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.

The Frobenius problem for homomorphic embeddings of languages into the integers

“…A paperfolding morphism θ with θ(a) = a... is called perfect if the four walks generated by the fixed point x = θ ∞ (a), and its three rotations over π/2, π and 3π/2 visit every integer point in the plane exactly twice (except the origin, which is visited 4 times). In [6] it is-not explicitly-proved that for any odd integer N that is the sum of two squares there exists a perfect symmetric paperfolding morphism of length N . To make the proof explicit, one uses that according to the paragraph at the end of Section 7 in [6] there exists a symmetric planefilling and self-avoiding string for each such N , and then one observes that the construction of such a string in the proof of [6,Theorem 4] always satisfies the perfectness criterion given in [6,Theorem 5].…”

Multidimensional paperfolding systems

“…A comprehensive and readable account was published in the Mathematical Intelligencer (Dekking et al, 1982a,b,c). Newer references are the seminal book by Allouche & Shallit (2003) and a recent paper by Dekking (2012).…”

Classification of Generalized Paper Folding Sequences

“…Recently, paper folding sequences have been investigated extensively by many researchers [1][2][3][4][5][6][7][8]. Davis and Knuth [4] introduced a paper folding sequence and they used 0 for a crease that makes the paper upward and 1 for a crease that makes the paper downward.…”