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We define a partition $\mathcal{P}_0$ and a $\mathbb{Z}^2$-rotation($\mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whoseassociated symbolic dynamical system is a minimal proper subshift of theJeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define anotherpartition $\mathcal{P}_\mathcal{U}$ and a $\mathbb{Z}^2$-rotation on$\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimaland aperiodic Wang shift defined by 19 Wang tiles. This proves that$\mathcal{P}_\mathcal{U}$ is a Markov partition for the $\mathbb{Z}^2$-rotationon $\mathbb{T}^2$. We prove in both cases that the toral$\mathbb{Z}^2$-rotation is the maximal equicontinuous factor of the minimalsubshifts and that the set of fiber cardinalities of the factor map is$\{1,2,8\}$. The two minimal subshifts are uniquely ergodic and are isomorphicas measure-preserving dynamical systems to the toral $\mathbb{Z}^2$-rotations.It provides a construction of these Wang shifts as model sets of 4-to-2 cut andproject schemes. A do-it-yourself puzzle is available in the appendix toillustrate the results.

Markov partitions for toral $\mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets