Nonexpansive directions in the Jeandel-Rao Wang shift

Figure 1.  Two resolutions of singular tilings by unmarked Penrose rhombs. The parts of the tilings where the two tilings are different is shaded and are called Conway worms

Figure 2.  An illustration of an unresolved Conway worm made of two kinds of hexagons together with its two resolutions within a Penrose tiling

Figure 3.  The aperiodic set $ \mathcal{T}_0 $ of 11 Wang tiles discovered by Jeandel and Rao in 2015 [22]

Figure 4.  Jeandel-Rao tiles can be encoded into a set of equivalent geometrical shapes in the sense that every tiling using Jeandel-Rao tiles can be transformed into a unique tiling with the corresponding geometrical shapes and vice versa

Figure 5.  A partial tiling of the plane with an unresolved Conway worm of slope 0

Figure 6.  Tilings of a $ 20\times20 $ square illustrating the Conway worms of slope $ \varphi+3 $, $ -3\varphi+2 $ and $ -\varphi+\frac{5}{2} $. The difference between the left and the right images is shown with a colored background

Figure 7.  Tilings of a $ 30\times30 $ square illustrating the four Conway worms. The difference between both images is shown with a colored background. This reminds of the cartwheel tiling in the context of Penrose tilings [18, Figure 10.5.1 (c)]

Figure 8.  The partition $ \mathcal{P}_0 $ of $ \mathbb{R}^2/\Gamma_0 $

Figure 9.  On the left, we illustrate the lattice $ \Gamma_0 = \langle(\varphi, 0), (1, \varphi+3)\rangle_ \mathbb{Z} $, where $ \varphi = \frac{1+\sqrt{5}}{2} $, with black vertices, a rectangular fundamental domain of the flat torus $ \boldsymbol{T} = \mathbb{R}^2/\Gamma_0 $ with a black contour and a polygonal partition $ \mathcal{P}_0 $ of $ \mathbb{R}^2/\Gamma_0 $ with indices in the set $ \{0, 1, \dots, 10\} $. For every starting point $ {\boldsymbol{p}}\in \mathbb{R}^2 $, the coding of the orbit $ \mathcal{O}_{R_0}(\boldsymbol{p}) $, which is just the shifted lattice $ {\boldsymbol{p}}+ \mathbb{Z}^2 $ (the white dots), under the polygonal partition yields a configuration $ w: \mathbb{Z}^2\to\{0, 1, \dots, 10\} $ which is a symbolic representation of $ {\boldsymbol{p}} $. The configuration $ w $ corresponds to a valid tiling of the plane with Jeandel-Rao's set of 11 Wang tiles. As shown on the right, when the orbit of $ {\boldsymbol{p}} $ hits the boundary of the partition $ \mathcal{P}_0 $, $ {\boldsymbol{p}} $ has more than one symbolic representations

Figure 10.  Illustration of the set $ W( {\boldsymbol{p}}) $ computed experimentally

Figure 11.  $ \Delta_{ \mathcal{P}_0} $-lines in $ \mathcal{P}_0 $

Figure 12.  All four nonexpansive directions shown in Theorem 1.1 are exhibited by the set $ W(\boldsymbol{0}) $

Figure 13.  The $ \mathbb{Z}^2 $ action induces an exchange of the intervals (or rotation of) $ B $ and $ G $. A point $ \boldsymbol{p} $ on the segment $ \overline{PQ} $ from $ (\varphi - 1, 0) $ to $ (1, 1) $ will return to $ \overline{PQ} $ in a manner captured by the rotation $ T $ of $ [0, 1] $

Figure 14.  The patterns $ b^+_{\varphi^2} $, $ g^+_{\varphi^2} $, $ b^-_{\varphi^2} $, and $ g^-_{\varphi^2} $

Figure 15.  An example of the Sturmian structure of a worm $ \mathcal{O}_{R_0}(\boldsymbol{p}) \cap \Delta_{ \mathcal{P}_0} $

Figure 16.  The dynamics of slope-$ \varphi $ segments in $ \Delta_{ \mathcal{P}_0} $ under $ \mathbb{Z}^2 $ translation

Figure 17.  The patterns $ b^+_{\varphi} $, $ g^+_{\varphi} $, $ b^-_{\varphi} $, and $ g^-_{\varphi} $

Figure 18.  The dynamics of vertical segments in $ \Delta_{ \mathcal{P}_0} $ under $ \mathbb{Z}^2 $ translation

Figure 19.  The patterns $ b^+_{\infty} $, $ g^+_{\infty} $, $ b^-_{\infty} $, and $ g^-_{\infty} $

Figure 20.  The dynamics of horizontal segments in $ \Delta_{ \mathcal{P}_0} $ under $ \mathbb{Z}^2 $ translation

Figure 21.  The patterns $ b^+_{0} $, $ g^+_{0} $, $ b^-_{0} $, and $ g^-_{0} $

Figure 22.  An octopod for the Jeandel-Rao Wang shift

Figure 23.  A valid tiling around an octopod in the Jeandel-Rao Wang shift. The hole can not be filled by copies of the Jeandel-Rao Wang tiles