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Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift $\Omega_0$ made of all valid tilings using the set $\mathcal{T}_0$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $X_0$ of $\Omega_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112 that are equivalent to a set of 19 self-similar and aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel-Rao tilings, as we believe that $\Omega_0\setminus X_0$ is a null set for any shift-invariant probability measure on $\Omega_0$. The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles. The 2 other steps involve the addition of decorations to deal with fault lines and changing the base of the $\mathbb{Z}^2$-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles., Comment: v1: 38p., 18 Fig. arXiv admin note: text overlap with arXiv:1802.03265 (the preliminaries). v2: 42p. Substitutions and Wang tile sets now appears directly in the proofs. Image of letters are now sorted by length and then lexicographically. v3+v4: 48p., corrections after review. Jupyter notebook available at https://nbviewer.jupyter.org/url/www.slabbe.org/Publications/arxiv_1808_07768_v4.ipynb

We study tilings of the plane by a single prototile with respect to the lattice and to the crystallographic group p2. We are interested in the connection between the neighbors of a tile in the tiling and its topology. We show that lattice and p2-tiles always have at least six neighbors. We characterize self-affine tiles that are homeomorphic to a disk in a rather easy way by the set and number of neighbors of the central tile in the tiling. This extends the work of Bandt and Wang devoted to lattice self-affine disk-like tiles of the plane.