Aperiodic Tiling - Confusion Regarding Terminology

Confusion Regarding Terminology

The terms non-periodic, quasiperiodic and aperiodic have been used in a wide variety of ways in a wide variety of fields, leading to considerable confusion. Moreover, the word "tiling" itself is quite problematic.

In the context of 'Aperiodic tiling', a non-periodic tiling is simply one with no period, as discussed above, and aperiodicity is a property of tiles: a set of tiles is aperiodic if and only if it admits only non-periodic tilings. There is no mathematical concept of aperiodic tiling per se. Quasiperiodic tilings, generally, mean those obtained by the cut-and-project method; however William Thurston's influential lecture notes used the term to mean repetitive tilings. The Penrose tiles themselves are a source of much of the confusion, for the tilings they admit are quasiperiodic, in both senses, and non-periodic, and they themselves are aperiodic.

Moreover the terms aperiodic, non-periodic and quasiperiodic are widely used in other fields, such as dynamical systems, with altogether different meanings; and there is much literature on tilings in which, inappropriately, the distinction is not made. It is important to note however, that the core results of the field simply are not meaningful without this careful delineation.

The word "tiling" is problematic as well, despite its straightforward definition. There is no single Penrose tiling, for example: the Penrose rhombs admit infinitely many tilings (which cannot be distinguished locally) and even established figures in the field informally refer to "aperiodic tiling", knowing full well that this is not technically defined. A common solution is to try to use the terms carefully in technical writing, but recognize the widespread use of the informal terms.