Paperfolding morphisms, planefilling curves, and fractal tiles

TL;DR

This paper explores the geometric and fractal properties of plane-filling curves generated by paperfolding sequences, providing classification criteria and linking their automatic structure to self-similarity and fractal boundaries.

Contribution

It generalizes previous results on paperfolding curves, offering new geometric criteria for their classification and demonstrating how their automatic sequences lead to fractal tiles with calculable Hausdorff dimensions.

Findings

Criteria for classifying paperfolding curves as self-avoiding or planefilling.

Demonstration of self-similarity in the curves due to automatic structure.

Explicit formulas for the Hausdorff dimension of certain fractal tile boundaries.

Abstract

An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and planefilling properties of these curves, giving simple geometric criteria for a complete classification. Finally, we show how the automatic structure of the sequences leads to self-similarity of the curves, which turns the planefilling curves in a scaling limit into fractal tiles. For some of these tiles we give a particularly simple formula for the Hausdorff dimension of their boundary.