  The recently discovered Hat tiling admits a 4-dimensional family of shape
deformations, including the 1-parameter family already known to yield alternate
monotiles. The continuous hulls resulting from these tilings are all
topologically conjugate dynamical systems, and hence have the same dynamics and
topology. We construct and analyze a self-similar element of this family called
the CAP tiling, and we use it to derive properties of the entire family. The
CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and
comes from a natural cut-and-project scheme with 2-dimensional Euclidean
internal space. All other members of the Hat family, in particular the original
version constructed from 30-60-90 right triangles, are obtained via small
modifications of the projection from this cut-and-project scheme.
