1st International School on Hypermaterials, June 21-25, 2021, online

A theoretical guide to quasiperiodic tilings
Nobuhisa Fujita
Institute of Multidisciplinary Research for Advanced Materials,
Tohoku University, Sendai 980-8577, Japane-mail: nobuhisa.fujita.a4@tohoku.ac.jp
The lecture starts with a general introduction to quasiperiodic tilings, exemplified by the celebrated
Penrose tilings in two dimensions [1]. The vertices of a quasiperiodic tiling is supported on a Z-module[2], whose rank, i.e., the number of generating bases, is higher than the number of space dimensions.
This is in contrast to that the rank of a periodic lattice is given by the number of space dimensions.
Four known techniques to generate quasiperiodic tilings, namely, the cut-and-project, section, grid,and inflation methods are summarized and compared [3].
Dimensionality and point group symmetry are the crudest criteria for classifying quasiperiodictilings. Down the classification hierarchy to Bravais classes, we restrict ourselves to octagonal,
decagonal and dodecagonal Bravais classes in two dimensions and primitive, face-centered and bodycentered icosahedral Bravais classes in three dimensions. The minimal choice of a Z-module is uniqueto each of the Bravais classes. Tilings in the same Bravais class would look dissimilar to each other ifthey have different sets of prototiles (i.e., shapes of the tiles). Even so, one of them might be mappedfrom another just by re-drawing the tile boundaries according to locally determined rules. In such acase, the former tiling is said to be locally derivable from the latter. If the two are mutually locallyderivable, it is said that they belong to the same mutual local derivability (MLD) class [4,2]. An atomic
decoration of a quasiperiodic tiling can be obtained as an atomic decoration of another tiling in thesame MLD class, so that the distinction between tilings in the same MLD class is physically unessential.
MLD relationships between different versions of Penrose tilings (P1, P2, P3 and Robinson-triangletilings) [1] are relatively well documented. Another MLD class in the decagonal family is represented
by the Tübingen tiling [2]. MLD classes can be differentiated from each other according to how thewindow vertices are embedded w.r.t. the Z-module in the perpendicular space [5]. The finest of theclassification hierarchy is given by local isomorphism (LI). Two instances of tiling with the same setof prototiles are said to be locally isomorphic to each other if an arbitrarily large patch in one of themcan be found in the other, and vice versa [6]. Locally isomorphic tilings are indistinguishable inarbitrarily large scale as long as the scale is finite. Tilings in the same LI class are related through aparallel shift of the window along the perpendicular space. The MLD relationships above are in factdefined as equivalence relationships between LI classes rather than between tilings [4].
The rest of the lecture will exploit a systematic inflation technique [7] to generate well-known tilings
with non-crystallographic symmetries in two and three dimensions. These will cover AmmannBeenker (octagonal), Penrose / Tübingen (decagonal), Shield / Stampfli (dodecagonal) and theAmmann-Kramer-Neri (icosahedral) tilings, which all have polygonal or polyhedral windows.
References:[1] B. Grünbaum & G. C. Shephard: Tilings & Patterns, 2nd ed. (Dover, 2016).
[2] M. Baake: A guide to mathematical quasicrystals. arXiv:math-ph/9901014.
[3] T. Janssen, G. Chapuis & M. de Boissieu: Aperiodic Crystals, 2nd ed. (Oxford Univ. Pr, 2018).
[4] M. Baake, M. Schlottmann & P. D. Jarvis: J. Phys. A 24 (1991) 4637.
[5] K. Niizeki & N. Fujita: J. Phys. Soc. Jpn. 71 (2002) 71; K. Niizeki: J. Alloys Compd. 342 (2002)
213.
[6] D. Levine & P. J. Steinhardt: Phys. Rev. B 34 (1986) 596; 617.
[7] K. Niizeki, J. Phys. A: Math. Theor. 41 (2008) 175208; Prog. Theor. Phys. 128 (2012) 629.