Homochiral inflation for the aperiodic monotile Tile(1,1)

TL;DR

This work develops a homochiral inflation scheme for the chiral monotile Tile(1,1), fixing the tile's chirality at all inflation steps to deterministically encode its 12 orientations. By decomposing the tiling into two clusters, $\Gamma$ and $\Omega$, and packaging their six orientations into twelve metatiles, the authors separate rotation from translation and map metatile adjacencies onto a triangular lattice. The inflation rules yield a 2×2 matrix for cluster counts and a 12×12 matrix for orientation-resolved metatiles, with a Perron root of $4+\sqrt{15}$, giving explicit asymptotic proportions: $N^{\infty}_{\Omega}=(3+\sqrt{15})N^{\infty}_{\Gamma}$. Orientation densities converge rapidly, yielding equal counts within each odd/even group and a global even/odd ratio of $4+\sqrt{15}$. The dual hexagonal representation via junction points and hex clusters offers a complementary view and links to related hex-cluster tilings, strengthening the understanding of the tiling's quasiperiodic structure and the role of chirality in inflation.

Abstract

The recently discovered chiral monotile Tile(1,1) is tiling the plane in a quasiperiodic fashion by taking twelve different orientations when applying $2π/12$ rotation. An homochiral inflation construction of such a quasiperiodic tiling is proposed where the chirality of the monotile is completely fixed at all inflation steps, avoiding to exchange its chirality between two successive steps. Doing so, the twelve possible orientations of the monotile are explicitly coded and the key difference between odd and even orientations is taken into account. The tiling is decomposed using only two different clusters, $Γ$ and $Ω$, each of them taking six possible orientations. This gives a total set of twelve metatiles, which assembly can be mapped onto a triangular lattice. This approach allows to properly separate rotation and translation symmetry elements relating monotiles together. As all possible orientations of the two clusters are already incorporated in the twelve metatiles, positions of adjacent metatiles are given by translations which are along three equivalent directions ($2π/3$ rotation) as evidenced by junction lines. Finally, thanks to the homochiral inflation, the orientation distribution of the monotile at each inflation step is computed.

Figures (10)

• Figure 1: Chiral monotile. (a) Left and right versions. Contour of Tile(1,1) is a sequence of 14 equal length edges and rotation angles of $\pi/12$. By convention, odd edges are depicted in blue and even ones in red. Tile(1,0) contains only the even edges and has a comet shape while Tile(0,1) is obtained from the blue edges and has a chevron shape. (b) Decoration of the left chiral monotile with light blue and magenta edges. Decoration edges have a length $\sqrt{2}$ larger than edges of Tile(1,1) and are rotated by $\pi/12$ with respect to them. Decoration is giving rise to one hexagon, one square and one thin rhombus. For comparison, a reference point (white circle) is depicted on all shapes.
• Figure 2: Monotile, clusters and metatiles (a) Color code for the 12 orientations of the Tile(1,1) monotile. A left handed chirality is chosen. Shades of gray correspond to odd monotiles (M$_1$, M$_3$, ...) and shades of green to even ones (M$_2$, M$_4$, ...). (b) Construction of a C cluster of seven monotiles. A monotile (here M$_1$) is surrounded by 6 monotiles (shades of green) and the border of the C cluster is depicted by a thick black line. The same C cluster is also shown using the edge color representation (blue and red) (c) The 3 types of cluster: C, $\Gamma$ and $\Omega$. A $\Gamma$ cluster (8 monotiles) is a C cluster plus one additional even duplicated monotile (here M$_2$) while an $\Omega$ cluster (9 monotiles) contains one more additional even duplicated monotile (here M$_{10}$). (d) The six orientations of the odd monotiles in the tiling along with their nine neighboring even monotiles. Glue points (yellow color) are placed at the junction point of three even monotiles where two of them are duplicated. (e) The set of twelve metatiles corresponding to the $\Gamma$ and $\Omega$ clusters in their six orientations which allow to tile the plane.
• Figure 3: Analysis of a typical finite region of the quasiperiodic tiling. (a) C clusters are identified by their contours (thick black lines). (b) Same region where only the orphan even monotiles outside the C clusters are evidenced (shades of green). (c) Attribution of the orphan monotiles to either an $\Omega$ (orange contour) or a $\Gamma$ (pink contour) metatile. (d) Mapping onto a triangular lattice of the metatiles. Attribution to a metatile needs the knowledge of enough adjacent clusters. Otherwise, at the border of the finite region, the number corresponds to the orientation of the metatile, as it might be either a $\Gamma$ or an $\Omega$ one.
• Figure 4: Translation between two adjacent metatiles. (a) Translation (here $T_2$) between two adjacent monotiles in the same orientation (here M$_{10}$). A glue point (yellow color) is placed as a marker. Translation to superimpose the two monotiles is from a yellow to a black point and conversely. (b) In the quasiperiodic tiling, around each odd monotile (here M$_1$) three even monotiles (here M$_{10}$, M$_{12}$ and M$_{2}$) are always duplicated twice. The three corresponding glue points (yellow color) are located on the border of the C cluster (thick back line). Translation vectors between two adjacent monotiles are $T_1$ (M$_2$/M$_8$), $T_2$ (M$_4$/M$_{10}$) and $T_3$ (M$_6$/M$_{12}$) with $T_3=T_1+T_2$. (c) Example with translation $T_2$ between $\Gamma_1$ and $\Omega_6$ metatiles with two M$_{10}$ duplicated monotiles.
• Figure 5: The 12 metatiles with connection lines towards adjacent metatiles. Monotiles with a dashed line border are located in an adjacent metatile and give its position. Connection lines are connecting duplicated monotiles.