The paperfolding sequence is one of the standard, paradigmatic substitution systems

[1,2].  We have solved the longstanding problem of generalizing the paperfolding sequence to higher dimensions.  For physical applications, such as templates for  fabrication of photonic, phononic and plasmonic nanostructures it is important to construct two- and three-dimensional analogs of substitution sequences.  Recently we have constructed a two-dimensional paperfolding structure by a recursive algorithm which, however, made the structure extremely fractal and hence inconvenient for physical applications [3].  We present a general recursive scheme for constructing genuine regular paperfolding structures in any dimension.  Here we explicitly construct the two-dimensional version, compute its symbolic complexity referred to rectangles and show its Fourier transform.  We also show the first generations of the three- and four-dimensional versions.  Moreover, we derive from the structures novel tilings.  Finally we discuss possible multidimensional generalizations of the dragon curve.

[1] M. Dekking, M. Mendès-France and A.J. van der Poorten, Folds! Math. Intelligencer 4, 130 -138; 173 -181;  190 -195 (1982).

[2] J.-P. Allouche and J. Shallit, Automatic sequences: theory, applications, generalizations, (Cambridge University Press, (2003).

[3] S.I. Ben-Abraham and A. Quandt, Aperiodic structures and notions of order and disorder,  Phil. Mag. 91, 2718 - 2727.