  Tiling models can reveal unexpected ways in which local constraints give rise
to exotic long-range spatial structure. The recently discovered Hat monotile
(and its mirror image) has been shown to be aperiodic~[Smith et al.,
arXiv:2303.10798 (2023)]; it can tile the plane with no holes or overlaps, but
cannot do so periodically. We show that the structure enforced by the local
space-filling constraints is quasiperiodic with hexagonal (C6) rotational
symmetry. Although this symmetry is compatible with periodicity, the
incommensurate ratio characterizing the quasiperiodicity stays locked to the
golden mean as the tile parameters are continuously varied. We analyze a
modification of the metatiles introduced by Smith et al. that yields a set of
``Key tiles'' that can be constructed as projections of a subset of
six-dimensional hypercubic lattice points onto the two-dimensional tiling
plane. We analytically compute the diffraction pattern of a set of unit masses
placed at the tiling vertices, establishing the quasiperiodic nature of the
tiling. We point out several unusual features of the family of Key tilings and
associated Hat tilings, including the tile rearrangements associated with the
phason degree of freedom associated with incommensurate density waves, which
exhibit novel features that may influence the elastic properties of a material
in which atoms or larger particles spontaneously exhibit the symmetries of the
Hat tiling.
