.

Contribution

It introduces a novel octonary tree framework for the butterfly fractal, connecting it to Diophantine equations and topological quantum properties.

Findings

The butterfly fractal is described by an octonary tree structure.

A Diophantine mapping encodes quantum numbers of the fractal.

The model generalizes the Pythagorean triplet tree.

Abstract

The hierarchical structure of the butterfly fractal -- the Hofstader butterfly, is found to be described by an octonary tree. In this framework of building the butterfly graph, every iteration generates sextuplets of butterflies, each with a tail that is made up of an infinity of butterflies. Identifying {\it butterfly with a tale} as the building block, the tree is constructed with eight generators represented by unimodular matrices with integer coefficients. This Diophantine description provides one to one mapping with the butterfly fractal, encoding the magnetic flux interval and the topological quantum numbers of every butterfly. The butterfly tree is a generalization of the ternary tree describing the set of primitive Pythagorean triplets.