# Paperfolding morphisms, planefilling curves, and fractal tiles

• Published in 2010
In the collections
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.

### BibTeX entry

@article{Paperfoldingmorphismsplanefillingcurvesandfractaltiles,
title = {Paperfolding morphisms, planefilling curves, and fractal tiles},
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.},
url = {http://arxiv.org/abs/1011.5788v2 http://arxiv.org/pdf/1011.5788v2},
year = 2010,
author = {Michel Dekking},
comment = {},
urldate = {2020-09-21},
archivePrefix = {arXiv},
eprint = {1011.5788},
primaryClass = {math.CO},
collections = {geometry,things-to-make-and-do}
}