Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding

TL;DR

This paper explores a novel connection between paper folding fractals, Mahler functions, and continued fractions, revealing new mathematical structures and relationships.

Contribution

It introduces a Mahler function derived from existing functions that links continued fractions with fractal shapes like the dragon curve.

Findings

Constructed a Mahler function with fractal and continued fraction properties

Established a link between irregular and regular continued fractions

Presented variations on the folding-fractal-continued fraction theme

Abstract

Repeatedly folding a strip of paper in half and unfolding it in straight angles produces a fractal: the dragon curve. Shallit, van der Poorten and others showed that the sequence of right and left turns relates to a continued fraction that is also a simple infinite series. We construct a Mahler function from two functions of Dilcher and Stolarsky with similar properties. It produces a predictable irregular continued fraction that admits a regular continued fraction and a shape resembling the dragon curve. Furthermore, we discuss numerous variations on this theme.