Before revision:  We also prove the existence of a topological conjugacy \eta:\Omega_6 \Omega_5 which shears Wang tilings by the action of the matrix and an embedding \pi:\Omega_5 \Omega_4 that is unfortunately not onto. The Wang shift \Omega _{12 subshift X_0 of the Jeandel-Rao tilings \Omega_0 .
After revision:  We also prove the existence of a topological conjugacy \eta:\Omega_6 \Omega_5 which shears Wang tilings by the action of the matrix and an embedding \pi:\Omega_5 i.e., unit squares with colored edges, that admit valid (contiguous edges of adjacent tiles have the same color) tilings of the plane none of them being invariant under a nontrivial translation. In this contribution, we study the Wang shift \Omega _{12 subshift X_0 of the Jeandel-Rao tilings \Omega_0 .