Multidimensional paperfolding systems

Abstract

Algorithms for constructing aperiodic structures produce templates for the nanofabrication of arrays for applications in photonics, phononics and plasmonics. Here a general multidimensional recursion rule is presented for the regular paperfolding structure by straightforward generalization of the one-dimensional rule. As an illustrative example the two-dimensional version of the paperfolding structure is explicitly constructed, its symbolic complexity referred to rectangles computed and its Fourier transform shown. The paperfolding structures readily yield novel 'paperfolding' tilings. Explicit formulas are put forward to count the number of folds in any dimension. Finally, possible generalizations of the dragon curve are discussed.

On triangular paperfolding patterns

“…The substitution sequence was generalized to higher-dimensional case by Ben-Abraham, Quandt and Shapiraa [4]. The paper [11] by Gähler and Nilsson provides substitution rules for the corresponding paperfolding structures and studies various properties of the corresponding patterns.…”

Multidimensional "Paperfolding" Structures - Three and Four Dimensions

“…The alphabet may be reduced by ignoring the orientation to A 2 = {+ + +, − − −} to produce a black-and--white version which may be preferable for physical applications (cf. [13]). …”

“…In [13] we derived a general formula for the number of folds |S(n)| in any generation n of the structure S in arbitrary dimension d:…”

“…[13]) is a 16-cell (sometimes also called hexadecachoron or 4-orthoplex) or its topological deformations. These tilings will be dealt with elsewhere.…”

“…Recently we put forward a general recursive scheme to produce paperfolding structures in any dimension [13].…”