The road to a geometric nonperiodic monotile
Abstract
Summary for Oberwolfach Reports:
The first nonperiodic tile set was discovered by Berger in 1966, in his resolution of Wang’s Domino Problem. I will survey the history of discoveries of nonperiodic sets of few tiles following this. In the 1970s, Penrose discovered his remarkable aperiodic set of just two tiles [3], naturally leading to the Monotile Problem(s) of finding a single nonperiodic tile. There are, in fact, many possible variants one may ask: should the tile be ‘perfect’ in the sense of local isomorphism, do we allow non-geometric matching rules, which isometries of the tile are allowed? etc. With examples such as the Schmitt–Conway–Danzer tile — a convex polyhedron in 3-space that tiles without translation symmetries but allows screw symmetries — one even has to be careful in what is meant by ’nonperiodic’. I will explain the subtleties involved here. The first answer to one variant came in the early 2010s, with the (LI-perfect) Taylor–Socolar monotile [6], represented either as a simple hexagon but with next-nearest neighbour matching rules, or purely geometrically as a tile that is not a topological ball. The hexagonal grid and, in particular, the arrowed half hex tiling, has been used several times as the scaffolding for interesting small nonperiodic tile sets [1,7], such as the Penrose $1+\epsilon+\epsilon^2$ tilings. This year, the hat monotile was discovered [4], receiving an unprecedented and welcome amount of public attention. Although it is some sense commensurate with the hexagonal lattice, its global structure is completely different from the above previous hexagonal examples. The hat settles the Monotile Problem for a purely geometric tile that is a is a topological ball and only tiles nonperiodically (but is not translationally LI-perfect), and shortly after the spectre [5], which settles the same problem but without requiring reflections of the tile. Within just a few months of discovery, lots is already known about the global dynamical and topological properties of these tilings [2], uncovered by an already advanced toolkit developed within the field of Aperiodic Order.
REFERENCES
[1] M. Baake, F. Gähler and U.G. Grimm, Hexagonal inflation tilings and planar monotiles, Symmetry 4 (2012), no.4, 581–602.
[2] M. Baake, F. Gähler and L. Sadun, Dynamics and topology of the Hat family of tilings, arXiv:2305.05639 (2023).
[3] R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Math. Intelligencer 2 (1979/80), no.1, 32–37.
[4] An aperiodic monotile, D. Smith, J.S. Myers, Craig S. Kaplan and C. Goodman-Strauss, arXiv:2303.10798 (2023).
[5] A chiral aperiodic monotile, D. Smith, J.S. Myers, Craig S. Kaplan and C. Goodman-Strauss, arXiv:2305.17743 (2023).
[6] J.E.S. Socolar and J.M. Taylor, An aperiodic hexagonal tile, J. Combin. Theory Ser. A 118 (2011), no.8, 2207–2231.
[7] J.J. Walton and M.F. Whittaker, An aperiodic tile with edge-to-edge orientational matching rules, J. Inst. Math. Jussieu 22 (2023), no.4, 1727–1755.
