Artificial aperiodic structures have recently been the subject of extensive and intensive research, resulting in layered quasiregular heterostructures, as well as photonic and phononic metamaterials with possible applications such as optical and acoustic bandpassfilters or photonic waveguides.  It is an experimental challenge to produce a physical realization of the Prouhet-Thue-Morse structure in one, two and three dimensions and subject it to a diffraction experiment.

We are interested in fundamental questions about determinism, order and "disorder" and their quantification.  Specifically, we study multidimensional generalizations of the standard substitution sequences.  Here we present and discuss some two-dimensional instances of the Prouhet-Thue-Morse and paperfolding systems.  We compute their rectangle complexities; these are at most polynomial implying zero entropy.  We suggest to concisely characterize the complexity by the exponent of its leading term.  We point out that the perfectly deterministic Champernowne sequence has entropy ln2 exactly like fair Bernoulli.  This example clearly shows that entropy, regardless of its definition, does not distinguish between deterministic and random systems.  There still remain many unanswered questions.