  The decades-long search for a shape that tiles the plane only aperiodically
under translations and rotations recently ended with the discovery of the
`spectre' aperiodic monotile. In this setting we study the dimer model, in
which dimers are placed along tile edges such that each vertex meets precisely
one dimer. The complexity of the tiling combines with the dimer constraint to
allow an exact solution to the model. The partition function is
$\mathcal{Z}=2^{N_{\textrm{Mystic}}+1}$ where $N_{\textrm{Mystic}}$ is the
number of `Mystic' tiles. We exactly solve the quantum dimer (Rokhsar Kivelson)
model in the same setting by identifying an eigenbasis at all interaction
strengths $V/t$. We find that test monomers, once created, can be infinitely
separated at zero energy cost for all $V/t$, constituting a deconfined phase in
a 2+1D bipartite quantum dimer model.
